Abstract:
A putter head has a front that strikes a golf ball during putting, a length a, a width b, a weight W, and a moment of inertia I. The width extends along a horizontal width axis perpendicularly intersecting the front of the putter head. The length extends along a horizontal length axis perpendicularly intersecting the horizontal axis. The dimensions are, for example, a≦7 inches, b≦a, and I/Wa 2 &gt;0.30.

Description:
RELATED APPLICATIONS  
       [0001]    The present application claims the benefit of Provisional Application Ser. No. 61/061,440 filed on Jun. 13, 2008, the entire disclosure of which is incorporated by reference herein. 
     
    
     TECHNICAL FIELD  
       [0002]    The present disclosure relates to design of a head of a putter used in the game of golf. 
       BACKGROUND  
       [0003]    When a putter head hits a golf ball, the putter exerts a force on the ball, and the ball exerts an equal force on the putter in the opposite direction. In general, the force exerted on the putter by the ball does two things. It slows down the forward linear motion of the putter, and it causes the putter head to rotate about the vertical axis through its center of mass (COM). 
         [0004]    This rotation of the putter head is undesirable because it produces an error in the direction and speed of the ball. If the face of the incident head is perpendicular to the desired initial direction of the ball, as it should be, then the error arises because the rotated head will point away from this desired direction. During the brief time that the ball is in contact with the face of the putter head, the putter head will have rotated through a small angle so that, when the ball leaves the face of the putter head, it will move in a direction which is approximately perpendicular to the rotated face instead of the direction perpendicular to the original face. Also, because some of the kinetic energy of the incident putter head goes into the rotational energy acquired by the putter head, the speed of the struck ball will be less than anticipated. 
         [0005]    However, if ball is hit directly in front of the COM of the putter head, then there will be no induced rotation about the COM axis, and the above direction and speed errors will be avoided. Of course, the ball is not often hit directly in front of the COM of the putter head. Thus, the moment of inertia (MOI) of the putter head about the vertical axis through the COM of the putter head is important. (This MOI is defined as Σm i r i   2 , where each mass element m i  is multiplied by the square of the perpendicular distance r i  between the position of the element and the chosen vertical axis that intersects the COM of the putter head.) For an impact that is not directly in front of the COM of the putter head, the larger the MOI, the smaller the angular error. In other words, the larger the MOI, the larger the area on the clubface that produces an acceptable hit. This relationship is why the MOI is so important. 
         [0006]    USGA regulations restrict the size of a putter head, but not the weight or MOI of the putter head. Professional golfers consistently hit the ball very close to the point on the putter face directly in front of the COM of the putter head. This point may be referred to as the COM-point or “sweet spot” on the face of the putter head. 
         [0007]    Many articles, books, and patents erroneously claim that the sweet spot is the point in front of the center of percussion (COP) of the putter head. The confusion arises because the COP of the putter head is the point where an impact does not induce a reaction at the shaft insertion point into the putter head. An impact at the COP of the putter head, therefore, does not eliminate a putter head rotation, but instead creates a rotation about the COM of the putter head, since this created rotation must cancel the translational motion at the shaft induced by the impact. This rotation causes the ball to leave the clubface in the wrong direction. The sweet spot of the head is therefore the COM, not the COP, of the putter head. 
         [0008]    Amateur golfers, on the other hand, usually hit the ball at a point on the clubface that is a fair distance (often 0.5″ and sometimes over 1″) from the COM point of the putter head. It is, therefore, in the interest of most golfers to use a club with as large a MOI as possible. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS  
         [0009]      FIG. 1  illustrates three putter head configurations useful in explaining the present invention; 
           [0010]      FIG. 2  illustrates a putter head configuration useful in explaining the present invention; 
           [0011]      FIG. 3  illustrates various prior art putter head designs; 
           [0012]      FIG. 4  illustrates shapes of loads and connecting elements that can be used in connection with putter heads; 
           [0013]      FIG. 5  illustrates a four-load putter head, a three-load putter head, and a two-load putter head that achieve very large values of MOI; 
           [0014]      FIG. 6  illustrates smoothed version of four-load and three-load putter heads of  FIG. 5 ; 
           [0015]      FIG. 7  illustrates other four-load and three-load putter heads; 
           [0016]      FIG. 8  illustrate two-load putter heads; 
           [0017]      FIG. 9  is a three-dimensional illustration of a four-load putter head; 
           [0018]      FIG. 10  is a three-dimensional illustration of a four-load putter head with shaft; 
           [0019]      FIG. 11  is a graph of I/W vs. length for various putter head types; and, 
           [0020]      FIG. 12  is a graph similar to the graph of  FIG. 11 . 
       
    
    
     DETAILED DESCRIPTION  
       [0021]    According to embodiments of putter heads described herein, the putter heads are characterized by extremely large moments of inertia (MOI). These MOI values are much larger than those of putter heads currently on the market or disclosed in prior art. These large MOI values may be achieved in one or more of four novel ways. 
         [0022]    First, the putter head includes two to four relatively heavy “load” elements placed in locations as far as possible from the putter head center of mass (COM) and interconnected by a minimal number of relatively light “connecting elements,” which including a face plate and shaft holder. 
         [0023]    Second, the shapes of these elements are chosen to increase the MOI of the putter head. These shapes, and their distributions within the putter head, result in novel appearances of the putter heads. 
         [0024]    Third, the dimensions of the load elements (large in the vertical direction and small in the horizontal directions) are chosen to increase the MOI of the putter head. These dimensions also give rise to novel appearances of the putter heads. 
         [0025]    Fourth, the weights of the load elements are determined, by mathematical optimization calculations, to maximize the MOI of the putter head, given the configuration, overall weight, and overall size of the putter head. (The sizes are consistent with USGA regulations.) 
         [0026]    One way to obtain a putter head with a large MOI is to give it a large weight. Golfers, however, typically prefer a head weight within a very limited range, such as between 11 and 16 ounces. (A too-light head requires a swing speed that is relatively large and difficult to control, whereas a too-heavy head requires a swing speed that is relatively small and difficult to adjust.) A large MOI, therefore, should be achieved by distributing the desired weight within the putter head so that it is as far as possible from the COM of the putter head. The relevant quantity to consider is, therefore, the ratio I/W wherein I represents MOI and W represents weight. Putters are disclosed herein that have large values of I/W, values that are much larger than those previously obtained. 
         [0027]    One way to achieve large I/W values is to give the putter head a relatively large size and placing most of its weight far from its COM. There are, however, practical and official limitations on the acceptable size of a putter head. The USGA limits the maximum head size (see below), and, in any case, a putter head that is too large looks and feels awkward and is difficult to control. 
         [0028]    The maximum linear dimension of a given putter head may be denoted as a. The most relevant quantity to consider is, therefore, the dimensionless ratio I/Wa 2 . Putter heads are described herein having the largest possible values of this ratio. 
         [0029]    The dimensions of the putters as disclosed herein are compliant with the USGA regulations. The USGA putter head dimension limits are on the overall length OL, the face length FL, the overall width OW, and the overall height OH. The restrictions are that OL is greater than OW but at most 7 inches, FL is at least two-thirds of OW and at least one-half of OL, and OH is at most 2.5 inches. The maximum linear dimension a is, therefore, the OL. OW is referred to herein as b, OH is referred to herein as h, and FL is referred to herein as f. Thus, the restrictions on the putter head are b≦a≦7″, f≧2b/3, f≧a/2, and h≦2.5″. A compliant putter head must, therefore, fit within a rectangular box of length a≦7″, width b≦a, and height h≦2.5″. 
         [0030]    The SI unit of MOI is kg-m 2 . However, because the USGA regulations, and the specifications given by most club manufacturers, are given in English units (ounces and inches), MOI units are specified herein as oz-in 2 . Thus, MOI as used herein, therefore, uses the weights, rather than the masses, of the material elements in the MOI definition. In other words, MOI as used herein is the product of the SI MOI and the acceleration of gravity (32 ft/s 2 ). 
         [0031]    It is shown in the following that the theoretical absolute maximum value of I/Wa 2  for a putter head is 0.50. For putter heads compliant with USGA regulations (a≦7″), this absolute maximum value of I/Wa 2  implies that the maximum value of I/W is 24.5 in 2 . 
         [0032]    These theoretical putter heads consist of point weights alone, without a faceplate, connecting elements, or a shaft holder. Realistic putter heads, which include these elements and have non-point weights, cannot of course attain these maximum values. However, a realistic putter head with I/Wa 2  as large as 0.42 and an I/W, therefore, as large as 21 in 2  is possible. 
         [0033]    For comparison, one of the larger putter heads on the market has an I/W of about 6 in 2 . Although many patents claim to disclose putter heads with large values of I/W, none includes calculated values of I/W for realistic putters as large as those described herein. 
         [0034]    To produce a desired putt with a conventional putter, a golfer must hit the ball with the correct swing speed, with the correct swing direction, and at the sweet-spot on the putter head face. For putter heads with MOI ratios as large as those described herein, this last requirement is unnecessary. The entire putter head face is the sweet-spot, and the hit ball will proceed in the intended direction for virtually any impact point. The golfer is thus free to concentrate on only the first two requirements. 
         [0035]    In order to determine the theoretical upper limit on the possible values of I/Wa 2  for putter heads, “theoretical” heads are considered. These heads are mathematical constructs consisting of nothing except point masses separated as far possible from each other and from the COM of the system. The connecting, faceplate, and shaft-attachment elements necessary for a realistic putter are absent in the theoretical putter. The presence of any such element would decrease the value of I/W because they would add weight closer to the COM. 
         [0036]    In order to comply with USGA regulations, the point masses must lie within a rectangular box of length a≦7″, width b≦a, and height h≦2.5″. The optimal choice is b=a, and only the case where the mass points lie on the perimeter of a square of side a need be considered. The point masses must reside at the corners of this square in order to maximize the separation distances. This construct is illustrated in  FIG. 1(   a ) and  FIG. 2 . The weights in the corners are w 1 , w 2 , w 3 , and w 4 , and the fixed total weight is W=w 1 +w 2 +w 3 +w 4 . 
         [0037]    In the coordinate system centered at w 3  of  FIG. 2 , the coordinates x 0  and y 0  of the COM are given by the following equations: 
         [0000]        x 0= a ( w 2 +w 3)/ W,    
         [0000]        y 0= a ( w 1 +w 2)/ W.    
         [0000]    The MOI of the system about the transverse axis through the COM is given by the following equation; 
         [0000]        I ( w 1 ,w 2, w 3)=Σ wi*ri   2 , 
         [0000]    where the sum is over i=1, 2, 3, 4 and ri is the distance between wi and the COM. The maximum value of I is given by the solution of the three simultaneous equations 
         [0000]      ∂ I/∂wi= 0, for  i= 1, 2, 3. 
         [0000]    The solution is given by the following equation: 
         [0000]        w 1 =w 2,  w 3 =w 4 =W/ 2 −w 1, 
         [0000]    and the corresponding maximum value is given by the following equation: 
         [0000]        I   max   =Wa   2 /2. 
         [0000]    This is the largest possible value of the MOI for a putter head of weight W and length a. The COM is at the center of the square (x 0 =y 0 =a/2). 
         [0038]    The maximum value of I/W is, therefore, a 2 /2, and the maximum value of I/Wa 2  is, therefore, ½=0.50. These values serve as upper limits on the MOIs of realistic putter heads. For USGA compliant putter heads, a is at most 7″, and so the following values are obtained: 
         [0000]      ( I/W ) max   =a   2 /2=49/2=24.5 in 2 , 
         [0000]      ( I/Wa   2 ) max =½=0.50. 
         [0000]    The goal is realistic putter heads, which include faceplates, connecting elements, and shaft holders, that have MOI values as close to these upper-limit values as possible. 
         [0039]    Of special interest is the equal weight case w 1 =w 2 =w 3 =w 4 =W/4. The theoretical putter head is then left-right symmetric, as is the case of most realistic putter heads on the market. Another special case of interest is the novel choice w 1 =w 3 =0, w 2 =w 4 =W/2. This choice, consisting of only two loads, results in a theoretical putter head that is far from left-right symmetric. It is illustrated in  FIG. 1(   b ). Although it looks unusual, this type of putter head has the advantage of requiring fewer connecting elements when it is made into a realistic putter head. 
         [0040]    There is no optimal theoretical putter head with only three weights because the optimal weight choices do not include the case when only one of the wi is zero. There is, however, an interesting class of a three-load putter head which has a very large, but not optimally large, MOI. The system is illustrated in  FIG. 1(   c ). Two equal weights w 2  reside in the upper two corners, and a single weight w 1  resides at the center of the lower side of the square. 
         [0041]    In this latter case, the total weight W=w 1 +2w 2  and the length a are fixed, and the value of the weight ratio s=w 1 /W that maximizes the ratio I/Wa 2  can be determined. The COM coordinates are given by the following equations: 
         [0000]        x 0= a/ 2, 
         [0000]        y 0 =a (1 −s ), 
         [0000]    and the MOI ratio is 
         [0000]        I/Wa   2 =¼+3 s/ 4− s   2   ≡f ( s ). 
         [0000]    The optimal choice of s is given by f′(s)=0. The solution is s=⅜, so that 
         [0000]        w 1=3 W/ 8, 
         [0000]        w 2=5 W/ 16, 
         [0000]        y 0=5 a/ 8, 
         [0000]      and 
         [0000]      ( I/Wa   2 ) max   =f (⅜)=25/64=0.391. 
         [0042]    The MOI ratio is thus 22%. less than the optimal value of 0.50 for the optimal four and two load putter heads, it will be shown below that the difference is rather less for the realistic putter heads based on these platforms. 
         [0043]    When a faceplate is added to the bottom side of the four or two load platforms of  FIGS. 1(   a ) and  1 ( b ), the ratio I/W is reduced because weight must be added closer to the COM. But when a faceplate is added to the bottom side of the three load platform of  FIG. 1(   c ), the ration I/W is reduced less because there already is a weight at this location. However, if that faceplate is instead added to the top side of the three load platform, there is no such advantage. 
         [0044]    The earliest attempts to increase putter head MOIs placed loads at the heal and toe ends of blade shaped designs. The corresponding theoretical putter head is depicted in  FIG. 3(   a ). The optimal choice for such a configuration is to have equal weights  10  and  12  (each having a weight of W/2) at opposite ends of an element  14  of length a. Then I/Wa 2 =¼=0.250. This is much less than the 0.500 value for the four and two load cases or the 0.391 value for the three load case discussed above. 
         [0045]    The benefits of putters with increased MOIs have long been known. The earliest attempts involved modified blade type putter heads with weights added at the toe and heel locations of the face. This type of arrangement is illustrated in  FIG. 3(   a ). Typical patents for putters of this type are Scarborough U.S. Pat. No. 3,516,674, Rozmas U.S. Pat. No. 3,966,210, and Finney U.S. Pat. No. 4,898,387. As explained above, the maxim MOI ratio that can be expected from such putter heads is I/Wa 2 &lt;0.25. Finney claims to achieve the value is I/Wa 2 =0.17 for an a=5″ length and W=10.6 oz weight head. This is reasonably close to the theoretical maximum value for his type of putter design. 
         [0046]    It was later realized that the MOIs could be further increased by adding weight rearward from the center of the putter face. Thus, the putter face has weights  16  and  18  at opposite ends of an element  20  of length a and a third weight  22  rearward from the center of the element  20 . This type of arrangement is illustrated in  FIG. 3(   b ). A well-researched example of this type of putter head is disclosed by Winchell in U.S. Pat. No. 5,080,365. Winchell claims to achieve an I/W value of 1.94 in 2  for an a=5″ length and a W=1 lb weight. 
         [0047]    As shown above, the largest possible theoretical value of I/W for a three-load putter head is 0.391a 2 , which is 9.78 in 2  for a=5″. Below is a discussion of why a realistic three-load a =5″ putter head can achieve an I/W=8-9 in 2 . There are four reasons that the three load putter heads described herein achieve significantly higher values than Winchell&#39;s. (1) The shapes of the loads and connecting elements are better chosen. (2) The dimensions of the loads are better chosen. (3) The weight ratios of the loads are optimally chosen. (4) The two-weight line at the rear of the head and the third weight at the center of the faceplate are placed more optimally. 
         [0048]    A four-load putter head was disclosed by Long in U.S. Pat. No. 4,010,958. A preferred embodiment of Long&#39;s head is illustrated in  FIG. 3(   c ). Square loads  24 ,  26 ,  28 , and  30  are placed in the four corners of a (typically 5″ by 5″) square and interconnected by three lower-density tubular struts  32 ,  34 , and  36  and a faceplate  38 . A shaft  40  of the putter is inserted at the center of the square and is connected to the back loads  24  and  26  and the faceplate  38  by the three struts  32   34 , and  36 . 
         [0049]    The weights of the loads  24 ,  26 ,  28 , and  30  are unspecified except for the requirement that the weights of the front loads  28  and  30  are less than the weights of the back loads  24  and  26  in order that the COM resides at the center  40  of the square. Long does not numerically evaluate MOI values, but it can be calculated that the construction of  FIG. 3(   c ) yields I/Wa 2  values of at most 0.30. 
         [0050]    While this value is impressive, it is not impressive enough and the four-load putter heads disclosed herein have I/Wa 2  values over 0.42, which is a 40% increase over Long&#39;s putter head. As discussed above, the theoretical upper-limit of I/Wa 2  for a four-load head was shown in Section 2 to be 0.50. 
         [0051]    The novel features of the putter heads disclosed herein, and the reasons that their values are much larger than Long&#39;s and are much closer to the theoretical maximum value, are the following: (1) the weights of the loads for the putter heads disclosed herein are mathematically chosen to maximize I/Wa 2 ; (2) the number, shapes, and locations of the connecting elements (struts) of the putter heads disclosed herein are chosen to maximize I/Wa 2 ; and, (3) the shapes and dimensions of the loads of the putter heads disclosed herein are chosen to maximize I/Wa 2 . 
         [0052]    A different type of putter head is described by Rohrer in U.S. Pat. No. 7,077,758. Rohrer asserted that his designs achieved higher MOIs than Long&#39;s, and in fact achieved maximum values of I/Wa 2 . This claim is incorrect. 
         [0053]    Rohrer describes a putter head in which the bulk (at least 70%) of the weight is concentrated within a circular ring concentric with the COM. Rohrer claims that, for a given size and weight, his putter head has a MOI that is 34% larger then Long&#39;s. This claim is based on the comparison illustrated in  FIG. 3(   d ). This figure compares a Long head, having of length a and in which the weights of the loads  24 ,  26 ,  28 , and  30  are concentrated in the four corners, to a Rohrer head, having face length a and in which the weights are concentrated in two protruding circular segments  42  and  44 . 
         [0054]    The theoretical limit (zero weight volumes, no connectors of faceplate) for the value of I/Wa 2  is the same, 0.50, for each of these heads. However, the comparison of the two heads is unfair because the size of the Rohrer head is obviously much larger than that of the Long head. 
         [0055]    The USGA places upper limits on the overall length a and width b of the head (b≦a≦7″), not on just the face length. Therefore, if a=7″, the illustrated Long putter is compliant with USGA requirements, but the Rohrer putter is not. The fair comparison between the putters must compare heads of equal overall size (and weight). This comparison is illustrated in  FIG. 3(   e ). The value of I/Wa 2  for the theoretical Long putter  46  is again 0.50, but that for the Rohrer putter  48  is only 0.25, which is the same as for the simple blade putter of  FIG. 3(   a ). 
         [0056]    Rohrer also claims that the MOI of his putter head is further increased relative to Long&#39;s putter head because his shaft insertion point is relatively far from the COM. This claim is also unfair because the contact time between a struck golf ball and a putter head is too short for the presence of most of the shaft to be felt. 
         [0057]    The conclusion from this analysis is, therefore, that, for a given weight and size, the Long head has a 100% larger MOI than the Rohrer head and a 40% smaller MOI than the four-load head described herein. 
         [0058]    A two-load putter head is described by Sato in U.S. Pat. No. 6,409,613 and is illustrated in  FIG. 3(   f ). This head is L-shaped, with cylindrical loads  50  and  52 , of unspecified weight, placed in opposite corners and connected by lower-density arms  54  and  56  and a faceplate  58 . Sato does not address the MOI concept, but does state that the rotation of his putter head caused by an impact away from the sweet spot is decreased relative to a blade-type head. He argues that this decreased rotation is because the exerted torque on his putter head is less because the COM is further back from the face. This argument is incorrect. 
         [0059]    Torque is the product of the applied force and the perpendicular distance between the force vector and the vertical axis through the COM, and thus torque depends on the distance between the impact point and the sweet spot on the face, but is independent of the distance of the COM behind the sweet spot on face. 
         [0060]    Although Sato&#39;s reasoning is incorrect, his conclusion about decreased rotation is correct because the MOI of his head is relatively large. He does not numerically evaluate MOI values, but the I/Wa 2  values for his construction can be calculated to be at most 0.27. This I/Wa 2  value is large, but the two-load putter heads described herein have I/Wa 2  values as high as 0.41, a 52% increase. The theoretical upper-limit of I/Wa 2  for a two-load head 0.50 as described above. 
         [0061]    The novel features of the two-load putter heads described herein, and the reasons that their I/Wa 2  values are much larger than Sato&#39;s, and much closer to the theoretical maximum value, are the following: (1) the weights of the loads for the two-load putter heads described herein are mathematically chosen to maximize I/Wa2; (2) the shapes and locations of connecting elements for the two-load putter heads described herein are chosen to maximize I/Wa 2 ; and, (3) the shapes and dimensions of the loads for the two-load putter heads described herein are chosen to maximize I/Wa 2 . 
         [0062]    In one embodiment, the putter heads under consideration herein incorporate four types of components: loads, a faceplate, connecting elements, and a shaft holder. The first three of these are discussed more fully herein. The shaft holder is a simple low-weight addition that will be discussed after the first three elements. These components are situated within a rectangular box of length a 0 ≦7″, width b 0 ≦a 0 , and height c 0 ≦2.5″. For simplicity, each component is assumed to have a constant density, although such an assumption is not required. 
         [0063]    Regarding the load components first, these must be placed as far as possible from the vertical axis through the putter head COM in order to achieve the largest possible MOI. The loads, therefore, will reside at the corners of the base a 0 ×b 0  rectangle and extend upwards in the perpendicular direction. Likewise, the COM of each load must be as far as possible from the putter head COM. Among the practical load shapes, the triangular shape is considered herein. Any other simple shape would move the load COM and the putter head COM closer together, as demonstrated below. 
         [0064]    A typical triangular load is illustrated in  FIG. 4(   a ). The base dimensions are a×b and the height of the load coming out of the page is denoted as c. Although this triangular shape is a preferred design for the base of the load, the putter heads disclosed herein will yield very large MOI ratios for a variety of other shapes. 
         [0065]    Another novel feature of the putter heads disclosed herein is the exploitation of the generous USGA limit of 2.5″ for the height of the putter head. Whatever the shape of the base of the load, the use of load heights near this upper limit significantly contributes to the achievement of very large MOI ratios. The loads used in prior art putter heads do not make use of this freedom. If more of the load is in the vertical direction, then the base of the load can be smaller, and so more of the load can be farther from the COM, leading to a larger MOI ratio. 
         [0066]    The simplest shape for the connecting elements is a solid rectangular box. The rectangular base of such an element is illustrated in  FIG. 4(   b ). Because they are closer to the putter head COM, these connectors must be as light as possible to minimize their effect in decreasing the overall MOI ratio I/W. They must, however, be sufficiently strong to securely connect the other elements. 
         [0067]    There are two possibilities for the orientation of these connectors, as illustrated in  FIG. 4(   c ) and  4 ( d ). In  FIG. 4(   c ), the shorter side of the connector, of length a, is in the horizontal direction, and the longer side, of length b, is in the vertical direction. These choices are reversed in  FIG. 4(   d ). Each of these connector possibilities has width c in the backward direction. The connector of  FIG. 4(   c ) would be farther from the putter head COM, but it has the smaller MOI about the vertical axis through it&#39;s COM, whereas the connector of  FIG. 4(   d ) would be closer to the putter head COM, but it has the larger MOI about the vertical axis through it&#39;s COM. It will shown below that it is the connector of  FIG. 4(   c ) that achieves the largest putter head MOI ratio. 
         [0068]    To specify distances, the x-axis (1-axis) is chosen to be along the (toe-heel) length of the putter face, the y-axis (2-axis) is chosen to be along the (front-back) width of the putter face, and the z-axis (3-axis) is chosen to be in the vertical direction. The COM of the triangular solid load having the base shown in  FIG. 4(   a ) is at x=a/3, y=b/3, and z=c/2. The MOI ratio of this solid about the vertical axis through it&#39;s COM is given by the following equation: 
         [0000]        I   T0   /W   T =( a   2   +b   2 )/18. 
         [0000]    The CoM of the rectangular solid connector with base shown in  FIG. 4(   b ) is at x=a/2, y=b/2, and z=c/2. The MOI ratio of this solid about the vertical axis through it&#39;s COM is given by the following equation: 
         [0000]        I   R0   /W   R =( a   2   +b   2 )/12. 
         [0000]    The MOI ratio of either component (C=T or R) about the vertical axis through the putter head COM is given by the parallel-axis theorem and is given by the following equation: 
         [0000]        I   C   /W   C   =I   C0   /W   C   +l   2 . 
         [0000]    where 1 is the distance between the vertical axis through the load or connector COM and the vertical axis through the putter head COM. 
         [0069]    It can now be confirmed that the triangular load achieves a larger MOI ratio about the putter head COM than a rectangular load of the same dimensions and weight W.  FIG. 4(   e ) depicts a triangular load (the lower right triangle “T” of base a and height b), and a rectangular load (the full rectangle “R” of base a and height b) at the same position in a corner of a putter head. The COM of the triangle is at x=a/3, y=b/3, and the COM of the rectangle is at x=a/2, y=b/2. The MOI ratios about the COMs are given by the following equations: 
         [0000]        I   T0   /W =( a   2   +b   2 )/18, 
         [0000]        I   R0   /W =( a   2   +b   2 )/12. 
         [0000]    The rectangular MOI ratio is larger, but it will be shown that this difference is more than made up for by the fact that the triangle is further from the COM of the putter head. 
         [0070]    Relevant distances are indicated in  FIG. 4(   e ), where d=√{square root over ((a 2 +b 2 ))}/3 denotes the distance between the origin and COM of the triangle and D denotes the distance between the outer corner of the rectangle and the COM of the putter head. The MOI ratio of the rectangle head about the COM of the putter head is, therefore given by the following equation: 
         [0000]        I   R   /W=I   R0   /W +( D+ 3 d/ 2) 2 =3 d   2 /4+( D+ 3d/2) 2    
         [0000]    and the MOI ratio of the triangle about the COM of the putter head is given by the following equation: 
         [0000]        I   T   /W=I   T0   /W +( D+ 2 d ) 2   =d   2 /2+( D+ 2 d ) 2    
         [0000]    The difference is given by the following equation: 
         [0000]        I   T   /W−I   R   /W=Dd+ 3 d   2 /2, 
         [0000]    which is always positive. The above proves that the MOI ratio of the triangular load is always greater than the MOI ratio for the rectangular load. 
         [0071]    In proving this result, a specific geometrical relationship between the loads and the COM of the putter head is chosen in  FIG. 4(   e ), but the result is completely general. For any realistic geometry, the MOI difference is always positive and close to the value given above. 
         [0072]    It can also be confirmed that the “vertical” rectangular connector of  FIG. 4(   c ) achieves a larger MOI ratio about the putter head COM than the “horizontal” rectangular connector of  FIG. 4(   d ) of the same dimension a×b×c and weight W. With reference to  FIG. 4 , the following equation is given for the vertical connector: 
         [0000]        I   c   /W=a   2 /3 +c   2 /12 +l   2   −a 1, 
         [0000]    and the following equation is given for the horizontal connector: 
         [0000]        I   d   /W=b   2 /3 +c   2 /12 +l   2   −b 1. 
         [0000]    The difference is given by the following equation: 
         [0000]      ( I   c   −I   d )/ W =( b−a )(1 −a/ 3 −b/ 3). 
         [0000]    For all parameter values of interest (0.1251″≦a≦0.25″, 0.5″≦b≦1″, l≧2″), this difference is positive and so the “vertical” rectangular connector of  FIG. 4(   c ) is seen to provide the larger MOI ratio. 
         [0073]    In a preferred embodiment, the load is chosen to be triangular with base dimensions a and b chosen to be between 0.25″ and 1″, depending on the chosen density and desired weight. The height dimension is chosen to be between about 1″ and 2.5″ (the maximum allowed by USGA regulations), depending on the load density, desired weight, and optimization specifications. As a novel contribution, in order to achieve the largest possible MOI ratios, load heights close to the 2.5″ limit can be chosen. To insure overall stability of the putter head, the short length a of the connecting elements is chosen to be preferably at least 0.125″, and the long height b is chosen to be preferably about 1″. 
         [0074]    For simplicity and economy, the face plate can be chosen to coincide with the connecting element between the forward loads. This choice also serves to minimize the consequent decrease of the overall value of I/W. The height of this element should be at least 1″ in order to avoid ball miss-hits in the vertical direction. The length of the face plate must be long enough to connect the forward loads and, to comply with USGA regulations, at least ⅔ of the overall length. The thickness (width) should be at least 0.125″, to provide a solid impact (momentum transformation) between the club and the ball. 
         [0075]    It will now by demonstrated how to combine the three putter head elements into a complete entity with as large a MOI ratio I/Wa 2  as possible. As much of the weight as possible is placed as far as possible from the putter head&#39;s COM. This construction is constrained by USGA size regulations and a desire for a pleasant and manageable appearance. The specific configurations that will by described are preferred embodiments, which incorporate these principles and which, when using the optimal weight ratios derived hereinafter, give rise to optimally large MOI ratios. Skilled persons in the art can use similar components and optimization calculations to arrive at other configurations with very large MOI ratios. 
         [0076]    A four-load putter head is initially described. For a given overall weight W and size a, this configuration gives rise to the absolute largest MOI value, a value as close as possible to the theoretical limit I=Wa 2 /2. The basic configuration of a putter head  60  is illustrated in  FIG. 5(   a ). The overall length is labeled a 0  and the overall width is labeled b 0 . USGA regulations require that b 0 ≦a 0 , so that the optimal choice is b 0 =a 0 . 
         [0077]    Four triangular loads  62 ,  64 ,  66 , and  68  are situated at the four corners of the base rectangle. The lengths of the triangles are c 1  (for the front loads  66  and  68 ) and d 1  (for the back loads  62  and  64 ). The widths are c 2  and d 2  and the heights (coming out of the page as viewed in  FIG. 5(   a )) are c 3  for the front loads  66  and  68  and d 3  for the back loads  62  and  64 . A faceplate  70 , which also serves as the forward connecting element, is a rectangle of length a 1 =a 0 −2c 1 , width (thickness) is a 2 , and the height (coming out of the page as viewed in  FIG. 5(   a )) is a 3 . Left and right connecting elements  72  and  74  are rectangles of length b 1 , width b 2 =b 0 −c 2 −d 2 , and height b 3  (coming out of the page as viewed in  FIG. 5(   a )). Apart from the shaft holder (not shown in  FIG. 5(   c )), which can be attached at any desired location, this minimal configuration is all that is needed. 
         [0078]    This configuration achieves the goals of locating the relatively heavy loads  62 ,  64 ,  66 , and  68  as far from the COM of the putter head 60 as possible, and locating the relatively light connecting elements  70 ,  72 , and  74  as far from the COM as possible, given the constraint that they must hold the loads  62 ,  64 ,  66 , and  68  and the faceplate  70  in place. 
         [0079]    The final steps in the construction specifications, which will be carried out hereinafter, will be to choose values for the free parameters (a 0 , b 1 , c 1 , etc.). These choices are determined by the following four conditions: (1) the connecting elements  70 ,  72 , and  74  are vertically oriented, as in  FIG. 4(   c ), in order to maximize their contribution to the overall MOI; (2) the various dimensions are chosen by the desired overall size of the putter head  60 ; (3) the base-areas of the loads  62 ,  64 ,  66 , and  68  are as small as practical, and their heights are as large as practical, in order to maximize their contribution to the overall MOI; and, (4) the relative weights of the front loads  66  and  68  and the back loads  62  and  64  are chosen by an optimization calculation to maximize the final overall MOI. 
         [0080]    A three-load putter head  80  is shown in  FIG. 5(   b ), and has one central front load  82  and two back loads  84  and  86 . The triangular corners could be eliminated, constructed out of low-density material, or replaced by curved sections so as to create a U-shaped connection. However, this design necessarily gives rise to a putter head with a lower MOI ratio than the four-load head. The three-load putter head  80  is a T-shaped design. The faceplate is incorporated into the central forward load  82 , and the rear triangular loads  84  and  86  are situated at the two rear corners of the base. This design is novel. Previously described three-load heads have the face situated at the other end of the base (the top of the T). The three-load putter head  80 , while achieving an MOI ratio not quite as large as the four-load head  60 , has the advantage of a more compact forward shape and, with the forward load incorporated into the faceplate, provides a more solid impact. 
         [0081]    The dimensions of the various components of the three-load putter head  80  are given as the overall length a 0  and the overall width b 0 . Optimally, b 0 =a 0  as before. The length of each of the back loads  84  and  86  is d 1 , the width of each of the back loads  84  and  86  is d 2 , and the height of each of the back loads  84  and  86  (coming out of the page as viewed in  FIG. 5(   b )) is d 3 . The faceplate, which also serves as the front load  82 , is a triangle of length a 1 ≧2b 2 /3, width a 2 , and height a 3  (coming out of the page as viewed in  FIG. 5(   b )). A back connecting element  88  is a rectangle of length c 1 =a 0 −2d 1 , width c 2 , and height c 3  (coming out of the page as viewed in  FIG. 5(   b )). A central connecting element  90  is a rectangle of length b 1 , width b 2 =b 0 −c 2 −a 2 , and height b 3  (coming out of the page as viewed in  FIG. 5(   b )). Apart from the shaft holder, which can be attached at any desired location, this minimal configuration is all that is needed. 
         [0082]    This configuration has achieved the goals of locating the relatively heavy loads as far from the COM as possible, and the relatively light connecting elements as far from the COM as possible given the constraints of the desired geometry. The final steps in the construction specifications, which is carried out hereinafter, is to choose values for the free parameters (a 0 , b 1 , c 1 , etc.). This choice is determined by the following four conditions: (1) the back connecting element  88  is vertically oriented, as in  FIG. 4(   c ), and the central connecting element  90  is horizontally oriented, as in  FIG. 4(   d ), in order to maximize their contributions to the overall MOI; (2) the various component dimensions are chosen by the desired overall size of the putter head  80 ; (3) the base-areas of the loads  82 ,  84 , and  86  are as small as practical, and their heights are as large as practical; and, (4) the relative weights of the front and back loads  82 ,  84 , and  86  are chosen by an optimization calculation. 
         [0083]    A two-load putter head  100  is illustrated in  FIG. 5(   c ). The putter head  100  is basically two-thirds of the four weight head shown in  FIG. 5(   a ), and the same length labels can be used. The two-load putter head  100  has dense triangular loads  102  and  104 , which are positioned at the upper right and lower left corners, and lighter connecting elements  106  and  108 , which are solid rectangular boxes as before. The lower connecting element  108  comprises the faceplate of the two-load putter head  100 , and a light lower right triangular element  110  provides structural support. This light triangular element  110  is a convenient place to insert the shaft, as indicated by the circular hole. Other possible two-load configurations are discussed below. 
         [0084]    With regard to the theoretical limit (point loads and weightless connections), the two-load putter head  100  has the same MOI (Wa 2 /2) as the four-load putter head  60 . However, the effects of using realistic load sizes and connector weights are mixed. On the one hand, the two-load putter head  100  is favored because it has one fewer connecting element (two instead of three), but on the other hand, the four-load putter head  60  is favored because, for a given total weight, the loads  62 ,  64 ,  66 , and  68  can be smaller than the loads  102  and  104  and, therefore, farther from the COM. The first effect increases I/W for the two-load putter head  100 , and the second effect increases I/W for the four-load putter head  60 . It turns out that the second effect dominates and so the realistic four-load head  60  has the (slightly) larger MOI ratio. 
         [0085]    The final steps in the construction of the two-load head, which will be carried out hereinafter, is to choose values for the free parameters (a 0 , b 1 , c 1 , etc.). This choice is determined by the following four conditions: (1) the connecting elements  106  and  108  are vertically oriented, as in  FIG. 4(   c ), in order to maximize their contributions to the overall MOI; (2) the various component dimensions are chosen by the desired overall size of the head; (3) the base-areas of the loads  102  and  104  are made as small as practical, and their heights are made as large as practical; and, (4) the relative weights of the front and back loads  104  and  102  are chosen by an optimization calculation to maximize the final overall MOI. 
         [0086]    The putter head configurations illustrated in  FIG. 5  consist of simple combinations of the most basic triangular loads and rectangular connectors. In order to achieve a better-looking and more marketable appearance, these configurations can be smoothed out without significantly decreasing their MOI ratios.  FIG. 6  illustrates some of the many possibilities. A smoothed version of the four-load putter head  60  is illustrated in  FIG. 6(   a ), and a smoothed version of the three-load putter head  80  is illustrated in  FIG. 6(   b ). A different version of a two-load putter head  120  is illustrated in  FIG. 6(   c ). The two-load putter head  120  has loads  122  and  124  interconnected by connecting elements  126 ,  128 , and  130 . Because the connecting elements  126 ,  128 , and  130  are farther from the COM, the two-load putter head  120  has a larger MOI ratio than the three-load putter head  80 , but is not as compact, and it has a smaller MOI ratio than the four-load putter head  60 . A smoothed version of the two-load head  100  looks like the smoothed version of the four-load putter head  60  without the lower arm. 
         [0087]    Some other possibilities for four-load putter heads are illustrated in  FIGS. 7(   a ),  7 ( b ), and  7 ( c ), some other possibilities for three-load putter heads are illustrated in  FIGS. 7(   d ),  7 ( e ), and  7 ( f ), and some other possibilities for two-load putter heads are illustrated in  FIGS. 8(   a )-( f ). A three-dimensional illustration of the four-load putter head  60  is provided by  FIG. 9(   a ), and an illustration of a smoothed version of this putter head is provided by  FIG. 9(   b ). 
         [0088]    For a given configuration, the MOI I depends on the dimensions and densities of the included components. For simplicity, it is assumed here that each putter head uses only two different densities, the density dh of the heavy load elements and the density dl of the light connecting elements. The MOI ratio I/W is then a function of the density ratio r=dh/dl. A possible material for the light connecting elements is aluminum, with a weight density dl of about 1.6 oz/in 3 . Possible materials for the heavy load elements include copper (dh=5.3 oz/in 3 ), lead (dh=6.7 oz/in 3 ), and tungsten (dh=11.4 oz/in 3 ). The resulting density ratios are r=3.3, 4.2, and 7.1. The choice of r depends on the desired weight, size, and MOI of the putter head. 
         [0089]    The first step in optimizing a putter head is to choose its optimization parameter(s). Possible choices for these parameters include the ratio(s) of the load weight, sizes, or densities. For illustrative purposes, a single parameter s, a size ratio of the forward and backward load elements, is used. 
         [0090]    The second step is to choose the component dimensions not determined by the optimization variable s. These dimensions are restricted by the desired size, weight, and MOI of the putter head. Each choice influences the optimal value s 1  of s, and some dimension values must be adjusted in order to achieve a desired weight and MOI. 
         [0091]    The third step is to express the COM (x(s),y(s)), total weight W(s), MOI I(s), and ratio f(s)=I(s)/W(s) as functions of s. The optimal value s 1  of s can then be determined by finding the appropriate solution of the following differential equation: 
         [0000]        df ( s )/ ds=f ′( s )=0. 
         [0000]    This procedure determines the optimal values of the COM (x 1 ,y 1 ), weight W 1 , MOI I 1 , and ratio f 1 =f(s 1 ) for the chosen densities and dimensions. 
         [0092]    If the resultant weight is not acceptable, then some of the dimensions and/or densities can be adjusted, and the optimization calculation repeated, to achieve the desired weight. Alternatively, a weight condition, W(s)=constant, can be solved simultaneously with f′(s)=0. 
         [0093]    As a first example of the above optimization procedure and resultant MOI values, the four-load putter head  60  is considered. The overall dimensions of the base rectangle are ab 0 =a 0 =b 0 . The heal-toe dimensions are a 1 , b 1 , etc., the front-back dimensions are a 2 , b 2 , etc., and the vertical dimensions are a 3 , b 3 , etc. The thickness of the connecting elements  70 ,  72 , and  74  are chosen to be a 2 =b 1 =⅛″=0.125″, and their heights to be a 3 =b 3 =1″. The parameter ab 0  is varied between 4″ and 7″ (the maximum allowed by the USGA) and the load/connector density ratio r is varied between 3 (e.g., copper/aluminum) and 7 (e.g., tungsten/aluminum). The optimization variable s=c 3 /d 3  is chosen. This ratio is the ratio of the front and back load heights. The back load height d 3  is chosen between 1″ and 2.5″ (the maximum allowed by the USGA). The load base dimensions (cd 12 =c 1 =c 2 =d 1 =d 2 ) are chosen between 0.5″ and 1″ to keep the total weight between 11 oz and 17 oz. 
         [0094]    The results of the optimization calculations for some of these choices relative to a four-load putter head are given in Table 1. 
         [0000]    
       
         
               
               
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 1 
               
               
                   
               
               
                 ab0 
                 r 
                 cd12 
                 d3 
                 s1 
                 c3 
                 y1 
                 I/W 
                 I/Wa 2   
                 I 
                 W 
                 wd 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 7 
                 3 
                 1 
                 1 
                 0.986 
                 0.99 
                 3.24 
                 18.6 
                 0.380 
                 233 
                 12.5 
                 4.6 
               
               
                 7 
                 3 
                 0.813 
                 1.5 
                 1.182 
                 1.77 
                 3.18 
                 19.05 
                 0.389 
                 238 
                 12.5 
                 4.6 
               
               
                 7 
                 3 
                 0.625 
                 1.5 
                 1.014 
                 2.43 
                 3.17 
                 19.63 
                 0.401 
                 239 
                 12.2 
                 4.4 
               
               
                 7 
                 5 
                 1 
                 1 
                 0.988 
                 0.99 
                 3.33 
                 19.14 
                 0.391 
                 352 
                 18.4 
                 7.8 
               
               
                 7 
                 5 
                 0.625 
                 1.5 
                 1.013 
                 1.52 
                 3.19 
                 19.70 
                 0.402 
                 248 
                 12.6 
                 4.6 
               
               
                 7 
                 5 
                 0.5 
                 2 
                 1.031 
                 2.06 
                 3.11 
                 19.82 
                 0.404 
                 226 
                 11.4 
                 3.9 
               
               
                 7 
                 5 
                 0.5 
                 2.4 
                 1.021 
                 2.45 
                 3.16 
                 20.13 
                 0.411 
                 262 
                 13.0 
                 4.7 
               
               
                 7 
                 7 
                 0.5 
                 1.5 
                 1.028 
                 1.54 
                 3.13 
                 19.90 
                 0.406 
                 235 
                 11.8 
                 4.1 
               
               
                 7 
                 7 
                 0.5 
                 2 
                 1.015 
                 2.03 
                 3.20 
                 20.36 
                 0.416 
                 295 
                 14.5 
                 5.5 
               
               
                 7 
                 7 
                 0.5 
                 2.4 
                 1.010 
                 2.42 
                 3.25 
                 20.62 
                 0.421 
                 344 
                 16.7 
                 6.6 
               
               
                 6 
                 3 
                 1 
                 1.5 
                 0.981 
                 0.98 
                 2.82 
                 13.42 
                 0.373 
                 156 
                 11.6 
                 4.7 
               
               
                 6 
                 3 
                 1 
                 2 
                 0.985 
                 1.48 
                 2.88 
                 13.69 
                 0.380 
                 223 
                 16.3 
                 7.0 
               
               
                 6 
                 3 
                 0.625 
                 2.4 
                 1.006 
                 2.41 
                 2.76 
                 14.34 
                 0.398 
                 166 
                 11.6 
                 4.4 
               
               
                 6 
                 5 
                 1 
                 1 
                 0.986 
                 0.99 
                 2.89 
                 13.75 
                 0.382 
                 245 
                 17.8 
                 7.8 
               
               
                 6 
                 5 
                 0.625 
                 1.5 
                 1.005 
                 1.51 
                 2.77 
                 14.38 
                 0.399 
                 171 
                 11.9 
                 4.6 
               
               
                 6 
                 5 
                 0.5 
                 2 
                 1.020 
                 2.04 
                 2.71 
                 14.55 
                 0.404 
                 157 
                 10.8 
                 3.9 
               
               
                 6 
                 5 
                 0.5 
                 2.4 
                 1.013 
                 2.43 
                 2.75 
                 14.75 
                 0.410 
                 181 
                 12.3 
                 4.7 
               
               
                 6 
                 7 
                 0.5 
                 1.5 
                 1.018 
                 1.53 
                 2.73 
                 14.60 
                 0.406 
                 164 
                 11.2 
                 4.1 
               
               
                 6 
                 7 
                 0.5 
                 2 
                 1.009 
                 2.02 
                 2.78 
                 14.91 
                 0.414 
                 207 
                 13.9 
                 5.5 
               
               
                 6 
                 7 
                 0.5 
                 2.4 
                 1.005 
                 2.41 
                 2.82 
                 15.07 
                 0.419 
                 243 
                 16.1 
                 6.6 
               
               
                 5 
                 7 
                 0.5 
                 2.4 
                 1.002 
                 2.40 
                 2.38 
                 10.35 
                 0.414 
                 160 
                 15.5 
                 6.6 
               
               
                 4 
                 7 
                 0.5 
                 2.5 
                 0.999 
                 2.50 
                 1.93 
                 6.49 
                 0.406 
                 100 
                 15.4 
                 6.8 
               
               
                   
               
             
          
         
       
     
         [0095]    In the first row of data, the overall head base dimension is 7″×7″ and the density ratio is r=3. The load triangles have 1″×1″ bases and the height of the back loads is also 1″. The optimization calculation gives s 1 =0.986 so that the front loads are also 1″ high (c 3 =s 1 *d 3 =0.99″). The COM location in the toe-heal direction is x 1 =3.5″ since the four-load putter heads are left-right symmetric. The COM location in the front-back direction is y 1 =3.24″. The value 18.6 in 2  for I/W is already much larger than for any previously disclosed putter head, and quite close to the theoretical limit of 24.5 in 2  for a 7″ head. Likewise, the value 0.38 for I/Wa 2  is extremely large and close to it&#39;s theoretical limit of 0.50. The head weight is W=12.5 oz, but this weight can be adjusted to any desired value without changing I/W. 
         [0096]    The value of I/W can be further increased by choosing the bases of the loads to be smaller. This places the COM of the loads farther from the head COM, and therefore increases the MOI. This can be accomplished in two ways: the load heights d 3  and c 3  and/or the load densities r can be increased. The effect of increasing d 3  is shown in the next two rows in Table 1. Increasing d 3  to 1.5″ and decreasing cd 12  to 13/16″ increases I/W to 19.1 in 2  without changing the head weight. Increasing d 3  to the USGA limit of 2.5″ in not acceptable because the corresponding optimal choice for c 3  would be greater than the 2.5″ limit, but setting d 3 =2.4″, with cd 12 =⅝″=0.625″, gives the compliant value c 3 =2.43″, and increases I/W to 19.6 in 2 . 
         [0097]    The effect of increasing r is shown in the next rows in Table 1. Increasing r to 5 and then to 7, while decreasing cd 12  to maintain reasonable weights W, increases the MOI ratio for each choice of d 3 . When both r and d 3  are increased, the MOI ratio increases even further. The largest value for I/W is 20.6 in 2 , obtained for the largest density ratio r=7 and largest load height d 3 =2.4″. 
         [0098]    This very large I/W value can be further increased by fine-tuning the various load and connector dimensions. Values over 21 in 2  are easily attainable, corresponding to I/Wa 2  values over 0.43. MOI values even close to these have never been previously attained. The 7″×7″×2.5″ putter head size is, of course, while USGA compliant, larger than desirable for most golfers. The methods described here will, however, yield the largest possible MOI ratios for any desired putter head size. This is illustrated in the remaining rows of Table 1. 
         [0099]    It can be seen in the table that as ab 0  is decreased to 6″, 5″, or 4″, the maximum values of I/W (obtained for r=7 and d 3 =2.4″−2.5″) decrease to 15.1, 10.4, and 6.5 in 2 , respectively, with I/Wa 2  decreasing from 4.2 to 4.1. These values are more all than three times those larger than those previously disclosed for putter heads of the same size. 
         [0100]    Considered next is the optimization calculations and MOI evaluations for the two-load putter heads. The two-load configurations have fewer connecting elements, which tends to increase I/W, but, for a given total weight, the loads must be larger, because there are fewer of them, and this tends to decrease I/W. It will be seen that this latter effect dominates, and so, for a given size and weight, I/W is less than it is for the four-load configurations. 
         [0101]    The two-load configuration is illustrated in  FIG. 5(   c ), with the same notation as that of the four-load configuration. The fixed dimensions are chosen to be a 2 =b 1 =⅛″ and a 3 =b 3 =1″ as above, and now also c 1 =c 2 =⅝″. I choose the optimization parameter to be s=c 3 /d 3  as above. The results of the optimization calculations, for various values of ab 0 , r, d 12 , and d 3 , are given in Table 2 below. Both COM coordinates (x 1 ,y 1 ) are now given because the head is no longer left-right symmetric. The I/W values are seen to be between 3% and 4.56 less than those of the four-load heads. The largest chosen values for d3 are limited by the requirement that c 3  is less than the USGA limit of 2.5″. 
         [0000]    
       
         
               
               
               
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 2 
               
               
                   
               
               
                 ab0 
                 r 
                 cd12 
                 d3 
                 s1 
                 c3 
                 x1 
                 y1 
                 I/W 
                 I/Wa 2   
                 I 
                 W 
                 wd 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 7 
                 3 
                 1 
                 1 
                 1.484 
                 1.48 
                 2.31 
                 3.12 
                 17.70 
                 0.361 
                 158 
                 8.9 
                 4.7 
               
               
                 7 
                 3 
                 1 
                 1.5 
                 1.307 
                 1.96 
                 2.50 
                 3.09 
                 18.21 
                 0.372 
                 211 
                 11.6 
                 7.0 
               
               
                 7 
                 3 
                 1 
                 1.5 
                 1.219 
                 2.44 
                 2.61 
                 3.07 
                 18.51 
                 0.378 
                 263 
                 14.2 
                 9.4 
               
               
                 7 
                 7 
                 1 
                 1 
                 1.227 
                 1.23 
                 2.82 
                 3.34 
                 19.06 
                 0.389 
                 288 
                 15.1 
                 10.9 
               
               
                 7 
                 7 
                 0.625 
                 1.5 
                 1.445 
                 2.17 
                 2.50 
                 3.38 
                 19.73 
                 0.403 
                 211 
                 10.7 
                 6.4 
               
               
                 7 
                 7 
                 0.5 
                 2 
                 1.370 
                 2.47 
                 2.59 
                 2.47 
                 19.96 
                 0.407 
                 242 
                 12.1 
                 7.7 
               
               
                 6 
                 7 
                 0.75 
                 2.1 
                 1.171 
                 2.46 
                 2.49 
                 2.85 
                 14.50 
                 0.403 
                 244 
                 16.8 
                 12.9 
               
               
                 6 
                 7 
                 0.625 
                 1.9 
                 1.289 
                 2.45 
                 2.32 
                 2.87 
                 14.57 
                 0.405 
                 173 
                 11.9 
                 8.1 
               
               
                 5 
                 7 
                 0.75 
                 2.2 
                 1.126 
                 2.48 
                 2.14 
                 2.37 
                 9.82 
                 0.393 
                 165 
                 16.8 
                 13.5 
               
               
                 5 
                 7 
                 0.625 
                 2 
                 1.217 
                 2.43 
                 2.02 
                 2.38 
                 9.97 
                 0.399 
                 117 
                 11.7 
                 8.5 
               
               
                 4 
                 7 
                 0.75 
                 2.3 
                 1.085 
                 2.50 
                 1.77 
                 1.89 
                 6.02 
                 0.376 
                 101 
                 16.8 
                 14.1 
               
               
                 4 
                 7 
                 0.625 
                 2.1 
                 1.151 
                 2.42 
                 1.69 
                 1.90 
                 6.20 
                 0.388 
                 71 
                 11.4 
                 9.0 
               
               
                   
               
             
          
         
       
     
         [0102]    The three-load putter head configurations are considered next. Described above are two distinct types: the U-type, illustrated in  FIG. 5(   d ), and the T-type, illustrated in  FIG. 5(   b ). First considered is the U-type putter head. The fixed dimensions are a 2 =b 1 =⅛″ and a 3 =b 3 =1″ as before, and also the thickness of the forward load will be fixed at e 2 =⅛″. The optimization parameter is chosen to be s=e 2 /d 2 , the front/back load width ratio. The results of the optimization calculations, for ab 0 =7, 6, 5, and 4, r=3 and 7, e 3 =1 and 0.5, and d 3 =1 and 2.5 are given in Table 3. The back load base size d 1 =d 2 =d 12  is adjusted to give reasonable weights. The I/W values are seen to be between 13% and 15% less than those of the four-load heads, but still much larger than those described in the prior art. 
         [0000]    
       
         
               
               
               
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 3 
               
               
                   
               
               
                 ab0 
                 r 
                 cd12 
                 d12 
                 d3 
                 s1 
                 e1 
                 y1 
                 I/W 
                 I/Wa 2   
                 I 
                 W 
                 wd 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 7 
                 3 
                 1 
                 1 
                 1 
                 0.706 
                 0.71 
                 4.54 
                 16.92 
                 0.345 
                 142.1 
                 8.4 
                 4.7 
               
               
                 7 
                 7 
                 1 
                 1 
                 1 
                 4.070 
                 4.07 
                 3.90 
                 17.01 
                 0.347 
                 343.6 
                 20.2 
                 10.9 
               
               
                 7 
                 7 
                 0.5 
                 0.75 
                 1 
                 4.543 
                 3.41 
                 4.00 
                 17.46 
                 0.356 
                 214.8 
                 12.3 
                 6.1 
               
               
                 7 
                 7 
                 0.5 
                 0.5 
                 2.5 
                 7.024 
                 3.51 
                 4.15 
                 17.96 
                 0.367 
                 235.3 
                 13.1 
                 6.8 
               
               
                 6 
                 7 
                 0.5 
                 0.5 
                 1 
                 10.73 
                 5.36 
                 2.28 
                 11.94 
                 0.332 
                 115.8 
                 9.7 
                 2.7 
               
               
                 6 
                 7 
                 0.5 
                 0.5 
                 2.5 
                 6.070 
                 3.04 
                 3.68 
                 13.01 
                 0.361 
                 158.7 
                 12.2 
                 6.8 
               
               
                 5 
                 7 
                 0.5 
                 0.5 
                 1 
                 8.516 
                 4.26 
                 2.08 
                 8.44 
                 0.338 
                 70.9 
                 8.4 
                 2.7 
               
               
                 5 
                 7 
                 0.5 
                 0.5 
                 2.5 
                 5.057 
                 2.53 
                 3.20 
                 8.80 
                 0.352 
                 99.44 
                 11.3 
                 6.8 
               
               
                 4 
                 7 
                 0.5 
                 0.75 
                 1 
                 2.584 
                 1.94 
                 2.57 
                 5.15 
                 0.322 
                 48.93 
                 9.5 
                 6.1 
               
               
                 4 
                 7 
                 0.5 
                 0.5 
                 2.5 
                 3.968 
                 1.98 
                 2.69 
                 5.37 
                 0.336 
                 55.31 
                 10.3 
                 6.8 
               
               
                 4 
                 7 
                 1 
                 0.5 
                 2.5 
                 4.645 
                 2.32 
                 2.31 
                 5.55 
                 0.347 
                 67.16 
                 12.1 
                 6.8 
               
               
                   
               
             
          
         
       
     
         [0103]    The final example is the T-type three-load configuration. The fixed dimensions are chosen as above: a 3 =b 1 =c 3 =1″ and b 3 =c 2 =⅛″. The head size ab 0  varies from 4″ to 7″, the density ratio r is 3 or 7, and the back load height is 1″ or 2.5″. The front face a 1  of the front load is chosen to have length 2a 0 /3, the smallest face length compliant with USGA regulations. I have chosen this length to be as small as possible because the main advantage of this configuration over the above ones is it&#39;s more compact size. The optimization parameter is again chosen to be s=a 2 /d 2 , the front/back load width ratio. The back load base size d 12  is again adjusted to give reasonable weights. The results of the optimization calculations are given in Table 4. 
         [0000]    
       
         
               
               
               
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 4 
               
               
                   
               
               
                 ab0 
                 r 
                 a1 
                 d12 
                 d3 
                 s1 
                 a2 
                 y1 
                 I/W 
                 I/Wa 2   
                 I 
                 W 
                 wd 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 7 
                 3 
                 4.7 
                 1 
                 1 
                 0.380 
                 0.38 
                 3.97 
                 14.73 
                 0.301 
                 170.9 
                 11.6 
                 4.7 
               
               
                 7 
                 3 
                 4.7 
                 0.63 
                 1 
                 0.590 
                 0.37 
                 4.03 
                 15.27 
                 0.312 
                 174.1 
                 11.4 
                 4.6 
               
               
                 7 
                 7 
                 4.7 
                 1 
                 1 
                 0.312 
                 0.31 
                 4.06 
                 15.73 
                 0.321 
                 341.3 
                 21.7 
                 10.9 
               
               
                 7 
                 7 
                 4.7 
                 0.75 
                 1 
                 0.272 
                 0.20 
                 3.97 
                 15.66 
                 0.320 
                 220.8 
                 14.1 
                 6.1 
               
               
                 7 
                 7 
                 4.7 
                 0.5 
                 2.5 
                 0.433 
                 0.22 
                 4.05 
                 16.21 
                 0.331 
                 244.8 
                 15.1 
                 6.8 
               
               
                 6 
                 7 
                 4 
                 0.75 
                 1 
                 0.303 
                 0.23 
                 3.43 
                 11.44 
                 0.318 
                 153.3 
                 13.4 
                 6.1 
               
               
                 6 
                 7 
                 4 
                 0.5 
                 2.5 
                 0.483 
                 0.24 
                 3.51 
                 11.89 
                 0.330 
                 171.2 
                 14.4 
                 6.8 
               
               
                 5 
                 7 
                 3.4 
                 0.75 
                 1 
                 0.337 
                 0.25 
                 2.88 
                 7.85 
                 0.314 
                 100.5 
                 12.8 
                 6.1 
               
               
                 5 
                 7 
                 3.4 
                 0.5 
                 2.5 
                 0.536 
                 0.27 
                 2.96 
                 8.21 
                 0.328 
                 113.3 
                 13.8 
                 6.8 
               
               
                 4 
                 7 
                 2.7 
                 0.75 
                 1 
                 0.392 
                 0.29 
                 2.33 
                 4.88 
                 0.305 
                 58.56 
                 12.0 
                 6.1 
               
               
                 4 
                 7 
                 2.7 
                 0.5 
                 2.5 
                 0.623 
                 0.31 
                 2.41 
                 5.15 
                 0.322 
                 66.95 
                 13.0 
                 6.8 
               
               
                   
               
             
          
         
       
     
         [0104]    The largest values of I/W are again correspond to the larger size of ab 0  and the larger density ratio r. For a given size and density, the largest values correspond to the largest back load height d 3 . The I/W values are seen to be between 20% and 21% less than those of the four-load heads, but again still much larger than those previously described in the prior art. 
         [0105]    Some of the above results are summarized is Table 5 that provides I/W values for the four lengths (7″, 6″, 5″, and 4″) of the four putter head types. 
         [0000]                                                                                    TABLE 5                           length                    type   7   6   5   4                            4S   20.6   15.1   10.4   6.5           2L   20.0   14.6   10.0   6.2           3U   18.0   13.0   8.8   5.6           3T   16.2   11.9   8.2   5.2                        
This table exhibits the maximum obtained value of the MOI ratio I/W (in in 2  units) for various head lengths (4″, 5″, 6″, and 7″) and for the four head types (4 load square-type 4S, 2 load L-type 2L, 3 load U-type 3U, and 3 load T-type 3T). For each head type, the I/W values decrease as the head size a decreases, and for each head size, the I/W value is largest for the 4 load square-configuration and smallest for the 3 load T-configuration. The decreases in I/W with size are much larger than the decreases in I/W with putter head type. For all sizes and types, the obtained I/W values are very much larger than those for all known prior art or marketed putters.
 
         [0106]    For each head type, the I/W values are approximately proportional to a 2 , and so are approximately constant when a 2  is divided out. The I/Wa 2  values are given in Table 6 for these same putter head types and sizes. These values are seen to change by only a few percent for each head type. 
         [0000]    
       
         
               
             
               
               
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 6 
               
             
             
               
                   
               
               
                 I/Wa 2  vs. length for four head types 
               
             
          
           
               
                   
                 length 
                   
               
             
          
           
               
                   
                 type 
                 7 
                 6 
                 5 
                 4 
               
               
                   
                   
               
               
                   
                 4S 
                 0.421 
                 0.419 
                 0.414 
                 0.406 
               
               
                   
                 2L 
                 0.407 
                 0.405 
                 0.399 
                 0.388 
               
               
                   
                 3U 
                 0.367 
                 0.361 
                 0.352 
                 0.347 
               
               
                   
                 3T 
                 0.331 
                 0.330 
                 0.328 
                 0.322 
               
               
                   
                   
               
             
          
         
       
     
         [0107]    The data in Table 5 are graphically illustrated in  FIG. 11 . The highest curve on the chart gives the I/W maximum possible values a2/2 for the theoretical 4-load head. The next two curves are for the 4-load and 2-load heads. The fourth curve gives the I/W maximum possible values 25a2/64 for the theoretical 3-load head, and the lowest two curves are for the 3U and 3T heads, respectively. The obtained values for each head type are as close as possible to the theoretical upper limits. The MOI ratios of prior art putter heads are not even close these obtained values. 
         [0108]    The data exhibited in tables 1-6 and the graph of  FIG. 11  enables each golfer to choose an ideal putter head size and type from among the configurations described in this document. The choices can be determined from the I/W verses size graph in  FIG. 11 . Each golfer can choose appropriate criterion in one of two ways. The golfer can state the largest head size with which the golfer is comfortable, or the golfer can state the smallest I/W value that the golfer finds necessary. The size criterion is based on the look and feel desired by the golfer, whereas the MOI criterion is based on the magnitude of the off-center error the golfer typically makes. The larger this error, the larger is the MOI needed to control the putt. The head weight W desired by the golfer determines the MOI ratio I/W appropriate for the golfer. 
         [0109]      FIG. 12  is the same as  FIG. 11  but with the theoretical 4-load putter head and 4-load putter head plots removed. 
         [0110]    To illustrate the procedure, suppose that a given golfer desires a MOI ratio of 11 in 2  to control off-center hitting errors. The smallest length putter head that will provide this ratio is given by the intersection of the 11 in 2  horizontal line with the appropriate curve in  FIG. 12 . This golfer can thus use the 4-load type head with size a=5.1″, the 2-load head with a=5.2″, the 3-load U head with a=5.5″, or the 3-load T head with a=5.8″. The choice among these possibilities depends on the size and shape of the head with which the golfer is most comfortable. 
         [0111]    Suppose instead that the golfer does not want to use a putter head with size a greater than 5.5″. The available range of maximum MOI ratios for this golfer is given by the intersection of the 5.5″ vertical line with the appropriate curve in  FIG. 12 . This golfer can thus use the 4-load head with I/W=12.5 in 2 , the 2-load head with I/W=12.25 in 2 , the 3-load U head with I/W=10.75 in 2 , or the 3-load T head with I/W=9.75 in 2 . The choice among these possibilities depends on the MOI ratio that the golfer needs to control off-center hitting errors. 
         [0112]    Whatever a given golfer requires, the putter head described herein provides the golfer with the largest possible MOI. If the golfer&#39;s size requirement comes first, the golfer can use the vertical lines in  FIG. 12 , or the data in Table 5, to choose an appropriate MOI value. If the golfer&#39;s MOI requirement comes first, the golfer can use the horizontal lines in  FIG. 12 , or the data in Table 7 below, to choose an appropriate length value. 
         [0000]    
       
         
               
               
               
             
               
               
               
               
               
               
             
               
               
               
               
               
               
             
           
               
                   
                 TABLE 7 
               
             
             
               
                   
                   
               
               
                   
                 IW 
                   
               
             
          
           
               
                   
                 type 
                 7 
                 11 
                 15 
                 19 
               
               
                   
                   
               
             
          
           
               
                   
                 4S 
                 4.07 
                 5.11 
                 5.97 
                 6.72 
               
               
                   
                 2L 
                 4.13 
                 5.19 
                 6.07 
                 6.83 
               
               
                   
                 3U 
                 4.38 
                 5.49 
                 6.41 
                 &gt;7 
               
               
                   
                 3T 
                 4.59 
                 5.75 
                 6.73 
                 &gt;7 
               
               
                   
                   
               
             
          
         
       
     
         [0113]    If, for example, a golfer requires I/W=7 in 2 , the golfer can use the 4S putter head with a=4.1″ or the 3T head with a=4.6″, etc. If the golfer instead uses a conventional head, a size of 6″ or 7″ would be required. If the golfer requires I/W  32  11 in 2 , the golfer can use the 4S putter head with a=5.1″, or the 3T head with a=5.75″, or etc. If the golfer instead wants to use a conventional head, the golfer would find nothing available. (No conventional head, of any USGA compliant size, can provide an MOI ratio as large as 11 in 2 .) These considerations illustrates a major advantage of the heads disclosed herein. They provide the largest MOI for a given weight and size, or, equivalently, they provide the heads of the smallest size for a given weight and MOI. 
         [0114]    The same goes for the larger I/W values. As an extreme case, if I/W=19 in 2  is required (for a golfer who hits off-center by several inches!), the head choices are limited to the 6.7″ 4S head or the 6.8″ 2L head. The 3-load heads require sizes beyond the USGA limit of 7″, and conventional heads would require sizes of over 10″. 
         [0115]    These tables and graphs display data for putter heads of maximum MOI ratio for each head type. These maximum values of I/W arise from use of load heights at or close to the maxim value of 2.5″ compliant with USGA regulations, and the use of load densities as large as practical. If one chooses to use lower load heights, the methods disclosed herein can be used to design heads with maximum MOI ratios for a given head length and load height. 
         [0116]    The very large MOI ratios disclosed herein are achieved from use of the following disclosed principles: 1) because of their placement and shape, the head loads and connecting elements are located as far as possible from the head COM; 2) because of their dimensions and shapes, the head loads are as heavy as possible, and the connecting elements are as light as possible; and, 3) the load weight ratios are optimally determined by mathematical maximization calculations. 
         [0117]    Embodiments of these principles as disclosed above are intended to illustrate these principles. Persons skilled in the art can easily use these principles to design large MOI putter heads with many different sizes, shapes, densities, and appearances. It is likewise easy to incorporate conventional elements such as a lofted face to help lift a golf ball, a face with ridges to provide more friction and spin, an embedded elastomer to provide better feel, a (possibly adjustable) shaft holder, and a visible line to indicate the COM position. (Because of the large MOI, the latter element is not really necessary.) A prototype four-load putter is shown in  FIG. 10 . 
       Structure Ratios for Large MOI Putters 
       [0118]    1. The connecting elements have a relatively small density, such as 1.6 oz/in 3  (aluminum), and the load elements have a relatively large density, such as 5.3 oz/in 3  (copper) and 11.6 oz/in 3  (tungsten). The density ratios thus vary from 3.3 to 7.3. 
         [0119]    2. The loads are much higher than wide, with heights preferably close to the USGA limit of 2.5″ and widths between 0.5″ and 0.75″. The ratio of load height h to width d is at least 3 and preferably about 5. 
         [0120]    3. The base of the head is preferably square (side length a), with the loads placed at the four corners for the four-load head, two opposite corners for the two-load head, and at the two rear corners and at the center of the front side for the three load head. The corner loads are preferably substantially right-triangular, with right sides of equal length d. The width d is preferably about 0.5″, so that the ratio of base width a to load width d is about 8 for a=4″ and 14 for a=7″. 
         [0121]    4. There are as few connecting elements as possible (three for the four-load and three-load heads, two for the two-load head), and they are placed at the perimeter of the head base. The connecting elements are much higher than wide, with heights about 1″ and widths about ⅛″ for stability. The ratio of connector height to width is thus at least about 8. 
         [0122]    5. The total weight wc of the (aluminum) connecting elements is about a*(0.6 oz/in), and the typical total head weight is about W=12 oz. The ratio wc/W is thus about a/20″, which is 0.20 for a=4″ and 0.35 for a=7″. 
         [0123]    6. The ratio s=w 1 /w 2  of the front-load weight w 1  to the back-load weight w 2  is such that the MOI ratio f(s)=I/Wa 2  is maximal. That is, s is the appropriate solution of df/ds=0. This ratio is found to vary between 1.0 and 1.5, depending on the size and configuration of the club head. 
         [0124]    Certain modifications of the present invention have been discussed above. Other modifications of the present invention will occur to those practicing in the art of the present invention. Accordingly, the description of the present invention is to be construed as illustrative only and is for the purpose of teaching those skilled in the art the best mode of carrying out the invention. The details may be varied substantially without departing from the spirit of the invention, and the exclusive use of all modifications which are within the scope of the appended claims is reserved.