Patent Publication Number: US-9839813-B2

Title: Low dimple coverage and low drag golf ball

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This patent application is a non-provisional and claims the priority of U.S. Prov. Pat. App. Ser. No. 62/188,436, having the same title, filed Jul. 2, 2015, and incorporated fully herein by reference. 
    
    
     FIELD 
     The present invention is generally related to the field of golf balls, and more specifically to a golf ball having low dimple coverage and low drag. 
     BACKGROUND 
     Golf has become an increasingly popular sport with both amateurs and professionals, which has spurred the development of a wide range of technologies related to the design and manufacture of golf balls to improve the flight performance of a golf ball. The flight performance of the golf ball is affected by a variety of factors including the weight, size, materials, dimple pattern, and external shape of the golf ball. The United States Golf Association (“USGA”) sets the limits for the maximum weight of a golf ball to 45.93 grams (1.62 oz) and the minimum diameter of a golf ball to 42.67 grams (1.680 oz). Golf ball manufacturers seek to improve the performance of golf balls by adjusting the materials and construction of the ball within USGA constraints, and adjusting the dimple pattern and dimple shape to enhance the aerodynamics. 
     There are two important dimensionless parameters related to golf ball aerodynamics: the Reynolds number (Re=UD/v) and the dimensionless spinning rate (α=ω*D/(2U)). In these equations, U is the speed of the golf ball, D is the diameter of the golf ball, v is the kinematic viscosity of the air, and ω is the angular velocity of the golf ball. The Reynolds number Re measures the effect of inertial to viscous forces and is generally in the range of Re=80,000 to Re=250,000 during the flight of a golf ball. The spinning rate a is the ratio of the tangential rotational velocity of the golf ball to its translation speed and measures how fast the golf ball is spinning compared to its translational speed, and is generally between α=0.1 to α=0.3. 
     During the flight of a golf ball the air exerts a force on the golf ball that affects its trajectory. The aerodynamic force has three components as shown in  FIG. 1 . One component is the drag F D , which is parallel and opposite in direction to the motion of the golf ball G M . The other two components are the lift F L , which is perpendicular to the drag and is almost aligned with the vertical direction, and the lateral force F S , which is perpendicular to both the drag and lift forces (i.e., normal to the drawing page). The drag and lift coefficients C D  and C L , respectively, are defined as C D =F D /(½ρU 2 ), and C L =F L /(½ρU 2 ), where ρ is the density of the air A M  in which the golf ball G M  is traveling. 
     SUMMARY 
     An aspect of the present disclosure provides for an improved golf ball having low dimple coverage and a low drag coefficient to improve flight characteristics of the golf ball. 
     In one implementation, the present disclosure provides a method of determining the optimum number of dimples on a spherical surface. The method includes providing at least two spheres each having a plurality of dimples on its surface. Then generating a drag curve for each sphere and a lift curve for each sphere. A trajectory analysis for each sphere is then conducted and the results are for each sphere are compared. 
     The method can further include generating the drag curve by determining the coefficient of pressure over the polar angle for each sphere and then taking the integral of coefficient of pressure over the polar angle for each sphere. 
     The method can further include where the trajectory analysis includes launch parameters for the spheres. The launch parameters can include selecting a launch speed, launch angle, and spin rate. 
     The method can also include providing at least two spheres have a different number of dimples on its surface. Alternatively, or in addition to, the two spheres could have different dimple shapes and/or sizes. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
       Embodiments of the invention will now be described, by way of example only, with reference to the accompanying drawings in which corresponding reference symbols indicate corresponding parts. 
         FIG. 1  is a schematic illustrating the drag, lift, and lateral aerodynamic forces acting on a ball. 
         FIG. 2  is a plot of the drag coefficient versus the Reynolds number for a stationary smooth sphere, a conventional golf ball with circular dimples, a Uniroyal Plus 6 golf ball, a Titleist DT-Distance golf ball, and a Srixon golf ball. 
         FIG. 3A  is a schematic illustrating boundary layer behavior and flow separation for a subcritical regime of a golf ball in flight. 
         FIG. 3B  is a schematic illustrating boundary layer behavior and flow separation for a supercritical regime of a golf ball in flight. 
         FIG. 3C  is a plot showing a distribution of the pressure coefficient over the sphere for the subcritical regime of  FIG. 3A  and the super critical regime of  FIG. 3B . 
         FIG. 4  is a schematic illustrating the polar and azimuthal angles on a ball in a spherical coordinate system. 
         FIG. 5A  is a plot of the contours of the coefficient of friction scaled by Re 0.5  on a traditional golf ball. 
         FIG. 5B  is a plot of the coefficient of friction averaged over the azimuthal angle versus the polar angle for a traditional golf ball and a smooth sphere. 
         FIG. 6A  is a plot of the average pressure coefficient plotted versus the polar angle. 
         FIG. 6B  is a plot of the pressure coefficient over the azimuthal coordinate. 
         FIG. 6C  is a plot of the integral of the coefficient of pressure over the projected surface area as a function of the polar angle. 
         FIG. 7  is a diagram of a low drag golf ball having 120 quasi-equally spaced spherical dimples. 
         FIGS. 8A and 8B  are plots of the contours of time averaged skin friction scaled by Re 0.5  on the traditional golf ball of  FIG. 5A  and the low drag golf ball of  FIG. 7 , respectively. 
         FIG. 9A  is a plot of the time averaged friction coefficient versus the polar angle along a first azimuthal line and a second azimuthal line for the low drag golf ball of  FIG. 7 . 
         FIG. 9B  is a plot of the time averaged friction coefficient averaged in the azimuthal and plotted versus the polar angle. 
         FIG. 10A  is a plot of the pressure coefficient averaged over time and azimuthal coordinate versus polar angle for a traditional golf ball and the low drag golf ball of  FIG. 7 . 
         FIG. 10B  is a plot of the integral of the pressure coefficient over the projected surface area as a function of the polar angle for a traditional golf ball and the low drag golf ball of  FIG. 7 . 
         FIG. 11A  is a diagram of the icosahedron used, in accordance with the present disclosure, to create the icosadeltahedral structures for the golf ball embodiments shown in perspective view in  FIGS. 11B-E : a golf ball embodiment having 92 dimples, as shown in  FIG. 11B ; a golf ball embodiment having 162 dimples, as shown in  FIG. 11C ; a golf ball embodiment having 272 dimples, as shown in  FIG. 11D ; and a golf ball embodiment having 392 dimples as shown in  FIG. 11E . 
         FIG. 12A  is a diagram of the octahedron used, in accordance with the present disclosure, to create the octahedral structures for the golf ball embodiments shown in perspective view in  FIGS. 12B and 12C : a golf ball embodiment having 110 dimples, as shown in  FIG. 12B ; and a golf ball embodiment having 194 dimples, as shown in  FIG. 12C . 
         FIG. 13  is a table showing the golf ball surface area coverage by dimples, dimple volume, and drag coefficients for the golf ball embodiments of  FIGS. 11B-E . 
         FIG. 14  is a plot of the drag coefficient versus Reynolds number for golf ball embodiments of  FIGS. 11B-E  which are not spinning. 
         FIG. 15  is a plot of the drag coefficient versus Reynolds number for the golf ball embodiments of  FIGS. 11B-E  spinning at 2500 rpm. 
         FIG. 16  is a plot of the coefficient of lift versus Reynolds number for the golf ball embodiments of  FIGS. 11B-E  spinning at 2500 rpm. 
         FIG. 17A  is a diagram of a golf ball embodiment having 120 spherical dimples that are 25% deeper than the embodiments in  FIGS. 11B-E . 
         FIG. 17B  is a diagram of golf ball embodiment having 120 spherical dimples that are 25% larger than the embodiments in  FIGS. 11B-E . 
         FIG. 17C  is a diagram of a golf ball embodiment having 120 dimples shaped as truncated cones with walls. 
         FIG. 18  is a plot of the drag coefficient versus Reynolds number for the non-spinning golf ball embodiments of  FIGS. 17A, 17B, and 17C . 
         FIG. 19  is a plot of the coefficient of drag versus Reynolds number for the golf ball embodiments of  FIGS. 17A, 17B, and 17C  spinning at 2500 rpm. 
         FIG. 20  is a plot of the coefficient of lift for the golf ball embodiments of  17 A,  17 B, and  17 C spinning at 2500 rpm. 
         FIG. 21  is a plot of the trajectory for the golf ball embodiments of  FIGS. 11B-E  for a high swing speed representative of a PGA tour driver. 
         FIG. 22  is a plot of the trajectory for the golf ball embodiments of  FIGS. 17A, 17B, and 17C  for a high swing speed representative of a PGA tour driver. 
         FIG. 23  is a plot of the trajectory for the golf ball embodiments of  FIGS. 11B-E  for a moderate swing speed representative of an amateur driver. 
         FIG. 24  is a plot of the trajectory for the golf ball embodiments of  FIGS. 17A, 17B, and 17C  for a moderate swing speed representative of an amateur driver. 
         FIG. 25A  is a diagram of a golf ball embodiment having 180 spherical dimples that are 25% deeper than those in the embodiments in  FIGS. 11B-E . 
         FIG. 25B  is a diagram of a golf ball embodiment having 180 spherical dimples that are 25% larger than the embodiment in  FIG. 25A  that are 25% wider than those in the embodiments in  FIGS. 11B-E . 
         FIG. 25C  is a diagram of a golf ball embodiment having 180 dimples shaped as cones with walls. 
         FIG. 26A  is a diagram of a golf ball embodiment having 180 spherical dimples with protrusions and recesses formed into the dimple rim. 
         FIG. 26B  is a close-up view of a dimple of the golf ball of  FIG. 26A , according to inset  26 B of  FIG. 26A . 
         FIG. 27A  is a diagram of an alternative embodiment of a dimple with protrusions formed into the dimple rim. 
         FIG. 27B  is a diagram of an alternative embodiment of a dimple with recesses formed into the dimple rim. 
     
    
    
     DETAILED DESCRIPTION 
     The present disclosure describes a golf ball having low dimple coverage and decreased drag compared to conventional golf balls and a method of designing a golf ball by placing dimples on a sphere. Dimples are used to lower the drag compared to a smooth sphere. However, each dimple imposes a drag penalty on the golf ball. Described in this disclosure are a number of embodiments that optimize the number and placement of dimples on a smooth sphere to reduce the amount of drag on the golf ball. 
     In one embodiment, the present disclosure provides a golf ball including a spherical outer surface having a ball diameter that is the golf ball&#39;s maximum diameter, and a plurality of dimples formed into the outer surface and arranged into a substantially symmetrical pattern entirety around the outer surface, each of the plurality of dimples having a corresponding dimple depth and a corresponding dimple diameter, less than 70% of the outer surface being covered by the plurality of dimples. The dimple depth may be less than or equal to about 0.01 times the ball diameter. The dimple diameter may be less than one-tenth of the ball diameter. The plurality of dimples may each be formed as a conical frustum, or may have a spherical geometry. The plurality of dimples may cover less than 44% and/or more than 18% of the outer surface. The plurality of dimples may express icosadeltahedral symmetry. 
     The plurality of dimples may number between 110 and 272, inclusive. Each dimple may be oriented at a corresponding vertex of an icosahedron having an equal number of vertices to the number of dimples. Or, the plurality of dimples number between 110 and 194, inclusive, and each dimple of the plurality of dimples may be oriented at a corresponding vertex of an octahedron having an equal number of vertices to the number of dimples. The plurality of dimples may consist of a first number of the dimples, and a solution to the Thomson problem for placing the first number of electrons on a sphere may be used to position the first number of dimples on the outer surface, each dimple of the first number of dimples having a corresponding electron of the first number of electrons. 
     In other embodiments, the present disclosure provides a golf ball including a spherical outer surface having a first diameter, and a plurality of dimples formed into the outer surface and arranged around the outer surface, between 18% and 61% of the outer surface being covered by the plurality of dimples. The plurality of dimples may number between 120 and 272, inclusive, and each dimple may be oriented at a corresponding vertex of an icosahedron having an equal number of vertices to the number of dimples. The plurality of dimples may express icosadeltahedral symmetry around the outer surface. The plurality of dimples may be spherical and may have a uniform second diameter less than one-tenth of the first diameter. Or, the plurality of dimples may number between 110 and 272, inclusive, and may have a uniform depth of at least 0.0087 times the first diameter. 
     An aggregate dimple volume, measured as the sum of a dimple volume of each dimple of the plurality of dimples, may be between 0.38% and 1.61%, inclusive, of a volume of a smooth sphere having the first diameter. The plurality of dimples may have a uniform depth greater than 0.0065 times the first diameter. A first dimple of the plurality of dimples may include a dimple surface that intersects the outer surface of the golf ball, and a rim formed at the intersection of the dimple surface with the outer surface, the rim including either or both of: one or more recesses formed into the rim; and one or more protrusions extending away from the rim. 
       FIG. 2  shows a plot  10  of the drag coefficient (C D ) versus Reynolds number for a smooth sphere  14 , a traditional golf ball  15  with circular dimples, a golf ball  16  with hexagonal dimples, and a golf ball  17  with 392 circular dimples. The coefficient of drag can generally be divided into three regions: a pre-drag crisis region, a drag crisis region, and a post-drag crisis region. For a smooth sphere  14 , the pre-drag crisis, also known as the subcritical regime, includes the lower range of Reynolds numbers up to approximately Re=300,000. The coefficient of drag for a smooth sphere  14  in this region remains almost constant, at approximately C D =0.52. As the Reynolds number increases, the coefficient of drag starts to drop quickly and reaches a minimum of C D =0.08 at around Re=400,000. The Reynolds number associated with the minimum coefficient of drag is known as the critical Reynolds number (Re cr ). The region around the critical Reynolds number is referred to as the drag crisis or critical regime. As the Reynolds number continues to increase, the coefficient of drag begins to gradually increase, reaching a value just below C D =0.2. This is the post-drag crisis or supercritical regime. Two schematics and a plot illustrating the main differences between the subcritical and supercritical regimes and why there is such a big drop in the drag coefficient are shown in  FIGS. 3A, 3B , and  FIG. 3C .  FIG. 3A  is a schematic illustrating boundary layer behavior and flow separation for the subcritical regime.  FIG. 3B  is a schematic illustrating boundary layer behavior and flow separation for the supercritical regime.  FIG. 3C  is a plot of the pressure coefficient over the sphere  14  for the two regimes. The pressure coefficient C p  is defined as p=(½ρU 2 ), where p is the static pressure. 
     Referring now to  FIG. 3A , the flow around a smooth sphere  14  in the subcritical regime is shown. The boundary layer is the layer of fluid in the immediate vicinity of a surface where the effects of viscosity are significant. The boundary layer around the sphere  14 , as shown in  FIG. 3A , is a laminar boundary layer  26 , that separates from the sphere  14  at a point  23  on the sphere which is around θ=85°, where θ is the polar angle measured from the stagnation point  22  on the front  19  of the sphere. That is, in the subcritical regime the separation point  23  is closer to the front  19  than the back  20  of the sphere, which creates a wide wake  21 . In the supercritical regime, as shown in  FIG. 3B , the boundary layer is initially a laminar boundary layer  26  but transitions at a point  25  before flow separation to a turbulent boundary layer  27  at around θ=60-90°. Turbulent activity in the turbulent boundary layer  27  creates momentum transport of high speed fluid (e.g., air) towards the ball, which helps overcome the adverse pressure gradient, due to a static pressure increase in the direction of the air flow over the sphere  14 , and shifts separation significantly toward the back  20  of the sphere  14 . The boundary layer separates at the separation point  23  at around θ=120°, resulting in a more streamlined flow with a smaller wake  21 . In both regimes, due to the high Reynolds number, the majority of the drag comes from the pressure distribution around the sphere  14 . A plot of the average pressure coefficient C p  versus polar angle θ from the front  19  of the sphere  14  to the back  20  of the sphere  14  for both the subcritical regime  28  and the supercritical regimes  29  is shown in  FIG. 3C . The plot of the change in the coefficient of pressure over the polar angle for the subcritical regime is shown as line  28 . The plot of the change in the coefficient of pressure over the polar angle for the supercritical regime is shown as line  29 . The pressure on the front part of the sphere  14  is lower in the supercritical regime as a result of higher velocities in the turbulent boundary layer  27 . In addition, the pressure at the back  20  of the sphere  14  recovers to higher values due to the delayed separation  24  contributing to a lowering of drag. 
     The golf ball plots  11 ,  12 , and  13  in  FIG. 2  show that the behavior of the drag coefficient is different than it is for the smooth sphere plot  10 . The same three regions, namely the subcritical, critical, and supercritical regimes, exist for the other golf ball plots  11 ,  12 , and  13 , but the drag crisis occurs at a lower Reynolds number. The Reynolds number where the drag crisis occurs can vary for golf balls depending on the dimple shape and dimple pattern of the particular golf ball. In general, however, the critical Reynolds number varies between Re cr =70,000-100,000 for the golf ball plots  11 ,  12 , and  13 , while for the sphere plot  10  it is Re cr =400,000. It is believed that the dimples cause the boundary layer to transition at a much lower Reynolds number and can therefore accelerate the drag crisis. However, the drag coefficient in the supercritical regime for the plots  11 ,  12 , and  13  of the golf balls are not as low as that for the sphere plot  10  in the supercritical regime. For the sphere plot  10 , the drag coefficient is below C D =0.2 and can be as low as C D =0.08. For the golf ball plots  11 ,  12 , and  13 , the drag coefficient varies between C D =0.22 and C D =0.28, depending on the dimple design of the golf balls of the plots  11 ,  12 , and  13 . Therefore, there is a difference in the drag coefficient of approximately AC D =0.1 between the golf balls plots  11 ,  12 , and  13  and the sphere plot  10  in the respective supercritical regimes. 
     Referring now to  FIGS. 5A and 5B , contours of the skin friction coefficient C f  scaled by Re 0.5  over a traditional golf ball  18  having 312 spherical dimples are shown. “Spherical,” in the sense used herein to describe the dimples of golf balls, means the dimple is concave and formed with a geometry as if an intersection between the golf ball outer surface and a second sphere were removed from the golf ball. The skin friction coefficient is defined as τ w /(½ρU 2 ), where τ w  is the local wall shear stress. The dimples (e.g., dimple  33 ) have a diameter of d=0.095 D and a maximum depth k=0.0067D, where D is the golf ball diameter. The golf ball  18  has a dimple coverage of approximately 70%, which is comparable to most commercially available golf balls on the market. The dimple coverage is defined as the percent of area covered by dimples to the total surface area of a smooth sphere of the same diameter as the golf ball. In  FIG. 5A  areas of flow separation  30  are indicated by solid curved lines. As shown in  FIG. 5A , the traditional golf ball  18  incurs localized separation at separation points  30  in the dimples as early as about θ=45°. Separation continues to occur within each dimple after that. The local separation points  30  are consistent with the formation of shear layers in the onset of transition triggered by the dimples. Complete global separation occurs along a jagged separation line  31  towards the rear of the golf ball  18 , as shown in  FIG. 5A . When the flow separates, the skin friction coefficient decreases in value and becomes negative inside the dimples, but then rises past the trailing edges of the dimples. The separation line varies significantly in the azimuthal depending on the dimple arrangement. The azimuthal is an angular measurement in a spherical coordinate system, where a vector between the center of the sphere and a point of interest is projected perpendicularly onto a reference plane, and where the angle φ between the projected vector and a reference vector on the reference plane is the azimuth as shown in  FIG. 4 . The separation line  31  can occur as late as θ=120° at some azimuthal locations, which is similar to the case of a sphere  14  in the supercritical regime. When averaged over the azimuthal coordinate though, global separation occurs around θ=113°, which is slightly earlier than for the sphere  34 , as shown in  FIG. 5B , which shows a plot of the skin friction coefficient averaged over the azimuthal angle versus the polar angle θ for the golf ball plot  35  and the smooth sphere plot  34 . 
     In the supercritical regime, most of the drag comes from the pressure and less than 10% comes from the skin friction on the surface of the golf ball. Therefore any differences in the total drag between the traditional golf ball  18  and the sphere  14  should be reflected in the pressure coefficient. A plot of the time averaged pressure coefficient C p  versus polar angle θ measured from the stagnation point  22  for a land area plot  41  of the golf ball  18 , dimpled area plot  40  of the golf ball  18 , and the smooth sphere plot  36  are shown in  FIG. 6A . The pressure coefficient on the land area plot  41  (the non-dimpled areas, see  FIG. 5A ) and the dimple area plot  40  are plotted separately to better understand the effect of the dimples  33  on the pressure. The distribution of the pressure coefficient for the sphere plot  36  in the supercritical regime is super-imposed on the same graph for comparison. The pressure on the traditional golf ball  18  fluctuates considerably, but compared to the smooth sphere  14 , the pressure coefficient is for the most part higher inside the dimples  33  and lower in the land area  32 . A golf ball plot  37  of the pressure coefficient on the traditional golf ball  18  averaged over the azimuthal angle is shown in  FIG. 6B , with an average depth plot  42  of dimples  33 . The average dimple  42  depth is the average depth of all the dimples  33  for a given polar angle. The fluctuations of the pressure coefficient are clearly observed in this plot too, they are relatively smaller near the stagnation point  22  but increase as the polar angle θ increases with the highest fluctuations taking place around θ=90°. Also, the peaks in the pressure coefficient correlate with peaks in the average depth. As the average depth plot  42  approaches zero, which is closer to the land area  32 , the pressure coefficient drops and approaches that of the sphere plot  36 . Overall, though, the pressure on the traditional golf ball  18  is consistently higher on the front part of the traditional golf ball  18  and up to the separation line  31  around θ=120°. This is important because it demonstrates that the dimples  33  on the traditional golf ball  18  incur a local pressure penalty compared to a smooth sphere  14 . 
     After separation occurs, the base pressure at the back  20  of the traditional golf ball  18  does not exhibit any fluctuations and remains close in value to that of the sphere  14 . This is consistent with the observation that the average separation line  31  on the traditional golf ball  18  is close to that of a sphere  14 . Finally, the integration of the pressure over the projected surface area of the traditional golf ball  18  and the sphere  14  are plotted as a golf ball plot  37  and a sphere plot  36  and a difference plot  39  showing the difference between the two plots is shown in  FIG. 6C . The integral is plotted versus polar angle θ, and when evaluated over the entire surface (i.e., θ=0−180°, gives the total drag coefficient due to pressure. The difference plot  46  helps identify the source of the drag penalty. Initially, the pressure integrals up to θ=45° show that the dimple  33  on the very front of the traditional golf ball  18  (i.e., θ=0° and at the stagnation point  22 ) does not impose a pressure penalty. The main drag penalty comes from the dimples  33  located between θ=45° and θ=90°. In this region the pressure overhead of the dimples  33  becomes very important and by this location the drag surplus of the dimples  33  is about 0.1, which is approximately the total drag difference between the traditional golf ball  18  and the sphere  14 . To put it in perspective, the cumulative pressure overhead of the dimples  33  in the front  19  of the traditional golf ball  18  is almost equal to the total drag of the sphere  14  in the super-critical regime. This analysis is very crucial in understanding the role of the dimples  33  in the drag reduction process. Although dimples  33  can accelerate the drag crisis and reduce drag by 50% compared to a smooth sphere  14 , at the same Reynolds number they also impose a drag ‘penalty’. It would be advantageous to identify the optimum number and position of dimples to minimize the drag penalty on a golf ball. 
     It should be noted that truly equally spaced points on a sphere  14  are limited to configurations with a predetermined number of points, such as octahedron, dodecahedron and icosahedron, which have 8, 12, and 20 vertices, respectively. For an arbitrary number of points, the methodology of placing the dimples  33  on a sphere  14  is very similar to the Thomson Problem. In the Thomson Problem, the location of the points is found by determining the minimum energy configuration of N electrons on the surface of a sphere that repel each other with a force given by Coulomb&#39;s law. The dimples  33  are then placed such that the dimple centers coincide with the location of the points (i.e., electrons). This method was used to place 120 spherical dimples  33  onto a sphere. The result is a design with dimples  33  that are quasi-equally spaced and quasi-symmetric. The latter is very important, since USGA sets a symmetry test for USGA sanctioned golf balls. The test involves measuring the deviation in flight performance when the golf ball is launched under specific conditions but is placed on the tee either in poles-over-poles or in poles horizontal orientation. It should be noted that other methods for accounting for the minimum energy configuration can be used such as replacing the electrostatic force with a spring model to produce similar results. 
       FIG. 7  illustrates a first exemplary embodiment for an improved golf ball  51  having a dimple coverage area that is lower than that of commercial golf balls, and in particular lower than about 70%. The golf ball  51  is a sphere having a diameter D, and contains 120 quasi-equally spaced spherical dimples  60 ; that is, the dimples  60  are laid out with an approximately even distribution, the distance between dimples  60  being as uniform as reasonably possible in a distribution of an arbitrary number of points. The golf ball  51  has a dimple diameter of d=0.079D and a dimple depth of k=0.0087D. The resulting dimple  60  coverage of the golf ball is 18.6%. 
     With a Reynolds number of Re=200,000 in the supercritical regime, the average drag coefficient of the golf ball  51  is 0.17, or 26% lower than the drag coefficient for the traditional golf ball  18  discussed in the background. Referring to  FIGS. 8A and 8B , the contours of the time averaged skin friction coefficient on both the traditional golf ball  18  ( FIG. 8A ) and the first embodiment golf ball  51  (FIG.  8 B 0  are shown. For the golf ball  51 , the peak values of skin friction coefficient are not as elevated as for the traditional golf ball  18 , however, the separation line  64  at the back of the golf ball  51  is shifted significantly toward the rear of the ball  71  and global separation is delayed. There is also a significant variation of the separation line  64  in the azimuthal direction.  FIG. 9A  shows a plot  67  of the skin friction coefficient versus polar angle θ on the azimuthal line  65  and a plot  68  of the skin friction coefficient versus polar angle θ on the azimuthal line  66  on the golf ball  51 , shown in  FIG. 8B , and clearly shows the variation of the separation line  64  along these lines  67  and  68  compared to the sphere plot  34 . For the plot  67 , the flow separates globally around θ=118°, which coincides with the presence of a dimple  60 . On the contrary, for the plot  68 , no dimples  60  are present on the line  66  around the same polar angle, and the separation line  31  is farther delayed at θ=131°. On average, the separation line  31  shown in  FIG. 9B  for the golf ball  51  occurs at θ=125°, compared to θ=118° for the sphere plot  34  and θ=113° for the golf ball plot  35 . 
     The effect of using fewer dimples  60  on the coefficient of pressure is shown in  FIGS. 10A and 10B .  FIG. 10A  plots the coefficient of pressure over the polar angle for a traditional golf ball  18  (plot  37 ) and the golf ball  51  of  FIG. 7  (plot  38 ). When the coefficient of pressure is averaged over the azimuthal direction, the oscillations for the golf ball  51  are significantly reduced compared to the traditional golf ball  18 . In addition the coefficient of pressure for the golf ball  51  is lower than that of the golf ball  18  at the front part of the ball. This confirms the concept that dimples  60  incur a pressure penalty; reducing the number of dimples  60  on the golf ball  51  causes the coefficient of pressure to be reduced too. Also, for the case of the golf ball  51 , the coefficient of pressure at the back of the golf ball  51  recovers more quickly as a result of the delayed separation, yielding a higher back pressure compared to the traditional golf ball  18 . In addition the coefficient of pressure for the golf ball  51  is consistently lower than that of the traditional golf ball  18 , especially in the front part and in the back up to the separation point  64 . 
     The integral of the coefficient of pressure over the projected surface area of the golf ball yields the coefficient of drag due to pressure. The difference plot  47  between the integral of the coefficient of pressure of the golf ball embodiment plot  45  and the traditional golf ball plot  44  is shown in  FIG. 10B . The decrease in drag is obvious in the difference plot  47 . In the front part of the golf ball  51 , the contribution to the drag drop is 0.04, while an additional 0.03 drop takes place due to the higher back pressure. This analysis clearly demonstrates that reducing the dimple  60  coverage on the golf ball  51  reduces the drag in two ways: first by reducing the pressure penalty from dimples of the ball on the front part, and second by shifting the separation point  64  rearward and allowing for a greater pressure recovery at the back  71  of the golf ball  51 . 
     Having clearly identified a source of drag penalty for golf balls and how to reduce the drag by reducing the number of dimples  60 , the drag and lift of the golf ball  51  can be controlled by adjusting the dimple  60  parameters, such as the number of dimples  60 , the shape of the dimples  60 , the size of the dimples  60 , and the depth of the dimples  60 . As shown in  FIGS. 11A-E , additional embodiments are shown having a different number of dimples  60 . For example: golf ball  50  has 92 spherical dimples  60 , golf ball  52  has 162 spherical dimples  60 , golf ball  53  has 272 spherical dimples  60 , and golf ball  54  has 392 spherical dimples  60 . In each embodiment the dimple diameter is d=0.079D and the dimple depth is k=0.0087D, which is the same as with the golf ball  51  of  FIG. 8B . All the embodiments are based on icosadeltahedral structures which are obtained by dividing the triangular faces of an icosahedron  48  into smaller, symmetrical triangles, thus maintaining icosahedral symmetry. The dimple coverage and depth percentage are listed in  FIG. 13 . The dimple coverage starts at 14.3% for the golf ball  50  and linearly increases to 60.8% for golf ball  54 . For the golf ball  54  of  FIG. 11E  having the highest dimple  60  coverage, the dimple  60  coverage is at the low end of today&#39;s commercial golf balls. The dimple  60  volume is defined as the percentage of volume of the dimples  60  divided by that of a smooth sphere  14  and varies from 0.38% to 1.61%. 
     Referring to  FIGS. 12A-C , golf ball embodiments with low dimple coverage can also be based on octahedral structures which are obtained by dividing the triangular faces of an octahedron  49  (see  FIG. 12A ) into smaller triangles. Two such golf ball embodiments  58  and  59 , with 110 and 194 spherical dimples  60  respectively, are shown in  FIGS. 12B and 12C , respectively. The advantage of this method is that the golf ball embodiments  58  and  59  maintain an icosahedral symmetry which is important for meeting the USGA symmetry test requirements. The test involves measuring the deviation in flight performance when the golf ball is launched under specific conditions but is placed on the tee either in poles-over-poles or in poles horizontal orientation. If one defines the poles of the golf ball embodiments  58  and  59  as any pair of opposite points  112 - 113 , or  114 - 115 , or  116 - 117  of the original octahedron, then the golf ball embodiments  58  and  59 , either in poles-over-poles or in poles horizontal orientation, are exactly identical and are therefore expected to have the same flight characteristics. In fact any rotation of the ball embodiments  58  or  59  by 90° about a vertical or horizontal axis results in an identical dimple configuration. 
       FIG. 14  illustrates the drag coefficient plotted against the Reynolds number for the plots  72 ,  73 ,  74 ,  75 , and  76  for the golf ball embodiments  50 ,  51 ,  52 ,  53 , and  54 , respectively, when tested in the wind tunnel in a non-spinning setup. The prototype with the lowest drag coefficient in the supercritical regime is the golf ball  50  with a drag coefficient of C D =0.166 at Re=185,000; this golf ball  50  has 92 spherical dimples  60 . As the number of dimples  60  increases, the coefficient of drag increases, and is the highest for the golf ball  54  with 392 spherical dimples. The behavior exhibited by these embodiments is in agreement with the concept that dimples  60  incur a drag penalty, and that for a given dimple  60  shape, minimizing the number of dimples  60  on a sphere  14  reduces the drag coefficient. Another important factor is that the critical Reynolds number occurs the earliest and the latest for the golf balls  54  and  50 , respectively, implying that more dimples  60  tend to accelerate the drag crisis. 
     When the embodiments start spinning, lift is generated and the drag curves change as well.  FIG. 15  illustrates the drag coefficient for the golf balls  50 ,  51 ,  52 ,  53 , and  54  spinning at 2500 rpm. The plots of drag coefficient versus Reynolds numbers for the golf ball embodiments  50 ,  51 ,  52 ,  53 , and  54  are respectively plots  77 ,  78 ,  79 ,  80 , and  81 . The spin has two main effects on drag. First, as the spin increases, the drag crisis becomes less steep and the critical Reynolds number occurs earlier. Second, in the supercritical regime, the drag increases. This phenomenon is commonly known as lift-induced drag and the amount of induced drag is related to the lift. In general however, the drag coefficient for the embodiments shows a similar trend as the corresponding one for the stationary cases. That is, the embodiments with the least dimple  60  coverage and volume percentage have the least drag in the supercritical regime. The lift coefficient curves for the embodiments spinning at 2500 rpm are shown in  FIG. 16 . The plots of lift coefficient versus Reynolds number for the golf ball embodiments  50 ,  51 ,  52 ,  53 , and  54  are respectively plots  82 ,  83 ,  84 ,  85 , and  86 . The lift is generally positive in the supercritical regime and may become negative in a narrow region just before the critical Reynolds number. As the dimple  62  coverage and volume percentage decreases, the lift curves shift to the right and the lift coefficient in the supercritical regime is higher. 
     Finally, the effect of dimple  60  shape was also investigated. A seventh golf ball  55  embodiment, an eighth golf ball  56  embodiment, and a ninth golf ball  57  embodiment, with 120 dimples each and using the same dimple width or depth as the golf ball  51  embodiment were fabricated and tested in the wind tunnel. The golf balls  55 ,  56 , and  57  are shown in  FIGS. 17A ,  17 B, and  17 C, respectively. The golf ball  55  has spherical dimples  61  with the same inscribed diameter, but the spherical dimples  61  are 25% deeper than the ones of the golf ball  51 . The golf ball  56  has spherical dimples  62  with the same depth but 25% larger inscribed diameter compared to the golf ball  51 . The golf ball  57  has dimples  63  with the same diameter and depth as the golf ball  51 , but formed in the shape of a truncated cone, or conical frustum, with a 45-degree angle. The depth and outer inscribed diameter of the dimples  63  of golf ball  57  is the same as the ones of the golf ball  51 . The dimple coverage for the golf balls  51 ,  55 , and  57  are the same, namely 18%, while coverage for the golf ball  56  is higher at 27%. The dimple volume percentage is the lowest for the golf ball  51  at 0.49% and increases to 0.65%, 0.71%, and 0.77% for the golf ball  55 ,  56 , and  57 , respectively. The drag coefficient curves for the stationary case are shown in  FIG. 18 . The plots of the drag coefficient versus Reynolds number for the golf ball embodiments  51 ,  56 ,  56 , and  57  are respectively shown as plots  73 ,  87 ,  88 , and  89 . Making the dimples deeper as in the golf ball  56  accelerates the drag crisis, but also increases the drag in the supercritical regime compared to the golf ball  51 . A similar behavior is observed with the golf ball  57 , although in this case, the effects are more pronounced. In contrast, making the dimple  60  diameter larger, as in the golf ball  56 , while keeping the depth constant accelerates the drag crisis a little without adding any drag in the supercritical regime. A similar trend for the drag exists when the embodiments where tested in the spinning setup. As shown in  FIGS. 19 and 20 , the drag and lift coefficients are shown at 2500 rpm. In  FIG. 19 , the coefficient of drag versus Reynolds number for the golf ball embodiments  51 ,  55 ,  56 , and  57  are shown as plots  78 ,  90 ,  91 , and  92 , respectively. In  FIG. 20 , the coefficient of lift versus Reynolds number for golf ball embodiments  51 ,  55 ,  56 , and  57  are shown as plots  83 ,  93 ,  94 , and  95  respectively. As shown in  FIG. 20 , the lift coefficient at the higher range of the supercritical regime is close amongst all the embodiments with the golf ball  56  having slightly higher lift than the others. The main difference occurs near the critical Reynolds number, where the lift decreases significantly and becomes negative for all but the golf ball  57 . 
     Using the drag and lift curves obtained from the wind tunnel tests, a trajectory analysis was performed. Two different swings were simulated: a high swing speed representative of PGA tour driver, shown in  FIGS. 21 and 22 , and a moderate swing representative of an amateur driver, shown in  FIGS. 23 and 24 . The launch conditions for the high swing correspond to a golf ball speed of 175 mph, a launch angle of 10°, and a spin of 2520 rpm. These launch conditions are also the ones used by USGA during the overall distance compliance tests. For the moderate swing the launch conditions correspond to a ball speed of 135 mph, a launch angle of 13°, and a spin of 3000 rpm. The trajectories account only for the carry distance, which is the distance covered from the tee to the first point of contact with the ground. The trajectories of the golf balls embodiments  50 ,  51 ,  52 ,  53 , and  54  are shown as plots  96 ,  97 ,  98 ,  99 , and  100 , respectively, for the high swing speed, in  FIG. 21 . The golf ball  54 , which has the largest dimple coverage and dimple volume percentage, traveled the shortest distance of golf balls embodiments  50 ,  51 ,  52 ,  53 , and  54 , due mainly to the golf ball&#39;s  54  relatively large drag coefficient in the supercritical regime. The golf ball  50  traveled the second shortest, due to the fact that the supercritical regime, where the drag coefficient is low, occurs at a relatively small portion of the trajectory in the beginning of the ball flight, and for the rest of the flight the drag coefficient increases considerably. Increasing the number of dimples  60  above golf ball  50 , significantly adds to the carry distance and the golf balls  51 ,  52 , and  53  all exceed 270 yards. The golf ball  53  reaches an estimated carry distance of 277 yards, which is about five yards further than the golf balls  51  and  52 . 
     The effect of dimple  60  shape on the trajectory of a high swing speed is shown in  FIG. 22 . The trajectory for the golf ball embodiments  51 ,  55 ,  56 , and  57  are shown as plots  97 ,  101 ,  102 , and  103 , respectively. The variation in carry distance and maximum height is small and making the dimples deeper, as in the golf ball  55  embodiment, can help the golf ball  55  travel a few extra yards. 
       FIG. 23  shows the trajectories for a moderate swing speed by an amateur player for the golf ball embodiments  50 ,  51 ,  52 ,  53 , and  54  as plots  104 ,  105 ,  106 ,  107 , and  108 , respectively. The Reynolds number at launch conditions is 170,000, which is 22% lower than that of high swing speed. As a result, the embodiment golf balls  51  and  50 , which experience a delayed drag crisis travel the shortest distance at just under 180 yards. The golf ball  54  travels farther at 191 yards while the golf balls  52  and  53  travel the farthest at 202 and 205 yards respectively. The number, coverage, and/or volume of dimples  60  seems to affect considerably the moderate swing speeds.  FIG. 24  shows the trajectories for a moderate swing speed for the golf ball embodiments  51 ,  55 ,  56 , and  57  as plots  105 ,  109 ,  110 , and  111 , respectively, under the launch conditions as in  FIG. 23 . The representative golf ball  51  travels the shortest distance of just under 180 yards, followed by the golf ball  56  having wider dimples  60  at about 190 yards, then the golf ball  57  having frustum-shaped dimples  60  at about 195 yards, and finally the golf ball  55  having deeper dimples  60  traveling the further distance of about 200 yards. The effect of dimple  60  shape seems to affect considerably the moderate swing speeds. 
       FIGS. 25A-C  illustrate additional embodiments of the present low dimple coverage golf balls with diameter D, in accordance with the above descriptions of ball and dimple characteristics. Golf ball  250  of  FIG. 25A , golf ball  260  of  FIG. 25B , and golf ball  270  of  FIG. 25C  each have 180 dimples arranged according to an icosahedral structure as described above. In  FIG. 25A , the golf ball  250  includes symmetrically arranged and uniformly configured spherical dimples  252  having a dimple diameter of 0.079D and a dimple depth of 0.011D. In  FIG. 25B , the golf ball  260  includes symmetrically arranged and uniformly configured spherical dimples  262  with a dimple diameter of 0.098D, about 25% larger than the dimples  252  of  FIG. 25A , and a dimple depth of 0.0087D. In  FIG. 25C , the golf ball  270  includes symmetrically arranged and uniformly configured conical dimples  272  with a dimple diameter of 0.078D and a dimple depth of 0.011D. 
     In the presently disclosed golf balls, the dimples may further have one or more structures recessed into or protruding from either or both of the surface and the rim of the dimple. Referring to  FIGS. 26A-B , another exemplary embodiment of a low dimple coverage golf ball  280  with diameter D has dimples  282  (e.g., 180, as illustrated), which are symmetrically arranged and have a uniform spherical profile, as described above. For example, the dimple diameter may be 0.078D and the dimple depth may be 0.087D. Additionally, each dimple  282  has an arrangement of recesses  288  and/or protrusions  290  formed into the rim  284  of the dimple  282 , where the concave dimple surface  286  meets the outer surface  281  of the golf ball  280 . In some embodiments, as illustrated, the recesses  288  and protrusions  290  may be alternated around the entire circumference defined by the rim  284 ; in other embodiments, only a portion of the rim  284  may include the recesses  288  and/or protrusions  290 , which may be arranged in repeating or other patterns besides alternating. For example,  FIG. 27A  illustrates an alternative dimple  300  having only protrusions  302  and no recesses, and  FIG. 27B  illustrates an alternative dimple  310  having only recesses  312  and no protrusions. The recesses  288  and/or protrusions  290  may be uniform in size, as illustrated, or may be any arrangement of different sizes. Thus, the size, shape, number, and spacing of the recesses  288  and protrusions  290  may be selected to optimize the performance of the golf ball  280 . This arrangement creates a more three dimensional effect around the dimple rim  284 , which can trigger transition to a turbulent flow at a lower Reynolds number. 
     Referring again to the illustrated arrangement of  FIG. 26B , a recess  288  may be any suitable volumetric shape formed into one or both of the outer surface  281  and the dimple surface  286  (e.g., in both when formed into the rim  284 ). For example, subtracting an intersecting volume between the golf ball  280  and a sphere (not shown), as described above, creates a spherical recess  288 . The diameter of the sphere used in the subtraction may be related to the diameter of the golf ball  280  and/or to the diameter of the dimple  282 , and may further depend on the number and/or spacing of structures around the rim  284 . In one embodiment, the diameter of the subtracting sphere is about 10% of the diameter of the sphere used to make the dimple  282  itself. 
     Similarly, a protrusion  290  may be any suitable volumetric shape formed into one or both of the outer surface  281  and the dimple surface  286  (e.g., in both when formed into the rim  284 ). For example, to create a “spherical” protrusion  290  as in the illustrated embodiment, a sphere (not shown) with a diameter of about 10% of the diameter of the dimple  282  may be intersected with the outer surface  281  of the golf ball  280 , and with the dimple surface  286 . The portion of the sphere outside of the outer surface  281  may be subtracted, leaving the outer surface  292  of the protrusion  290  flush with the outer surface  281 . The spherical inner surface  294  of the protrusion  290  may project toward the center of the dimple  282 . 
     Another test of golf balls in accordance with the present disclosure included performance comparisons of the golf balls  250 ,  260 ,  270 ,  280  of  FIGS. 25A-26B  against a popular USGA golf ball; the results of this test are reflected in Table 1, below. In particular, prototypes of the four embodiments were tested against the present market-leading golf ball, the PRO V1 by TITLEIST, each being hit multiple times by the same professional golfer under reasonably reproducible conditions. Launch conditions correspond to the lower range of a professional golfer&#39;s swing. One embodiment, the golf ball  250 , is shown to be hittable with a carry distance that is within about two yards, or under 1%, of the current leading golf ball. 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 Outdoor Testing of Embodiments 
               
            
           
           
               
               
               
            
               
                   
                 Launch Conditions 
                   
               
            
           
           
               
               
               
               
               
               
            
               
                   
                 Ball 
                 Launch 
                   
                 Spin 
                 Performance 
               
            
           
           
               
               
               
               
               
               
               
            
               
                 Golf Ball 
                 speed 
                 Angl. 
                 Spin rate 
                 axis 
                 Carry dist. 
                 Total dist. 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 Pro V1 
                 145.2 
                 7.0 
                 4222 
                 9.1 
                 216.9 
                 234.9 
               
               
                   
                 148.9 
                 6.7 
                 3762 
                 1.0 
                 227.8 
                 248.2 
               
               
                   
                 149.8 
                 9.1 
                 4268 
                 9.3 
                 229.2 
                 243.6 
               
               
                   
                 141.7 
                 4.7 
                 3734 
                 4.1 
                 203.8 
                 230.4 
               
               
                   
                 147.9 
                 11.4 
                 2209 
                 2.6 
                 239.2 
                 269.8 
               
               
                 Ball 250 
                 149.3 
                 9.3 
                 2694 
                 2.2 
                 237.6 
                 264.9 
               
               
                   
                 138.5 
                 7.3 
                 3965 
                 3.3 
                 205.6 
                 226.2 
               
               
                   
                 139.6 
                 8.6 
                 2935 
                 −11.6 
                 210.1 
                 238.6 
               
               
                 Ball 260 
                 144.6 
                 9.2 
                 3549 
                 4.8 
                 225.5 
                 245.1 
               
               
                   
                 139.2 
                 8.4 
                 4252 
                 8.4 
                 207.1 
                 224.6 
               
               
                 Ball 270 
                 147.9 
                 9.1 
                 2985 
                 7.0 
                 233.5 
                 258.1 
               
               
                   
                 139.6 
                 8.6 
                 2935 
                 −11.6 
                 210.1 
                 238.6 
               
               
                 Ball 280 
                 144.8 
                 6.4 
                 4052 
                 6.7 
                 216.2 
                 236.0 
               
               
                   
                 148.6 
                 6.9 
                 2652 
                 −5.3 
                 222.9 
                 256.4 
               
               
                   
                 150.4 
                 7.5 
                 3326 
                 −0.2 
                 234.7 
                 257.6 
               
               
                   
               
            
           
         
       
     
     Although the subject matter has been described in language specific to structural features and/or methodological acts, it is to be understood that the subject matter defined in the appended claims is not necessarily limited to the specific features or acts described. Rather, the specific features and acts are disclosed as illustrative forms of implementing the claims. 
     One skilled in the art will realize that a virtually unlimited number of variations to the above descriptions are possible, and that the examples and the accompanying figures are merely to illustrate one or more examples of implementations. 
     It will be understood by those skilled in the art that various other modifications may be made, and equivalents may be substituted, without departing from claimed subject matter. Additionally, many modifications may be made to adapt a particular situation to the teachings of claimed subject matter without departing from the central concept described herein. Therefore, it is intended that claimed subject matter not be limited to the particular embodiments disclosed, but that such claimed subject matter may also include all embodiments falling within the scope of the appended claims, and equivalents thereof. 
     In the detailed description above, numerous specific details are set forth to provide a thorough understanding of claimed subject matter. However, it will be understood by those skilled in the art that claimed subject matter may be practiced without these specific details. In other instances, methods, apparatuses, or systems that would be known by one of ordinary skill have not been described in detail so as not to obscure claimed subject matter. 
     Reference throughout this specification to “one embodiment” or “an embodiment” may mean that a particular feature, structure, or characteristic described in connection with a particular embodiment may be included in at least one embodiment of claimed subject matter. Thus, appearances of the phrase “in one embodiment” or “an embodiment” in various places throughout this specification is not necessarily intended to refer to the same embodiment or to any one particular embodiment described. Furthermore, it is to be understood that particular features, structures, or characteristics described may be combined in various ways in one or more embodiments. In general, of course, these and other issues may vary with the particular context of usage. Therefore, the particular context of the description or the usage of these terms may provide helpful guidance regarding inferences to be drawn for that context.