Abstract:
The present invention provides a shoe having a multi-layer, multi-density midsole where the surfaces between midsole layers have one or more convexities and one or more concavities which collectively contribute to simulating the effect, and imparting the fitness benefits, of walking on a sandy beach or on a giving or uneven surface regardless of the actual hardness of the surface.

Description:
[0001]    This application claims the benefit of priority based on Provisional Application No. 61/122,911 filed Dec. 16, 2008. 
     
    
     BACKGROUND OF THE INVENTION 
       [0002]    1. Field of the Invention 
         [0003]    The present invention relates to footwear, in particular, to a shoe with fitness benefits. The fitness benefits are experienced through a unique walking action in which the foot strike mimics the effect of walking on a sandy beach or on an uneven surface. This is accomplished through a multi-layer, multi-density midsole where the surfaces between midsole layers have one or more convexities and one or more concavities. 
         [0004]    2. Description of the Related Art 
         [0005]    Shoes are designed for many purposes-from protection on the job to performance on the track or court to special occasions and everyday lifestyle. Shoes have also been used to promote physical health and activity. Increasingly, shoes have given users fitness benefits. Many shoes have attempted to provide users the benefit of improving the user&#39;s fitness by simply walking while wearing such shoes. However, there continues to be a need for such shoes that improve the user&#39;s health yet are comfortable and easy to use. 
         [0006]    Walking is one of the easiest and most beneficial forms of exercise. When done properly and with the appropriate footwear, it strengthens the heart, improves cardiovascular health, increases one&#39;s stamina and improves posture. It also helps to strengthen one&#39;s muscles and maintain joint flexibility. 
         [0007]    Prior art shoes have attempted to improve the user&#39;s fitness by mimicking walking barefoot. These shoes have included a midsole made of hard material throughout the entire midsole except for a recess in the rear region of the shoe in which a softer, cushioning material is placed. See, for example, U.S. Pat. No. 6,341,432 to Muller. Such shoes include an abrupt, discrete pivot point on the bottom surface of the midsole in the middle region of the shoe where the cushioning material ends and the hard material of the midsole begins. Consequently, in every step taken during normal walking while wearing such shoes, the user is forced to overcome this abrupt, discrete pivot point. This can result in significant pain and discomfort. See also, for example, U.S. Pat. No. 6,782,639 to Muller. 
         [0008]    The present invention aims to provide a way of mimicking walking on a sandy beach or on a giving or uneven surface, while not inducing any significant pain or discomfort from doing so. By mimicking walking on a sandy beach and/or on an uneven surface, the present invention aims to significantly increase the fitness and health benefits of everyday walking by requiring the user to exert additional effort and energy while walking and to use muscles that the user otherwise would not use if wearing ordinary footwear, again all without inducing any substantial pain or discomfort. 
       SUMMARY OF THE INVENTION 
       [0009]    It is an object of the present invention to provide a shoe that mimics the effects, and imparts the fitness benefits of, walking on a sandy beach or on a giving or uneven surface without inducing any significant pain or discomfort from doing so. The present invention is a shoe comprising an upper, an outsole, and a midsole, each having a medial side and a lateral side. In a preferred embodiment, the midsole is affixed to the upper and the outsole is affixed to midsole. The upper, midsole, and outsole each has a frontmost point and a rearmost point substantially opposite the frontmost point. When the shoe is being worn by a user, each frontmost point and each rearmost point is oriented with respect to one another such that each frontmost point is closer to the user&#39;s toes than each rearmost point while at the same time each rearmost point is closer to the user&#39;s heel than each frontmost point. 
         [0010]    The shoe has a front portion and a rear portion substantially opposite the front portion. When the shoe is being worn by a user, the front portion and the rear portion are oriented with respect to one another such that the front portion is closer to the user&#39;s toes than the rear portion while at the same time the rear portion is closer to the user&#39;s heel than the front portion. 
         [0011]    The shoe has a front tip that is located at the farthest forward point of the shoe when moving from the rear portion to the front portion. The shoe has a rear tip that is located at the farthest rearward point of the shoe when moving from the front portion to the rear portion. In a preferred embodiment, the front tip coincides with the frontmost point of the upper, the frontmost point of the midsole, or the frontmost point of the outsole while the rear tip coincides with the rearmost point of the upper, the rearmost point of the midsole, or the rearmost point of the outsole. In a preferred embodiment the frontmost point of the upper, the frontmost point of the midsole, and the frontmost point of the outsole are all located relatively close to one another while the rearmost point of the upper, the rearmost point of the midsole, and the rearmost point of the outsole are all located relatively close to one another. 
         [0012]    The upper, midsole, and outsole each has a toe region. The toe region includes the region that extends substantially from the medial side to the lateral side at a location that begins in the vicinity of the front tip of the shoe and extends from there to a location that is approximately one third of the distance toward the rear tip of the shoe. 
         [0013]    The upper, midsole, and outsole each has a heel region. The heel region includes the region that extends substantially from the medial side to the lateral side at a location that begins in the vicinity of the rear tip of the shoe and extends from there to a location that is approximately one third of the distance toward the front tip of the shoe. 
         [0014]    The upper, midsole, and outsole each has a middle region. The middle region includes the region that extends substantially from the medial side to the lateral side at a location that extends approximately between the toe region and the heel region. 
         [0015]    The midsole further comprises an upper layer and a lower layer, the upper layer having a first density and the lower layer having a second density different from the first density, and the upper layer having a top surface and a bottom surface substantially opposite the top surface wherein the bottom surface has two or more convexities, or two or more concavities, or a single convexity and a single concavity. 
         [0016]    In a preferred embodiment, the invention includes an outsole that, when no load is applied, curves continuously upward in a direction toward the upper beginning at a location near the middle region of the outsole and ending at a location near the rearmost point of the upper. In this preferred embodiment, the midsole has two layers, an upper layer and a lower layer, and the upper layer and the lower layer each extend from at least the vicinity of the front tip of the shoe to at least the vicinity of the rear tip of the shoe. The upper layer is made from a material having a first density sufficiently dense to support and stabilize the foot. Typically, the upper layer has a density between about 0.400 and about 0.500 grams per cubic centimeter and a durometer between about 50 and about 75 on Shore A (ASTM D2240). The upper layer typically has a relatively low compressibility so that it compresses a relatively low, or small, amount under a given load. The lower layer, which may or may not be made of the same material as the upper layer, has a second density that is different from the first density and is sufficiently low in density and high in compressibility so as to allow the lower layer to compress and deform a higher, or greater, amount under a given weight than the upper layer would compress and deform under that same weight. Typically, the lower layer has a density between about 0.325 and about 0.419 grams per cubic centimeter and a durometer between about 15 and about 38 on Shore A (ASTM D2240). The density of the lower layer is sufficiently low and the compressibility of the lower layer is sufficiently high so that under normal walking conditions the user&#39;s foot, first in the heel region, then in the middle region, and then finally in the toe region, sinks toward the ground as the lower layer compresses and deforms due to the lower layer&#39;s relatively low density and/or high compressibility. 
         [0017]    Thus, during walking while wearing a preferred embodiment of the instant invention, when the curved heel region of the outsole strikes the ground, the heel region of the lower layer, which is less dense and more easily compressed than the upper layer, deforms to a relatively large degree compared to the upper layer. After each such initial heel region contact with the ground, the user&#39;s heel continues to sink or move toward the ground more than it would sink or move in a conventional shoe. This sinking or downward movement is due primarily to deflection of the heel region of the outsole and compression of the heel region of the midsole as they each respond to the increasing weight being transmitted through the user&#39;s heel as the step progresses and the user&#39;s heel continues to bear an increasing amount of the user&#39;s weight until it reaches a maximum. The impact is akin to a heel striking a sandy beach or a giving or uneven surface. Then, as the user&#39;s weight begins to shift toward the middle region of the shoe, the shoe rolls forward in a smooth motion, without the user having to overcome any abrupt or discrete pivot points. Then the lower layer of the midsole in the middle region and then in the toe region compresses and deforms under the increasing weight of the user&#39;s foot in those regions as the step progresses. This compression and deformation allows the user&#39;s foot to sink further toward the ground than would be the case with a conventional shoe. The user then completes the step by pushing off with the forefoot ball area of the user&#39;s foot. This push-off further compresses and deforms the lower layer in the toe region. 
         [0018]    The convexities and concavities in the instant invention are all identified as being on, and being a part of, the bottom surface of the upper layer. Under this convention, each convexity identified herein is, to some degree, an outward bulge of the bottom surface of the upper layer and each concavity identified herein is, to some degree, an inward depression in the bottom surface of the upper layer. Each convexity&#39;s outward bulge means that the upper layer is relatively thick wherever it has a convexity. This increased thickness of the upper layer corresponds to a decrease in thickness of the lower layer at each location where the lower layer is opposite a convexity. Similarly, each concavity&#39;s inward depression means that the upper layer is relatively thin wherever it has a concavity. This increased thinness of the upper layer corresponds to a decreased thinness, i.e., a thickening, of the lower layer at each location where the lower layer is opposite a concavity. 
         [0019]    Each convexity and concavity has at least five primary variables that control the effect of each convexity and each concavity. These primary variables are (1) the location where each convexity and concavity is located on the bottom surface of the upper layer, (2) the sharpness or shallowness of the convexity or concavity, i.e., its radius or radii of curvature, (3) the length or wavelength of each convexity or concavity as measured from a point where it begins to a point where it ends, (4) the amplitude, i.e., the greatest height of each convexity or the greatest depth of each concavity, and (5) the firmness or compressibility of the upper layer material with which each convexity or concavity is formed. These variables are some of the primary means by which the effects of the shoe on the user are controlled. These effects comprise primarily the degree of softness or hardness felt by the foot throughout each step while wearing the shoe, the amount of energy and effort needed for the user to complete each step, and the amount of muscle use, control and coordination necessary for the user to maintain the user&#39;s balance throughout each step. 
         [0020]    The degree of softness or hardness felt by the foot immediately after the heel strike is controlled primarily by a concavity located in the heel region. This concavity is typically relatively large overall, i.e., it typically has a long length, a large radius or radii of curvature, and a large amplitude. This relatively large concavity allows a relatively thick lower layer to be used in the heel region that can absorb and soften the initial heel strike of each step. Such a concavity could also be located in the middle region or the toe region of the upper layer. Whereas each concavity imparts a relatively soft feel to the user&#39;s foot while walking, each convexity imparts a relatively hard feel to the user&#39;s foot while walking. This relative hardness is due to the decreased thickness of the soft, highly compressible lower layer at each location where a convexity occurs. 
         [0021]    The amount of energy and effort required by the user in each step is related to the degree of softness or hardness felt by the user as discussed in the preceding paragraph insofar as each concavity corresponds to a softer feel which, in turn, requires more energy and effort to overcome in each step. 
         [0022]    The amount of muscle use, control and coordination necessary for the user to maintain the user&#39;s balance throughout each step increases in direct proportion to each one of the following: (1) increased concavity size, and (2) increased compressibility of the lower layer. Increased concavity size, primarily in the form of length and amplitude, corresponds to a thicker lower layer. The compressibility of the lower layer is a physical property inherent in the material out of which the lower layer is made. It is a measure of the readiness with which the lower layer compresses under a given load. A high compressibility means that the lower layer is highly compressible and can be compressed a high amount with relative ease. As the compressibility increases, the user must use more muscle control and coordination to maintain the user&#39;s balance during each step as the weight of the user compresses the lower layer. This compression is accompanied by a downward movement of the user&#39;s foot as it compresses the lower layer during each step. This downward compression movement requires balancing by the user to accommodate the inherent lateral and transverse instability that accompanies the compression. This inherent lateral and transverse instability is also affected by the thickness of the lower layer. This thickness, as mentioned above, increases as concavity size increases. As this thickness increases, the inherent lateral and transverse instability also increases. Thus, concavities contribute to a less stable walking nature of the shoe. The relative opposite effect is achieved with a convexity. Each convexity in the upper layer corresponds to a relative thinness in the lower layer. This relative thinness in the lower layer means that the user is not required to undergo as much balancing as when the lower layer is thick, primarily because the relatively unstable lower layer is relatively minimized where each convexity occurs in the corresponding upper layer. Thus, convexities contribute to a more stable walking nature of the shoe. 
         [0023]    One of the primary objectives of shoes having midsoles as disclosed herein is to provide fitness benefits to the user by requiring the user, by merely walking, to exert more energy and effort than would otherwise be required when walking while wearing conventional shoes, and to require the user to use, control, and coordinate muscles in ways that such muscles would not be used, controlled or coordinated when walking while wearing conventional shoes. Just as walking on a sandy beach requires more energy and effort than walking on a hard, flat surface, the relatively thick, highly compressible lower layer of the midsole in the area of the concavities requires that a user wearing such shoes exert more energy and effort to walk than is required while wearing conventional shoes. The extra thickness and high compressibility of the lower layer in the area of the concavities further allows the shoes to flex more, both transversely and laterally, than conventional shoes. In order for the user to maintain the user&#39;s balance and a normal walking gait under such flexure conditions, the user is required to use muscles and to control and coordinate muscles to an extent greater than is required when walking while wearing conventional shoes. The use of such muscles in such a manner further imparts a fitness benefit to the user. These and other fitness benefits of the instant shoe include, among others: muscle strengthening and toning, better posture, improved cardiovascular health, less stress on joints, and improved circulation. 
     
    
     
       I BRIEF DESCRIPTION OF THE DRAWINGS 
         [0024]    For a further understanding of the objects and advantages of the present invention, reference should be had to the following detailed description, taken in conjunction with the accompanying drawings, in which like parts are given like reference numbers and wherein: 
           [0025]      FIG. 1  is a side elevation view in cross section of an embodiment of the midsole and outsole of the shoe. 
           [0026]      FIG. 1A  an exploded view of  FIG. 1 . 
           [0027]      FIG. 2  is a front elevation view in cross section of the midsole and outsole shown in  FIG. 1  along line  2 - 2  in the direction of the appended arrows. 
           [0028]      FIG. 3  is a side elevation view in cross section of an alternative embodiment of the midsole and outsole of the shoe. 
           [0029]      FIG. 3A  an exploded view of  FIG. 3 . 
           [0030]      FIG. 4  is a front elevation view in cross section of the midsole and outsole of the shoe in  FIG. 3  along line  4 - 4  in the direction of the appended arrows. 
           [0031]      FIG. 5  is a front elevation view in cross section of the midsole and outsole of the shoe in  FIG. 3  along line  5 - 5  in the direction of the appended arrows. 
           [0032]      FIG. 6  is a front elevation view in cross section of the midsole and outsole of the shoe in  FIG. 3  along line  6 - 6  in the direction of the appended arrows. 
           [0033]      FIG. 7  is a front elevation view in cross section of the midsole and outsole of the shoe in  FIG. 3  along line  7 - 7  in the direction of the appended arrows. 
           [0034]      FIG. 8  is a side elevation view in cross section of a second alternative embodiment of the midsole and outsole of the shoe. 
           [0035]      FIG. 8A  an exploded view of  FIG. 8 . 
           [0036]      FIG. 9  is a front elevation view in cross section of the midsole and outsole of the shoe in  FIG. 8  along line  9 - 9  in the direction of the appended arrows. 
           [0037]      FIG. 10  is a front elevation view in cross section of the midsole and outsole of the shoe in  FIG. 8  along line  10 - 10  in the direction of the appended arrows. 
           [0038]      FIG. 11  is a front elevation view in cross section of the midsole and outsole of the shoe in  FIG. 8  along line  11 - 11  in the direction of the appended arrows. 
       
    
    
     DESCRIPTION OF PREFERRED EMBODIMENTS 
       [0039]    The invention will now be described with reference to  FIGS. 1 and 1A , which illustrate a side elevation view in cross section of the midsole  103 . The outsole  105  is not part of the midsole  103 . A sockliner  101  is not part of the midsole  103 . The midsole  103  is shown beneath the sockliner  101 . The outsole  105  of the shoe is beneath the midsole  103 . The dual density midsole is located between the shoe upper (not shown) and the outsole  105 . 
         [0040]    The midsole  103 , as shown in  FIG. 1A , comprises an upper layer  107  and a lower layer  109 . The upper layer  107  and/or the lower layer  109  may themselves each be comprised of two or more sub-layers. The upper layer  107  has a top surface  113  substantially opposite a bottom surface  115 . The lower layer  109  has a top surface  117  substantially opposite a bottom surface  121 . 
         [0041]    The shoe has a front tip  140  located at the farthest point toward the front of the shoe and a rear tip  142  located at the farthest point toward the rear of the shoe. The upper layer  107  includes a toe region  151  that extends substantially from the medial side of the shoe to the lateral side of the shoe at a location that begins in the vicinity of the front tip  140  and extends from there to a location that is approximately one third of the distance toward the rear tip  142 . The lower layer  109  includes a toe region  161  that extends substantially from the medial side of the shoe to the lateral side of the shoe at a location that begins in the vicinity of the front tip  140  and extends from there to a location that is approximately one third of the distance toward the rear tip  142 . The outsole  105  includes a toe region  171  that extends substantially from the medial side of the shoe to the lateral side of the shoe at a location that begins in the vicinity of the front tip  140  and extends from there to a location that is approximately one third of the distance toward the rear tip  142 . 
         [0042]    The upper layer  107  includes a heel region  153  that extends substantially from the medial side of the shoe to the lateral side of the shoe at a location that begins in the vicinity of the rear tip  142  and extends from there to a location that is approximately one third of the distance toward the front tip  142 . The lower layer  109  includes a heel region  163  that extends substantially from the medial side of the shoe to the lateral side of the shoe at a location that begins in the vicinity of the rear tip  142  and extends from there to a location that is approximately one third of the distance toward the front tip  140 . The outsole  105  includes a heel region  173  that extends substantially from the medial side of the shoe to the lateral side of the shoe at a location that begins in the vicinity of the rear tip  142  and extends from there to a location that is approximately one third of the distance toward the front tip  140 . 
         [0043]    The upper layer  107  includes a middle region  152  that extends substantially from the medial side of the shoe to the lateral side of the shoe at a location that extends approximately between the toe region  151  and the heel region  153 . The lower layer  109  includes a middle region  162  that extends substantially from the medial side of the shoe to the lateral side of the shoe at a location that extends approximately between the toe region  161  and the heel region  163 . The outsole  105  includes a middle region  172  that extends substantially from the medial side of the shoe at a location that extends approximately between the toe region  171  and the heel region  173 . 
         [0044]    Typically, the lower layer  109  is on average thicker in the heel region  163  than it is in the toe region  161 . Typically, the thickness of the lower layer  109  is less than about  45  millimeters in the heel region  163  and has an average thickness in the heel region  163  of at least about 6.5 millimeters, and is less than about  25  millimeters in the middle region  162  and the toe region  161  and has an average thickness in the middle region  162  and the toe region  161  of at least  3  millimeters. The upper layer  107  has a first density and the lower layer  109  has a second density that is different from the first density and is typically less dense than the first density. The upper layer  107  has a first compressibility and the lower layer  109  has a second compressibility that is different from the first compressibility. The compressibility of the lower layer  109  is typically relatively high. Due to this relatively high compressibility, the lower layer  109  undergoes a relatively high amount of deformation when subjected to a given load. The upper layer  107  is typically made from polyurethane, polyvinyl chloride, rubber or thermal plastic rubber. However, the upper layer  107  can be made from any other material without departing from the scope of the present invention. Typically the upper layer  107  will have a density of between about 0.400 and about 0.500 grams per cubic centimeter and a durometer between about 50 and about 75 Shore A (ASTM D2240). The lower layer  109  is made of a compressible and deformable yet resilient material which may or may not be the same material of which the upper layer  107  is made. Typically the lower layer  109  will have a density of between about 0.325 and about 0.419 grams per cubic centimeter and a durometer between about 15 and about 38 Shore A (ASTM D2240). The upper layer  107  has a top surface  113  that is typically positioned below an insole board (not shown) which is typically positioned below the sockliner  101 . The upper layer  107  also has a bottom surface  115  that is secured to and in substantially continuous contact with the top surface  117  of the lower layer  109  by either friction and/or an adhesive and/or other similar means. Alternatively, substantially the entire bottom surface  115  of the upper layer  107  may be molded to substantially the entire top surface  117  of the lower layer  109 . The outsole  105  has a top surface  119 . The bottom surface  121  of the lower layer  109  is positioned above the top surface  119  of outsole  105 . 
         [0045]    When viewed while moving from the frontmost point  150  of the upper layer  107  to the rearmost point  154  of the upper layer  107 , the bottom surface  115  of the upper layer  107 , as shown in a preferred embodiment in  FIG. 1A , has a convexity  180  that comprises at least a downward curve  190  located in at least a portion of the toe region  151 . All convexities identified by an element number in this specification are convexities that, to some degree, protrude from, and are part of, their respective bottom surface  115 ,  315 , or  815  of the respective upper layer  107 ,  307  or  807 . Downward curve, as used here and throughout this specification, unless otherwise noted, refers to a direction that moves toward the ground from any specified location on the shoe when viewed while moving from a front tip  142 ,  342 , or  842  to a respective rear tip  140 ,  340 , or  840  and while the shoe is oriented in its typical upright position where a bottom surface  123 ,  323  or  823  of the respective outsole  105 ,  305  or  805  is in unloaded contact with the ground. The downward curve  190  of convexity  180  begins at, or near the vicinity of, the frontmost point  150  of the upper layer  107  and gradually and continuously descends downwardly from there through at least a portion of the toe region  151 . The portion of the upper layer  107  indicated by lines extending from, and associated with, element number  180  indicates the approximate range wherein convexity  180  is typically primarily located. Convexity  180  may, or may not, be entirely located within the range indicated by the lines extending from, and associated with, element number  180 . Convexity  180 , as shown in a preferred embodiment in  FIG. 1A , is relatively shallow due to its large radius, or radii, of curvature. Convexity  180  may comprise a curve or curves in addition to downward curve  190 . The radius of curvature throughout convexity  180  may be completely constant, may have one or more constant portions mixed with one or more non-constant portions, or may be completely non-constant. Downward curve  190 , as well as any other curve or curves that are part of convexity  180 , may, at any point on any of those curves, have a slope somewhere between negative infinity and positive infinity and can include a slope that is zero, gradual, moderate, steep, vertical or somewhere between any of those amounts. Although the downward curve  190  of convexity  180  is shown in  FIG. 1A  as beginning near the frontmost point  150 , downward curve  190  of convexity  180  may instead begin at some other location on the upper layer  107 . Only a portion of convexity  180  may be located in the toe region  151 . Alternatively, all or substantially all of convexity  180  may be located in the toe region  151 . Convexity  180 , or a portion thereof, may occupy all of the toe region  151 . Alternatively, convexity  180 , or a portion thereof, may occupy a substantial portion of the toe region  151 . Convexity  180  has a first wavelength and a first amplitude. 
         [0046]    The bottom surface  115  of the upper layer  107 , as shown in  FIG. 1A , has a convexity  181  that comprises at least a downward curve  191  located in at least a portion of the middle region  152 . In this preferred embodiment, convexity  181  further comprises at least an upward curve  192 . Upward curve, as used here and throughout this specification, unless otherwise noted, refers to a direction that moves away from the ground from any specified location on the shoe when viewed while moving from a front tip  142 ,  342 , or  842  to a respective rear tip  140 ,  340 , or  840  and while the shoe is oriented in its typical upright position where a bottom surface  123 ,  323  or  823  of the outsole  105 ,  305  or  805  is in unloaded contact with the ground. Downward curve  191  may or may not be contiguous with upward curve  192 . Downward curve  191  descends downwardly in at least a portion of the middle region  152 . Upward curve  192  ascends upwardly in at least a portion of the middle region  152 . The portion of the upper layer  107  indicated by lines extending from, and associated with, element number  181  indicates the approximate range wherein convexity  18 i is typically primarily located. Convexity  181  may, or may not, be entirely located within the range indicated by the lines extending from, and associated with, element number  181 . Convexity  181  has a relatively pronounced bulge due to its relatively small radius, or radii, of curvature. Convexity  181  may comprise a curve or curves in addition to downward curve  191  and upward curve  192 . The radius of curvature throughout convexity  181  may be completely constant, may have one or more constant portions mixed with one or more non-constant portions, or may be completely non-constant. Downward curve  191 , upward curve  192 , as well as any other curve or curves that are part of convexity  181 , may, at any point on any of those curves, have a slope somewhere between negative infinity and positive infinity and can include a slope that is zero, gradual, moderate, steep, vertical or somewhere between any of those amounts. Although the downward curve  191  of convexity  181  is shown in  FIG. 1A  as beginning near the middle of the middle region  152  and ending at a location closer to the heel region  153  than the middle of the middle region  152 , downward curve  191  of convexity  181  may instead begin at some other location on the upper layer  107  and end at some other location on the upper layer  107 . Although the upward curve  192  of convexity  181  is shown in  FIG. 1A  as beginning near the middle of the middle region  152  and ending in the middle region at a location near the heel region  153 , upward curve  192  of convexity  181  may instead begin at some other location on the upper layer  107  and end at some other location on the upper layer  107 . Only a portion of convexity  181  may be located in the middle region  152 . Alternatively, all or substantially all of convexity  181  may be located in the middle region  152 . Convexity  181 , or a portion thereof, may occupy all of the middle region  152 . Alternatively, convexity  181 , or a portion thereof, may occupy a substantial portion of the middle region  152 . Convexity  181  has a second wavelength that is typically different from the first wavelength of convexity  180 . Convexity  181  has a second amplitude that is typically different from the first amplitude of convexity  180 . Line  2 - 2  is at or near the lowest point of convexity  181 . The primary purpose of convexity  181  is to reduce—but not eliminate—compression and deformity of the lower layer  109  in the region of the convexity  181  and to provide stability.  FIG. 2  shows how convexity  181  extends substantially from the lateral to medial side of the upper layer  107 . Convexity  180  may or may not be contiguous with convexity  181 . 
         [0047]    The bottom surface  115  of the upper layer  107 , as shown in  FIG. 1A , has a concavity  182  that comprises at least an upward curve  193  located in at least a portion of the heel region  153 . All concavities identified by an element number in this specification are concavities that, to some degree, form a depression in, and are part of, the respective bottom surface  115 ,  315 , or  815  of the respective upper layer  107 ,  307  or  807 . In this preferred embodiment, concavity  182  further comprises at least a downward curve  194 . Upward curve  193  may or may not be contiguous with downward curve  194 . Upward curve  193  ascends upwardly in at least a portion of the heel region  153 . Downward curve  194  descends downwardly in at least a portion of the heel region  153 . The portion of the upper layer  107  indicated by lines extending from, and associated with, element number  182  indicates the approximate range wherein concavity  182  is typically primarily located. Concavity  182  may, or may not, be entirely located within the range indicated by the lines extending from, and associated with, element number  182 . Concavity  182  has a relatively moderate depression due to its relatively moderate radius, or radii, of curvature. Concavity  182  may comprise a curve or curves in addition to upward curve  193  and downward curve  194 . The radius of curvature throughout concavity  182  may be completely constant, may have one or more constant portions mixed with one or more non-constant portions, or may be completely non-constant. Upward curve  193 , downward curve  194 , as well as any other curve or curves that are part of concavity  182 , may, at any point on any of those curves, have a slope somewhere between negative infinity and positive infinity and can include a slope that is zero, gradual, moderate, steep, vertical or somewhere between any of those amounts. Although the upward curve  193  of concavity  182  is shown in  FIG. 1A  as beginning at a location where the heel region  153  and the middle region  152  transition into one another, the upward curve  193  of concavity  182  could instead begin at some other location on the upper layer  107 . Although the upward curve  193  of concavity  182  is shown in  FIG. 1A  as ending at a location near the middle of the heel region  153 , upward curve  193  may instead end at some other location on the upper layer  107 . Although the downward curve  194  of concavity  182  is shown in  FIG. 1A  as beginning near the middle of the heel region  153  and ending in the vicinity of the rearmost point  154  of the upper layer  107 , downward curve  194  of concavity  182  may instead begin at some other location on the upper layer  107  and end at some other location on the upper layer  107 . Convexity  181  may or may not be contiguous with concavity  182 . Only a portion of concavity  182  may be located in the heel region  153 . Alternatively, all or substantially all of concavity  182  may be located in the heel region  153 . Concavity  182 , or a portion thereof, may occupy all of the heel region  153 . Alternatively, concavity  182 , or a portion thereof, may occupy a substantial portion of the heel region  153 . Concavity  182  has a third wavelength that is typically different from both the first wavelength of convexity  180  and the second wavelength of convexity  181 . Concavity  182  has a third amplitude that is typically different from both the first amplitude of convexity  180  and the second amplitude of convexity  181 . 
         [0048]    In preferred embodiments, the top surface  117  of the lower layer  109  of the midsole  103  is in substantially continuous contact with the bottom surface  115  of the upper layer  107  of the midsole. Due to this substantially continuous contact between top surface  117  and bottom surface  115  in these preferred embodiments, each convexity in the bottom surface  115  has a corresponding concavity in the top surface  117  and each concavity in the bottom surface  115  has a corresponding convexity in the top surface  117 . In other embodiments, such substantially continuous contact between top surface  117  and bottom surface  115  may not be present. 
         [0049]    The outsole  105  has a top surface  121  and a bottom surface  123 . The outsole  105  may curve upwardly in the heel region. When the shoe is in its typical upright, unloaded state, the frontmost point  170  is relatively high above the ground. From a point at or near the vicinity of the frontmost point  170 , the outsole  105  has a gradual downward curve  195  that continues through at least a portion of the toe region  171  of the outsole  105  until it becomes straight or nearly straight at some point in the middle region  172  of the outsole  105 . Starting in this middle region  172 , the outsole  105  has a gradual, upward curve  196  that continues to curve upward through at least a portion of the heel region  173  of the outsole  105 . This gradual upward curve  196  typically continues until the outsole  105  approaches the vicinity of the rear tip  142  of the shoe. This upward curve  196  is typically sharper than downward curve  195  in the toe region  171 . Upward curve  196  may be substantially sharper than shown in  FIG. 1A  or substantially shallower than shown in  FIG. 1A . The bottom surface  123  of the outsole  105  typically contains grooves and/or patterns for optimal traction and wear. 
         [0050]      FIG. 2  illustrates a front elevation view in cross section of  FIG. 1  along line  2 - 2  in the direction of the appended arrows.  FIG. 2  shows the construction and placement of the upper layer  107  on top of the lower layer  109  with the convexity  181  sitting in the congruent curved recess or depression  111 . The cross sectional shape of the bottom surface  115  of the upper layer  107  and the top surface  117  of the lower layer  109  at line  2 - 2  is shown in  FIG. 2  as a single line that is horizontal at one end, then dips downwardly toward the middle, is horizontal in the middle, then slopes upwardly at the other end and is horizontal at the other end. 
         [0051]    The invention will now be described with reference to a preferred embodiment shown in  FIGS. 3 and 3A . This embodiment shows a side elevation view in cross section of the midsole  303  and the outsole  305  of the shoe. 
         [0052]    The midsole  303 , as shown, comprises two layers. Typically, the lower layer  309  of the midsole  303  is on average thicker in the heel region  363  of the shoe than it is in the toe region  361 . Typically, the thickness of the lower layer  309  is less than about  45  millimeters thick in the heel region  363  of the shoe and has an average thickness in the heel region  363  of at least about 6.5 millimeters, and is less than about 25 millimeters thick in the middle region  362  and the toe region  361  of the shoe and has an average thickness in the middle region  362  and the toe region  361  of at least about 3 millimeters. The upper layer  307  has a first density and the lower layer  309  has a second density different from the first density and is typically less dense than the first density. The upper layer  307  has a first compressibility and the lower layer  309  has a second compressibility that is different from the first compressibility. The compressibility of the lower layer  309  is typically relatively high. Due to this relatively high compressibility, the lower layer  309  undergoes a relatively high amount of deformation when subjected to a given load. The upper layer  307  is typically made from polyurethane, polyvinyl chloride, rubber or thermal plastic rubber. However, the upper layer  307  can be made from any other material without departing from the scope of the present invention. Typically the upper layer  307  will have a density of between about 0.400 and about 0.500 grams per cubic centimeter and a durometer between about 50 and about 75 Shore A (ASTM D2240). The lower layer  309  is made of a compressible and deformable yet resilient material which may or may not be the same material of which the upper layer  307  is made. Typically the lower layer  309  will have a density of between about 0.325 and about 0.419 grams per cubic centimeter and a durometer between about 15 and about 38 Shore A (ASTM D2240). The top surface  313  of the upper layer  307  is typically positioned below an insole board (not shown) which is typically positioned below the sockliner  301 . The upper layer  307  has a bottom surface  315  that is located above the top surface  317  of the lower layer  309 . The lower layer  309  has a bottom surface  321 . The outsole  305  has a top surface  319 . The bottom surface  321  of the lower layer  309  is located above the top surface  319  of the outsole  305   
         [0053]    The bottom surface  315  of the upper layer  307 , as shown in a preferred embodiment in  FIG. 3A , has a convexity  380  that comprises at least a downward curve  390  located in at least a portion of the toe region  351 . The downward curve  390  of convexity  380  begins at, or near the vicinity of, the frontmost point  350  of the upper layer  307  and gradually and continuously descends downwardly from there through at least a portion of the toe region  351 . The portion of the upper layer  307  indicated by lines extending from, and associated with, element number  380  indicates the approximate range wherein convexity  380  is typically primarily located. Convexity  380  may, or may not, be entirely located within the range indicated by the lines extending from, and associated with, element number  380 . Convexity  380 , as shown in a preferred embodiment in  FIG. 3A , is relatively shallow due to its large radius, or radii, of curvature. Convexity  380  may comprise a curve or curves in addition to downward curve  390 . The radius of curvature throughout convexity  380  may be completely constant, may have one or more constant portions mixed with one or more non-constant portions, or may be completely non-constant. Downward curve  390 , as well as any other curve or curves that are part of convexity  380 , may, at any point on any of those curves, have a slope somewhere between negative infinity and positive infinity and can include a slope that is zero, gradual, moderate, steep, vertical or somewhere between any of those amounts. Although the downward curve  390  of convexity  380  is shown in  FIG. 3A  as beginning near the frontmost point  350 , downward curve  390  of convexity  380  may instead begin at some other location on the upper layer  307 . Although convexity  380  is shown in  FIG. 3A  as ending at a location in the middle region  352  or the location where the middle region  352  transitions into the heel region  353 , convexity  380  may end at some other location on the upper layer  307 . 
         [0054]    The bottom surface  315  of the upper layer  307 , as shown in  FIG. 3A , has a concavity  382  that comprises at least an upward curve  393  located in at least a portion of the heel region  353 . In this preferred embodiment, concavity  382  further comprises at least a downward curve  394 . Upward curve  393  may or may not be contiguous with downward curve  394 . Upward curve  393  ascends upwardly in at least a portion of the heel region  353 . Downward curve  394  descends downwardly in at least a portion of the heel region  353 . The portion of the upper layer  307  indicated by lines extending from, and associated with, element number  382  indicates the approximate range wherein concavity  382  is typically primarily located. Concavity  382  may, or may not, be entirely located within the range indicated by the lines extending from, and associated with, element number  382 . Concavity  382  has a relatively moderate depression due to its relatively moderate radius, or radii, of curvature. Concavity  382  may comprise a curve or curves in addition to upward curve  393  and downward curve  394 . The radius of curvature throughout concavity  382  may be completely constant, may have one or more constant portions mixed with one or more non-constant portions, or may be completely non-constant. Upward curve  393 , downward curve  394 , as well as any other curve or curves that are part of concavity  382 , may, at any point on any of those curves, have a slope somewhere between negative infinity and positive infinity and can include a slope that is zero, gradual, moderate, steep, vertical or somewhere between any of those amounts. Although the upward curve  393  of concavity  382  is shown in  FIG. 3A  as beginning at a location where the middle region  352  and the heel region  353  transition into one another, the upward curve  393  of concavity  382  could instead begin at some other location on the upper layer  307 . Although the upward curve  393  of concavity  382  is shown in  FIG. 3A  as ending at a location near the transition between the middle region  352  and the heel region  353 , upward curve  393  may instead end at some other location on the upper layer  307 . Although the downward curve  394  of concavity  382  is shown in  FIG. 3A  as beginning at a location near the transition between the middle region  352  and the heel region  353  and ending in the vicinity of the rearmost point  354  of the upper layer  307 , downward curve  394  of concavity  382  may instead begin at some other location on the upper layer  307  and end at some other location on the upper layer  307 . Convexity  380  may or may not be contiguous with concavity  382 . 
         [0055]    The outsole  305  has a top surface  319  and a bottom surface  323 . The outsole  305  may curve upwardly in the heel region. When the shoe is in its typical upright, unloaded state, the frontmost point  370  is relatively high above the ground. From a point at or near the vicinity of the frontmost point  370 , the outsole  305  has a gradual downward curve  395  that continues through at least a portion of the toe region  371  of the outsole  305  until it reaches a virtually flat surface in the middle region  372  of the outsole  305 . Starting in this middle region  372 , the outsole  305  has a gradual, upward curve  396  that continues to curve upward through at least a portion of the heel region  373  of the outsole  305 . This gradual upward curve  396  typically continues until the outsole  305  approaches the vicinity of the rear tip  342  of the shoe. This upward curve  396  is typically sharper than the curve in the toe region  371 . Upward curve  396  may be substantially sharper than shown in  FIG. 3A  or substantially shallower than shown in  FIG. 3A . The bottom surface  323  of the outsole  305  typically contains grooves and/or patterns for optimal traction and wear. 
         [0056]      FIG. 4  shows a front elevation view in cross section of the midsole  303  shown in  FIG. 3  along line  4 - 4  in the direction of the appended arrows. As shown in  FIG. 4 , the bottom surface  315  of the upper layer  307  is in substantially continuous contact with the top surface  317  of the lower layer  309 . The cross sectional shape of the bottom surface  315  and the top surface  317  at line  4 - 4  is shown in  FIG. 4  by a substantially horizontal line that extends from the lateral side of the midsole  303  to the medial side. 
         [0057]      FIG. 5  shows a front elevation view in cross section of the midsole  303  shown in  FIG. 3  along line  5 - 5  in the direction of the appended arrows. As shown in  FIG. 5 , the bottom surface  315  of the upper layer  307  is in substantially continuous contact with the top surface  317  of the lower layer  309 . The cross sectional shape of the bottom surface  315  and the top surface  317  at line  5 - 5  is shown in  FIG. 5  by a substantially horizontal line that extends from the lateral side of the midsole  303  to the medial side. 
         [0058]      FIG. 6  shows a front elevation view in cross section of the midsole  303  shown in  FIG. 3  along line  6 - 6  in the direction of the appended arrows. As shown in  FIG. 6 , the bottom surface  315  of the upper layer  307  is in substantially continuous contact with the top surface  317  of the lower layer  309 . The cross sectional shape of the bottom surface  315  and the top surface  317  at line  6 - 6  is shown in  FIG. 6  by a substantially horizontal line that extends from the lateral side of the midsole  303  to the medial side. 
         [0059]      FIG. 7  shows a front elevation view in cross section of the midsole  303  shown in  FIG. 3  along line  7 - 7  in the direction of the appended arrows. As shown in  FIG. 7 , the bottom surface  315  of the upper layer  307  is in substantially continuous contact with the top surface  317  of the lower layer  309 . The cross sectional shape of the bottom surface  315  and the top surface  317  at line  7 - 7  is shown in  FIG. 7  by a substantially horizontal line that extends from the lateral side of the midsole  303  to the medial side. 
         [0060]    As shown in  FIGS. 4-7 , the cross sectional of the midsole  303  is of varying thickness, with there generally being a progression in thickness as the midsole  303  moves from the toe region to the heel region. 
         [0061]    In preferred embodiments, the top surface  317  of the lower layer  309  of the midsole  303  is in substantially continuous contact with the bottom surface  315  of the upper layer  307  of the midsole. Due to this substantially continuous contact between top surface  317  and bottom surface  315  in these preferred embodiments, each convexity in the bottom surface  315  has a corresponding concavity in the top surface  317  and each concavity in the bottom surface  315  has a corresponding convexity in the top surface  317 . In other embodiments, such substantially continuous contact between top surface  317  and bottom surface  315  may not be present. 
         [0062]    The invention will now be described with reference to an alternative embodiment shown in  FIGS. 8 and 8A . This embodiment shows a side elevation view in cross section of the midsole  803  and the outsole  805  of the shoe. 
         [0063]    The midsole  803 , as shown, comprises two layers. Typically, the lower layer  809  of the midsole is on average thicker in the heel region  863  of the shoe than it is in the toe region  861 . Typically, the thickness of the lower layer  809  is less than about  45  millimeters thick in the heel region  863  of the shoe and has an average thickness in the heel region  863  of at least about 6.5 millimeters, and is less than about 25 millimeters thick in the middle region  862  and the toe region  861  of the shoe and has an average thickness in the middle region  862  and the toe region  861  of at least about 3 millimeters. The upper layer  807  has a first density and the lower layer  809  has a second density different from the first density and is typically less dense than the first density. The upper layer  807  has a first compressibility and the lower layer  809  has a second compressibility that is different from the first compressibility. The compressibility of the lower layer  809  is typically relatively high. Due to this relatively high compressibility, the lower layer  809  undergoes a relatively high amount of deformation when subjected to a given load. The upper layer  807  is typically made from polyurethane, polyvinyl chloride, rubber or thermal plastic rubber. However, the upper layer  807  can be made from any other material without departing from the scope of the present invention. Typically the upper layer  807  will have a density of between about 0.400 and about 0.500 grams per cubic centimeter and a durometer between about 50 and about 75 Shore A (ASTM D2240). The lower layer  809  is made of a compressible and deformable yet resilient material which may or may not be the same material of which the upper layer  807  is made. Typically the lower layer  809  will have a density of between about 0.325 and about 0.419 grams per cubic centimeter and a durometer between about 15 and about 38 Shore A (ASTM D2240). The top surface  813  of the upper layer  807  is typically positioned below an insole board (not shown) which is typically positioned below the sockliner  801  The upper layer  807  has a bottom surface  815  that is located above the top surface  817  of the lower layer  809 . The lower layer  809  has a bottom surface  821 . The outsole  805  has a top surface  819 . The bottom surface  821  of the lower layer  809  is located above the top surface  819  of the outsole  805 . 
         [0064]    The bottom surface  815  of the upper layer  807 , as shown in a preferred embodiment in  FIG. 8A , has a convexity  880  that comprises at least a downward curve  890  located in at least a portion of the toe region  851 . The downward curve  890  of convexity  880  begins at, or near the vicinity of, the frontmost point  850  of the upper layer  807  and gradually and continuously descends downwardly from there through at least a portion of the toe region  851 . The portion of the upper layer  807  indicated by lines extending from, and associated with, element number  880  indicates the approximate range wherein convexity  880  is typically primarily located. Convexity  880  may, or may not, be entirely located within the range indicated by the lines extending from, and associated with, element number  880 . Convexity  880 , as shown in a preferred embodiment in  FIG. 8A , is relatively shallow due to its large radius, or radii, of curvature. Convexity  880  may comprise a curve or curves in addition to downward curve  890 . The radius of curvature throughout convexity  880  may be completely constant, may have one or more constant portions mixed with one or more non-constant portions, or may be completely non-constant. Downward curve  890 , as well as any other curve or curves that are part of convexity  880 , may, at any point on any of those curves, have a slope somewhere between negative infinity and positive infinity and can include a slope that is zero, gradual, moderate, steep, vertical or somewhere between any of those amounts. Although the downward curve  890  of convexity  880  is shown in  FIG. 8A  as beginning near the frontmost point  850 , downward curve  890  of convexity  880  may instead begin at some other location on the upper layer  807 . Although convexity  880  is shown in  FIG. 8A  as ending at a location in the toe region  851 , convexity  880  may instead end at some other location on the upper layer  807 . Only a portion of convexity  880  may be located in the toe region  851 . Alternatively, all or substantially all of convexity  880  may be located in the toe region  851 . Convexity  880 , or a portion thereof, may occupy all of the toe region  851 . Alternatively, convexity  880 , or a portion thereof, may occupy a substantial portion of the toe region  851 . Convexity  880  has a first wavelength and a first amplitude. 
         [0065]    The bottom surface  815  of the upper layer  807 , as shown in  FIG. 8A , has a concavity  881  that comprises at least an upward curve  891  located in at least a portion of the toe region  851 . In this preferred embodiment, concavity  881  further comprises at least a downward curve  892 . Upward curve  891  may or may not be contiguous with downward curve  892 . Upward curve  891  ascends upwardly in at least a portion of the toe region  851 . Downward curve  892  descends downwardly in at least a portion of the toe region  851 . The portion of the upper layer  807  indicated by lines extending from, and associated with, element number  881  indicates the approximate range wherein concavity  881  is typically primarily located. Concavity  881  may, or may not, be entirely located within the range indicated by the lines extending from, and associated with, element number  881 . Concavity  881  has a relatively shallow depression due to its relatively long radius, or radii, of curvature. Concavity  881  may comprise a curve or curves in addition to upward curve  891  and downward curve  892 . The radius of curvature throughout concavity  881  may be completely constant, may have one or more constant portions mixed with one or more non-constant portions, or may be completely non-constant. Upward curve  891 , downward curve  892 , as well as any other curve or curves that are part of concavity  881 , may, at any point on any of those curves, have a slope somewhere between negative infinity and positive infinity and can include a slope that is zero, gradual, moderate, steep, vertical or somewhere between any of those amounts. Although the upward curve  891  of concavity  881  is shown in  FIG. 8A  as beginning at a location near where the toe region  851  and the middle region  852  transition into one another, the upward curve  891  of concavity  881  could instead begin at some other location on the upper layer  807 . Although the upward curve  891  of concavity  881  is shown in  FIG. 8A  as ending at a location near the transition between the toe region  851  and the middle region  852 , upward curve  891  may instead end at some other location on the upper layer  807 . Although the downward curve  892  of concavity  881  is shown in  FIG. 8A  as beginning near the transition between the toe region  851  and the middle region  852  and ending in the vicinity of the middle region  852 , downward curve  892  of concavity  881  may instead begin at some other location on the upper layer  807  and end at some other location on the upper layer  807 . Convexity  880  may or may not be contiguous with concavity  881 . Only a portion of concavity  881  may be located in the in toe region  851 . Alternatively, all or substantially all of concavity  881  may be located in the toe region  851 . Concavity  881  has a second wavelength that is typically different from the first wavelength of convexity  880 . Concavity  881  has a second amplitude that is typically different from the first amplitude of convexity  880 . 
         [0066]    The bottom surface  815  of the upper layer  807 , as shown in  FIG. 8A , has a convexity  882  that comprises at least a downward curve  893  located in at least a portion of the middle region  852 . In this preferred embodiment, convexity  882  further comprises at least an upward curve  894 . Downward curve  893  may or may not be contiguous with upward curve  894 . Downward curve  893  descends downwardly in at least a portion of the middle region  852 . Upward curve  894  ascends upwardly in at least a portion of the middle region  852 . The portion of the upper layer  807  indicated by lines extending from, and associated with, element number  882  indicates the approximate range wherein convexity  882  is typically primarily located. Convexity  882  may, or may not, be entirely located within the range indicated by the lines extending from, and associated with, element number  882 . Convexity  882  has a relatively moderate bulge due to its relatively moderate radius, or radii, of curvature. Convexity  882  may comprise a curve or curves in addition to downward curve  893  and upward curve  894 . The radius of curvature throughout convexity  882  may be completely constant, may have one or more constant portions mixed with one or more non-constant portions, or may be completely non-constant. Downward curve  893 , upward curve  894 , as well as any other curve or curves that are part of convexity  882 , may, at any point on any of those curves, have a slope somewhere between negative infinity and positive infinity and can include a slope that is zero, gradual, moderate, steep, vertical or somewhere between any of those amounts. Although the downward curve  893  of convexity  882  is shown in  FIG. 8A  as beginning near the middle of the middle region  852  and ending near the middle of the middle region  852 , downward curve  893  of convexity  882  may instead begin at some other location on the upper layer  807  and end at some other location on the upper layer  807 . Although the upward curve  894  of convexity  882  is shown in  FIG. 8A  as beginning near the middle of the middle region  852  and ending in the middle region at a location near the heel region  853 , upward curve  894  of convexity  882  may instead begin at some other location on the upper layer  807  and end at some other location on the upper layer  807 . Convexity  882  may or may not be contiguous with concavity  881 . Only a portion of convexity  882  may be located in the in middle region  852 . Alternatively, all or substantially all of convexity  882  may be located in the middle region  852 . Convexity  882 , or a portion thereof, may occupy all of the middle region  852 . Alternatively, convexity  882 , or a portion thereof, may occupy a substantial portion of the middle region  852 . Convexity  882  has a third wavelength that is typically different from both the first wavelength of convexity  880  and the second wavelength of concavity  881 . Convexity  882  has a third amplitude that is typically different from both the first amplitude of convexity  880  and the second amplitude of concavity  881 . 
         [0067]    The bottom surface  815  of the upper layer  807 , as shown in  FIG. 8A , has a concavity  883  that comprises at least an upward curve  895  located in at least a portion of the heel region  853 . In this preferred embodiment, concavity  883  further comprises at least a downward curve  896 . Upward curve  895  may or may not be contiguous with downward curve  896 . Upward curve  895  ascends upwardly in at least a portion of the heel region  853 . Downward curve  896  descends downwardly in at least a portion of the heel region  853 . The portion of the upper layer  807  indicated by lines extending from, and associated with, element number  883  indicates the approximate range wherein concavity  883  is typically primarily located. Concavity  883  may, or may not, be entirely located within the range indicated by the lines extending from, and associated with, element number  883 . Concavity  883  has a relatively moderate depression due to its relatively moderate radius, or radii, of curvature. Concavity  883  may comprise a curve or curves in addition to upward curve  895  and downward curve  896 . The radius of curvature throughout concavity  883  may be completely constant, may have one or more constant portions mixed with one or more non-constant portions, or may be completely non-constant. Upward curve  895 , downward curve  896 , as well as any other curve or curves that are part of concavity  883 , may, at any point on any of those curves, have a slope somewhere between negative infinity and positive infinity and can include a slope that is zero, gradual, moderate, steep, vertical or somewhere between any of those amounts. Although the upward curve  895  of concavity  883  is shown in  FIG. 8A  as beginning at a location in the middle region  852 , the upward curve  895  of concavity  883  could instead begin at some other location on the upper layer  807 . Although the upward curve  895  of concavity  883  is shown in  FIG. 8A  as ending at a location near the middle of the heel region  853  of the upper layer  807 , upward curve  895  may instead end at some other location on the upper layer  807 . Although the downward curve  896  of concavity  883  is shown in  FIG. 8A  as beginning near the middle of the heel region  853  and ending in the vicinity of the rearmost point  854  of the upper layer  807 , downward curve  896  of concavity  883  may instead begin at some other location on the upper layer  807  and end at some other location on the upper layer  807 . Convexity  882  may or may not be contiguous with concavity  883 . Only a portion of concavity  883  may be located in the in heel region  853 . Alternatively, all or substantially all of concavity  883  may be located in the heel region  853 . Concavity  883 , or a portion thereof, may occupy all of the heel region  853 . Alternatively, concavity  883 , or a portion thereof, may occupy a substantial portion of the heel region  853 . Concavity  883  has a fourth wavelength that is typically different from the first wavelength of convexity  880 , the second wavelength of concavity  881 , and the third wavelength of convexity  882 . Concavity  883  has a fourth amplitude that is typically different from the first amplitude of convexity  880 , the second amplitude of concavity  881 , and the third amplitude of convexity  882 . 
         [0068]    As further shown in the embodiment in  FIG. 8 , the top surface  817  of the lower layer  809  of the midsole  803  is in substantially continuous contact with the bottom surface  815  of the upper layer  807  of the midsole. Due to this substantially continuous contact between top surface  817  and bottom surface  815  in this embodiment, each convexity in the bottom surface  815  has a corresponding concavity in the top surface  817  and each concavity in the bottom surface  815  has a corresponding convexity in the top surface  817 . In other embodiments, such substantially continuous contact between top surface  817  and bottom surface  815  may not be present. 
         [0069]    The outsole  805  has a top surface  819  and a bottom surface  823 . The outsole  805  may curve upwardly in the heel region  873 . When the shoe is in its typical upright, unloaded state, the frontmost point  870  is relatively high above the ground. In this embodiment, from a point at or near the vicinity of the frontmost point  870 , the outsole  805  has a gradual downward curve  897  that continues through at least a portion of the toe region  861  of the outsole  805 , then continues to curve gradually downward in the middle region  872  of the outsole and then begins to curve upwardly forming an upward curve  898  in the heel region  873  of the outsole  805 . This gradual upward curve  898  typically continues until the outsole  805  approaches the vicinity of the rear tip  842  of the shoe. This upward curve  898  is typically sharper than the curve in the toe region  871 . Upward curve  898  may be substantially sharper than shown in  FIG. 8A  or substantially shallower than shown in  FIG. 8A . The bottom surface  823  of the outsole  805  typically contains grooves and/or patterns for optimal traction and wear. 
         [0070]      FIG. 9  shows a front elevation view in cross section of the midsole  803  shown in  FIG. 8  along line  9 - 9  in the direction of the appended arrows. As shown in  FIG. 9 , the bottom surface  815  of the upper layer  807  is in substantially continuous contact with the top surface  817  of the lower layer  809 . The cross sectional shape of the bottom surface  815  and the top surface  817  at line  9 - 9  is shown in  FIG. 9  by a substantially horizontal line that extends from the lateral side of the midsole  803  to the medial side. 
         [0071]      FIG. 10  shows a front elevation view in cross section of the midsole  803  shown in  FIG. 8  along line  10 - 10  in the direction of the appended arrows. As shown in  FIG. 10 , the bottom surface  815  of the upper layer  807  is in substantially continuous contact with the top surface  817  of the lower layer  809 . The cross sectional shape of the bottom surface  815  and the top surface  817  at line  10 - 10  is shown in  FIG. 10  by a substantially horizontal line that extends from the lateral side of the midsole  803  to the medial side. 
         [0072]      FIG. 11  shows a front elevation view in cross section of the midsole  803  shown in  FIG. 8  along line  11 - 11  in the direction of the appended arrows. As shown in  FIG. 11 , the bottom surface  815  of the upper layer  807  is in substantially continuous contact with the top surface  817  of the lower layer  809 . The cross sectional shape of the bottom surface  815  and the top surface  817  at line  11 - 11  is shown in  FIG. 11  by a substantially horizontal line that extends from the lateral side of the midsole  803  to the medial side. 
         [0073]    As shown in  FIGS. 9-11 , the midsole  803  is of varying thickness, with there generally being a progression in thickness as the midsole  803  moves from the toe region  851  to the heel region  853 . 
         [0074]    In normal use of the shoe, the user steps forward with the rear portion of the user&#39;s heel stepping on the ground first. When this happens, the lower layer  809  of the midsole  803  in the heel region  853  that is made of less dense and more readily compressible material, compresses and deforms, causing the heel of the user&#39;s foot to sink toward the ground to a greater extent than it would sink while wearing a conventional shoe. Due to the concavity  883 , the lower layer  809  is relatively thick in the heel region  863 . Since this relatively thick heel region  863  of the lower layer  809  is also relatively soft and highly compressible, it mimics the effect of walking on a sandy beach, thereby requiring the user to exert more energy while walking than would be required when walking while wearing conventional shoes. Additionally, since the heel region  863  of the lower layer  809  is relatively thick and highly compressible, it has a degree of inherent lateral and transverse instability that is not present in conventional shoes. This inherent instability forces the user to make a balancing effort and use muscles and muscle control and coordination to maintain a normal walking gait that would not be required with conventional shoes. 
         [0075]    As the step continues, the user&#39;s weight shifts to the center of the shoe and the shoe rolls forward in a smooth motion without the user having to overcome any abrupt pivot points. The lower layer  809  of the midsole  803  in the middle region  862  and then in the toe region  861 , compresses and deforms, allowing the user&#39;s foot in those regions to sink toward the ground more than it would sink if the user were wearing conventional shoes. The convexities  880 ,  882  in the toe region  861  and/or middle region  862 , limit compression of the lower layer  809  in those areas and thereby provide stability. The user then completes the step by pushing off with the forefoot ball region of the user&#39;s foot. All of this simulates the effects and the fitness benefits of walking on a sandy beach or on a giving or uneven soft surface regardless of the actual hardness of the surface. 
         [0076]    While the foregoing detailed description sets forth exemplary embodiments of a shoe in accordance with the present invention, it is to be understood that the above description is illustrative only and not limiting of the disclosed invention. Indeed, it will be appreciated that the embodiments discussed above and the virtually infinite embodiments that are not mentioned could easily be within the scope and spirit of the present invention.