WO2019140090A1

WO2019140090A1 - Polyhedra golf ball with lower drag coefficient

Info

Publication number: WO2019140090A1
Application number: PCT/US2019/013052
Authority: WO
Inventors: Nikolaos Beratlis; Elias Balaras; Kyle Squires
Original assignee: Nikolaos Beratlis; Elias Balaras; Kyle Squires
Priority date: 2018-01-12
Filing date: 2019-01-10
Publication date: 2019-07-18
Also published as: US20210197029A1; JP2021510578A

Abstract

A golf ball having an outer surface with a pattern forming a polyhedron. The pattern can be flat faces forming sharp edges and sharp points therebetween. In one embodiment, the polyhedron is a Goldberg polyhedron.

Description

POLYHEDRA GOLF BALL WITH LOWER DRAG COEFFICIENT

BACKGROUND

Related Applications

[0001] This application claims the benefit of U.S. Provisional Application No. 62/616,861, filed Jan. 12, 2018, the entire contents of which are incorporated herein by reference.

Technical Field

[0002] The present invention relates to golf balls having a polyhedra design that can yield lower drag coefficient than a dimpled sphere. The drag reduction is applicable to other sports equipment in general having a bluff body, such as the head of a golf club or a bike helmet.

Background of the Related Art

[0003] For the past 100 years the vast majority of commercial golf balls have used designs with dimples. For the purpose of this invention, a dimple generally refers to any curved or spherical depression in the face or outer surface of the ball. The traditional golf ball, as readily accepted by the consuming public, is spherical with a plurality of dimples, where the dimples are generally depressions on the outer surface of a sphere. The vast majority of commercial golf balls use dimples that have a substantially spherical shape. Some examples of such golf balls can be found in U.S. Patent No. 6290615, U.S. Patent No. 6923736, and U.S. Publ. No. 20110268833.

[0004] It is well established that such depressions can lower the drag coefficient of a ball compared to that of a smooth sphere at the same speed. The drag coefficient is a

dimensionless parameter that is used to quantify the drag force of resistance of an object in a fluid. The drag force is always opposite to the direction in which the object travels. The drag coefficient for a golf ball is defined as CD = 2*Fd/(p*U²*A), where Fd is the drag force, p is the density of the fluid in which the object is moving, U is the speed of the object and A is the cross sectional area. For a sphere the cross-sectional area is pϋ²/4, where D is the diameter of the ball.

[0005] FIG. 1 shows the variation of the drag coefficient, CD, as a function of Reynolds number, Re, for smooth and dimpled spheres. The data were obtained by performing wind tunnel experiments of non-spinning spheres. The Reynolds number is a dimensionless parameter used in fluid mechanics and is defined as Re=U*D/v, where v is the kinematic viscosity in which the object moves. For a smooth sphere the drag coefficient (shown by the solid black line) remains constant (CD~0.5) until the Reynolds number approaches a critical value (Re_Cr ~ 300,000). At this point, which is usually referred to as drag crisis, CD decreases rapidly and hits a minimum, which is an order of magnitude lower CD ~ 0.08. With further increase in the Reynolds number the flow enters the post-critical regime characterized by turbulent boundary layers on the surface of the sphere. In this regime the drag coefficient rises slowly with increasing Reynolds number.

[0006] In dimpled spheres (shown by dashed and dotted black lines) the drag crisis happens at a much lower critical Reynolds number (Re< 100,000). In general, a dimpled sphere will have a 50% or more reduction in drag compared to a smooth sphere in the range of Reynolds number from 100,000 to 250,000. A golf ball in flight during a driver shot can experience a Reynolds number ranging from 220,000 at the beginning of the flight to 60,000 at the end of the flight. It is therefore very important that a golf ball design can achieve a very low drag coefficient in that range. The critical value of the Reynolds number, as well as the attained minimum drag coefficient in the post-critical regime depend on the dimple geometry and arrangement. In general, as the aggregate dimple volume, measured as the sum of a dimple volume of each dimple of the plurality of dimples, decreases the critical Reynolds is increased and the drag coefficient in the post-critical regime increases. As the aggregate dimple volume approaches zero the drag curve approaches that of a smooth sphere. Drag is one of the two forces (the other being lift) that influence the aerodynamic performance of a golf ball. To increase the carry distance on a driver shot, the distance a golf ball travels during the flight, the dimple design is a balance between achieving the lowest critical Reynolds number and the lowest drag coefficient in the post-critical regime.

[0007] However, dimples are the only configuration known to provide a golf ball having a reduced drag coefficient.

SUMMARY

[0008] Accordingly, it is an object of the invention to provide a golf ball having minimal drag coefficient. It is a further object of the invention to provide a golf ball having reduced drag coefficient compared to a golf ball having only dimples. It is another object to provide a golf ball with minimal drag coefficient that is non-dimpled. In one embodiment, the golf ball has a plurality of flat faces with sharp edges and points that collectively form a polyhedron. These and other objects of the invention, as well as many of the intended advantages thereof, will become more readily apparent when reference is made to the following description, taken in conjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE FIGURES

[0009] FIG. 1 shows a plot of the drag coefficient CD VS Reynolds number Re for smooth and dimpled spheres. The solid black line represents a smooth sphere (Achenbach, 1972); the double-dashed lines represent a dimpled sphere (J. Choi, 2006); and the dash-dot lines represent a dimpled sphere (Harvey, 1976). The shaded area represents the typical range of Reynolds experience by a golf ball in flight during a driver shot (50,000 - 200,000).

[0010] FIG. 2(a) shows a golf ball in accordance with one embodiment of the invention. [0011] FIG. 2(b) shows an outline of a golf ball.

[0012] FIG. 2(c) shows an outline of a golf ball with non-sharp rounded edges.

[0013] FIG. 2(d) shows an icosahedron, a well-known Platonic solid used to derive the golf ball 100.

[0014] FIG. 2(e) shows an example of splitting a hexagonal face into 6 triangular faces.

[0015] FIG. 3 shows the Goldberg polyhedron with 192 faces.

[0016] FIG. 4 is a graph of the drag coefficient versus Reynolds for the Goldberg polyhedra with 162 faces and 192 faces.

[0017] FIG. 5 is a graph showing the CD versus Reynolds for a Goldberg polyhedron with 192 faces and a dimpled sphere.

[0018] FIG. 6 is a graph showing the CD versus Reynolds for a Goldberg polyhedron with 162 faces and a dimpled sphere.

[0019] FIG. 7 shows a geodesic polyhedron made from 320 triangles.

[0020] FIG. 8 shows a geodesic cube with 174 faces.

[0021] FIG. 9 shows a polyhedron with 162 faces and 162 dimples.

[0022] FIG. 10 is a graph showing the drag coefficient CD versus Reynolds number Re for a Goldberg polyhedron with 162 faces and a Goldberg polyhedron with 162 faces and 162 dimples.

[0023] FIG. 11 is a comparison of drag coefficient CD versus Reynolds number for one of the embodiments of the present invention shown in FIG. 9 against a commercial ball Callaway Superhot.

[0024] FIG. 12 is an alternative embodiment of a golf ball based on an icosahedron with 312 faces and 312 spherical dimples. [0025] FIG. 13 is a comparison of drag coefficient CD versus Reynolds number Re for one of the embodiments of the present invention based on a polyhedron with 312 faces and 312 spherical dimples of FIG. 12 against a commercial golf ball Bridgestone Tour.

DETAILED DESCRIPTION

[0026] In describing certain illustrative, non-limiting embodiments of the invention illustrated in the drawings, specific terminology will be resorted to for the sake of clarity. However, the invention is not intended to be limited to the specific terms so selected, and it is to be understood that each specific term includes all technical equivalents that operate in similar manner to accomplish a similar purpose. Several embodiments of the invention are described for illustrative purposes, it being understood that the invention may be embodied in other forms not specifically shown in the drawings.

[0027] The present invention is directed to a golf ball design based on polyhedra that can have reduced drag coefficient compared to a dimpled sphere. In one embodiment, a family of golf ball designs are made up of convex polyhedra whose vertices he on a sphere. A polyhedron is a solid in three dimensions with flat polygonal faces, straight sharp edges and sharp comers or vertices.

[0028] FIG. 2(a) shows a golf ball 100 in accordance with one embodiment of the present invention. The golf ball 100 has a body 110 with an inner core and an outer shell with an outer surface 112. A plurality of faces 120 are formed in the outer surface, creating a pattern 116. All the faces are formed in the outer surface and each of such faces is flat and lies in a plane. The faces are bound by straight or linear edges 124. Here, the golf ball 100 is a polyhedron with 162 polygons.

[0029] The body 110 defines a circumscribed sphere 102, which is the smallest sphere containing the polyhedron, and is shown with a dashed line about the outermost points 104 of the body 110. The sphere and the diameter provide a reference for the size of the golf ball. The Rules of Golf, jointly governed by the R&A and the USGA, state that the diameter of a "conforming" golf ball cannot be any smaller than 1.680 inches. For the purpose of a golf ball the diameter of the circumscribed sphere is at least l.68in. The vertices l22a, l22b of the polyhedron are the only points 104 on the polyhedron that lie on the sphere. Any point along the edges l24a, l24b or on the faces l20a, l20b of the polygons lies below the surface of the circumscribed sphere.

[0030] The golf ball body 110 is a polyhedron that is made from first faces l20a and second faces l20b. As shown, the first faces l20a have a first shape, namely pentagons, and the second faces l20b have a second shape different than the first shape, namely hexagons. All of the faces are flat and each such face forms a single plane. In one embodiment, there are 12 pentagons l20a and 150 hexagons l20b (a hexagon-to-pentagon ratio of 12.5: 1), each having comers or points l22a, l22b connected by boundaries such as straight lines or edges l24a, l24b. In various other embodiments, other quantities and/or ratios of such pentagons l20a and hexagons l20b can be used. However, the number of polygons and the angle between them determine when the drag coefficient will start to drop and how low it will become. In general, as the number of faces is increased the drag crisis occurs at higher Reynolds number and the drag coefficient decreases. The first and second faces l20a, l20b form the pattern 116.

[0031] The edges 124 are sharp, in that the faces are at an angle with respect to one another.

FIG. 2(b) shows a cross sectional cut through the body 110 of ball 100 of FIG. 2(a) along the line 150. In this embodiment the edges 124 are sharp, in that the radius of curvature 140 of an edge is less than 0.001D, where D is the diameter of the circumscribed sphere. Ideally all edges should be the sharpest possible. A radius of curvature of 0 is the case with the sharpest possible edges, though golf ball edges might become a little rounded during manufacture or follow use. FIG. 2(c) shows a cross section cut of two faces with rounded edges, whose radius of curvature is more than 0.001D. The resulting edge is not sharp and the reduction in drag is not maximized, which can be detrimental to the aerodynamic performance of the golf ball as the shape would approach that of a smooth sphere. Both a sharp edge and a non-sharp edge is shown in that embodiment for illustrative purposes. The angle Q formed between two adjacent flat / planar faces 120 is always smaller than 180 degrees. The geometric shape of the embodiment illustrated in FIG. 2(a) falls into the class of convex polyhedral where the angle between any pair of adjacent faces is always less than 180 degrees. In this

embodiment, the angle between a pentagon face l20a and an adjacent hexagonal face l20b is 166.215 degrees. The angle between two adjacent hexagon faces l20b varies from 161.5 degrees to 162.0 degrees.

[0032] Each face 120 is immediately adjacent and touching a neighboring face 120, such that each edge 124 forms a border between two neighboring faces 120 and each point 122 is at the intersection of three neighboring faces 120. And, each point 122 is at an opposite end of each linear edge 124 and is at the intersection of three linear edges 124. Accordingly, there is no gap or space between adjacent neighboring faces 120, and the faces 120 are contiguous and form a single integral, continuous outer surface 112 of the ball 100.

[0033] A golf ball usually has a rubber core and at least one more layer surrounding the core. The pattern 116 is formed on the outermost layer. The pattern is based on an icosahedron shown in FIG. 2(d). The icosahedron 170 is a well-known convex polyhedron made up of 20 equilateral triangle faces 180, 12 vertices 182 and 30 edges 184. An icosahedron is one of the five regular Platonic solids, the other four being the cube, the tetrahedron, the octahedron and the dodecahedron (see https://en.wikipedia.org/wiki/Platonic_solid).

[0034] In the icosahedron 170 five equilateral triangles 180 meet at each of its twelve vertices 182. The 12 pentagons l20a of the golf ball 100 shown in FIG. 2(a) are centered on the vertices of an icosahedron. Therefore, a pair of 3 pentagons l20a forms an equilateral triangular pattern 180. Along each of the edges of the triangles 180 there are 3 hexagons l20b. Finally, inside each triangular pattern 180 there are three hexagons l20b. The pentagons l20a are all equilateral, that is the 5 edges l24a all have the same length equal to 0.151D, where D is the diameter of the circumscribed sphere. The hexagons l20b are not equilateral and the lengths of the edges l24b vary from 0.151D to 0.1834D.

[0035] Table 1 lists the coordinates of all the vertices and how each face is formed, which defines the geometry of the golf ball. Table 1 lists the coordinates x, y, and z of all of the vertices 122 of golf ball 100. Faces are constructed by connecting the vertices with straight lines. The coordinates x, y, z are normalized by the diameter D of the circumscribed sphere 104 and are with respect to the center of the sphere. The golf ball 100 contains 320 vertices 122, 480 straight edges 120 and 162 polygonal faces 120.

[0036] The particular polyhedron can be any suitable configuration, and in one embodiment is a class of solids called Goldberg polyhedra that comprises convex polyhedra that are made entirely from a combination of rectangles, pentagons and hexagons. Goldberg polyhedra are derived from either an icosahedron (a convex polyhedron made from 12 pentagons), or an octahedron (a convex polyhedron made from 8 triangles) or a tetrahedron (a convex polyhedron made from 4 triangles). An infinite number of Goldberg polyhedra exist, as shown in

https://en.wikipedia.org/wiki/List_of_geodesic_polyhedra_and_Goldberg_polyhedra.

However, it will be recognized that the invention can utilize any convex polyhedron (that is, a polyhedron made up of polygons whose angle is less than 180 degrees), though in one embodiment on such polyhedron is a convex polyhedron with sharp edges, flat faces forming a single plane, and adjacent faces having an angle of less than 180 degrees between them. [0037] The particular configuration (Goldberg polyhedral 162 faces and icosahedral symmetry) can only be achieved with a combination of hexagons and pentagons. However other geometries with around 162 faces may be possible to do using only pentagons or only hexagons. Other embodiments of the invention can include a pattern with various geometric configurations. For example, the pattern can be comprised of more or fewer of hexagons and pentagons than shown. Or it can comprise all hexagons, all pentagons, or no hexagons or pentagons but instead one or more other shapes or polyhedrals having flat faces and sharp edges. One other shape can be formed, for example, by splitting each hexagon into 6 triangles or each pentagon into 5 triangles, which provides a similar drag coefficient. One embodiment can include any of the Goldberg polyhedra with a combination of pentagons and hexagons or even a convex polyhedral made of triangles or squares.

[0038] It is further noted that flat faces give lower drag and have the uniqueness of not being dimples (curved indentations). The flat faces only provide points of the faces that he on the circumscribed sphere. The sharp edges are defined by the angle between two adjacent faces. In addition, the edges forming the boundaries between the two adjacent faces are flat and not excessively rounded. An example of a non-sharp edge is shown in FIG. 2(c). The angle between the two edges is the same as is in FIG. 2(b) but it is rounded such that the edge is not sharp.

[0039] As the number of polygons increases (i.e. from 162 faces to 312 faces) the angle between the faces increases too and approaches 180 degrees. For the embodiments that are based purely on polyhedral shape such as those shown in FIGS. 2(a), 2(b) with 162 and 192 faces the range of angles is between 160 and 165 degrees. For the other embodiments in which dimples are added inside each face it is possible to go to as many as 312 faces and the angle between the faces can increase to 172 degrees. In one embodiment, the maximum angle could be close to 175 degrees and a range of angles between 160 and 175 degrees may be suitable for the purpose of a golf ball. Thus, a convex polyhedra is one having faces with angles substantially with the values and in the ranges noted herein.

[0040] In FIG. 2(a), the ratio of pentagons to hexagons is 12: 150, though any suitable ratio can be provided. For example, out of the 150 hexagons one of the hexagons can be split into 6 triangles and a have a polyhedron with 12 pentagons, 149 hexagons and 6 triangles and obtain substantially the same drag coefficient. FIG. 2(e) illustrates how such a splitting can be performed on one of the hexagonal faces 120. A vertex 140 can be chose anywhere inside the hexagon 120. For illustrative purposes, the vertex 140 is near the center of the hexagon although any other location can be used. Six new edges 142 can be formed by connecting the each of the vertices 122 with the new vertex 140. A triangular face 144 is formed by one edge 124 of the hexagon and two adjacent edges 142. The exact shape of the faces making up the polyhedral can vary but one important feature of the polyhedral pahem is the angle between faces.

[0041] FIG. 3 shows an example of a golf ball in accordance with another embodiment of the invention. The golf ball 200 has a body 210 with an inner core and an outer shell with an outer surface 212. A plurality of faces 220 are formed in the outer surface, creating a pahem 216. All the faces are formed in the outer surface and each of such faces is flat and lies in a plane. The faces are bound by straight or linear edges 224. Here, the golf ball 200 is a polyhedron with 192 polygons.

[0042] The body 210 defines a circumscribed sphere 202, which is the smallest sphere containing the polyhedron, and is shown with a dashed line about the outermost points 204 of the body 210. The vertices 222a, 222b of the polyhedron are the only points 204 on the polyhedron that lie on the sphere. Any point along the edges 224a, 224b or on the face of the polygons lies below the surface of the circumscribed sphere. [0043] The golf ball body 210 is a polyhedron that is made from first faces 220a and second faces 220b. As shown, the first faces 220a have a first shape, namely pentagons, and the second faces 220b have a second shape different than the first shape, namely hexagons. All of the faces are flat and each such face forms a single plane. In one embodiment, there are 12 pentagons 220a and 180 hexagons 220b (a hexagon-to-pentagon ratio of 15: 1), each having comers or points 222a, 222b connected by boundaries such as straight lines or edges 224a, 224b. In various other embodiments, other quantities and/or ratios of such pentagons 220a and hexagons 220b can be used. The first and second faces 220a, 220b form the pattern 216. The edges 224 are sharp, in that the radius of curvature of an edge is less than 0.001D, where D is the diameter of the circumscribed sphere. Ideally all edges should be the sharpest possible. A radius of curvature of 0 is the case with the sharpest possible edges, though golf ball edges might become a little rounded during manufacture or follow use.

[0044] The geometric shape of the embodiment illustrated in FIG. 3 falls into the class of convex polyhedral where the angle between any pair of adjacent faces is always less than 180 degrees. In this embodiment, the angle between a pentagon face 220a and an adjacent hexagonal face 220b is 167.6 degrees. The angle between two adjacent hexagon faces l20b varies from 163.4 degrees to 164.2 degrees. When comparing this embodiment with the golf ball 100 illustrated in FIG. 2 it is obvious that as the number of faces on a convex polyhedron increases the angle between faces increases too.

[0045] Each face 220 is immediately adjacent and touching a neighboring face 220, such that each edge 224 forms a border between two neighboring faces 220 and each point 222 is at the intersection of three neighboring faces 220. And, each point 222 is at an opposite end of each linear edge 224 and is at the intersection of three linear edges 224. Accordingly, there is no gap or space between adjacent neighboring faces 220, and the faces 220 are contiguous and form a single integral, continuous outer surface 212 of the ball 200. [0046] The patern is based on an icosahedron shown in FIG. 2(c). In the icosahedron 170 five equilateral triangles 180 meet at each of its twelve vertices 182. The 12 pentagons 220a of the golf ball 200 shown in FIG. 3 are centered on the vertices of an icosahedron.

Therefore, any pair of 3 pentagons 220a form an equilateral triangle 280. The pentagons 220a are all equilateral, that is the 5 edges 224a all have the same length equal to 0.136D, where D is the diameter of the circumscribed sphere. The hexagons 220b are not equilateral and the lengths of the edges 224b vary from 0.136D to 0.168D.

[0047] Table 2 lists the coordinates of all the vertices and how each face is formed, which defines the geometry of the golf ball. Table 1 lists the coordinates x, y, and z of all of the vertices 220 of polyhedron of golf ball 200. Faces are constructed by connecting the vertices with straight lines. The coordinates x, y, z are normalized by the diameter D of the circumscribed sphere 204 and are with respect to the center of the sphere. The golf ball 200 contains 380 vertices 222, 570 straight edges 220 and 192 polygonal faces 220.

[0048] From a visual perspective the above designs of FIGS. 2, 3 have the unique characteristics of not having any dimples. From a utility perspective the behavior of the drag coefficient is very interesting. FIG. 4 shows a graph of the drag coefficient, CD, versus the Reynolds number, Re, for the polyhedron with 162 and 192 faces. The drag coefficient was obtained by wind tunnel experiments of non-spinning models. Overall the drag curve is qualitatively very similar to that of a dimpled sphere. Namely there is a drag crisis that occurs around Re=60,000. For the polyhedron with 162 faces CD reaches a minimum value of 0.16 at Re=90,000 and remains almost constant as the Reynolds increases. For the polyhedron with 192 faces CD reaches a minimum value of 0.14 at Re=l l0,000 and remains almost constant as the Reynolds increases.

[0049] The graph reveals that as the number of faces increases the drag crisis shifts towards a higher Reynolds number and the CD in the post-critical regime decreases. This feature can be taken into advantage when designing a golf ball to tailor the needs of a golfer. For an amateur golfer a golf ball in flight during a driver shot can experience a Reynolds number ranging from 180,000 at the beginning of the flight to 60,000 at the end of the flight. For a professional golfer a golf ball in flight during a driver shot can experience a Reynolds number ranging from 220,000 at the beginning of the flight to 80,000 at the end of the flight. In other words, the range of Reynolds number experienced by a golf ball for an amateur golfer is lower than that for a professional golfer. A golf ball with more polygon faces might suit the needs of a professional golfer while a golf ball with less polygon faces might suit the needs of an amateur golfer.

[0050] The advantage of a design based on a convex polyhedron such as golf ball 200 with respect to a dimpled golf ball is now discussed. A comparison of the drag curve of the polyhedron with 192 faces, namely golf ball 200, against a dimpled sphere is shown in FIG.

5. The dimpled sphere has 322 spherical dimples and is representative of a commercial golf ball. The drag crisis for the polyhedron with 192 faces, namely golf ball 200, occurs at approximately the same range of Reynolds numbers as the dimpled sphere. The minimum CD for both balls is reached at Re=l 10,000. For the dimpled sphere CD=0.16 while for the golf ball 200 CD=0. l4, that is 12.5% drag reduction. At Re=l40,000 Co=0. l74 for the dimpled sphere while for the golf ball 200 CD=0T47, that is 15% drag reduction. Indeed, the drag coefficient for golf ball 200 illustrated in FIG. 3 is consistently lower than that of a dimpled golf ball in the range of Re=90, 000-220, 000. A lower drag coefficient is important because it plays an important role in achieving more carry distance during a driver shot.

[0051] A comparison of the drag curve of the golf ball 100 against the same dimpled sphere is shown in FIG. 6. While CD in the post-critical regime is almost identical for the two balls, the drag crisis for the golf ball 200 occurs at a lower Reynolds number. As a result, CD for Re<l 10,000 is consistently lower for the polyhedron with 192 faces than for the dimpled sphere. Thus, the embodiment of FIG. 2(a) has a lower drag coefficient than a dimpled sphere. A lower drag coefficient is important because it plays an important role in achieving more carry distance during a driver shot.

[0052] FIGS. 7, 8 are additional non-limiting embodiments of the invention. Those golf balls 300, 400 have similar structure as the embodiments of FIGS. 2, 3, and those structures have similar purpose. Those structures have been assigned a similar reference numeral and similar structure with the differences noted below. For example, FIG. 7 shows an example of a golf ball 300 with a body 310 with an inner core and an outer shell with an outer surface 312. A plurality of faces 320 are formed in the outer surface, creating a pattern 316. All the faces are formed in the outer surface 312 and each of such faces is flat and lies in a plane. The faces are bound by straight or linear edges 324 and comers vertices 322. The body 310 defines a circumscribed sphere 302, which is the smallest sphere containing the polyhedron, and is shown with a dashed line about the outermost points 304 of the body 310. And FIG. 8 shows an example of a golf ball 400 with a body 410 with an inner core and an outer shell with an outer surface 412. A plurality of faces 420 are formed in the outer surface, creating a pattern 416. All the faces are formed in the outer surface 412 and each of such faces is flat and lies in a plane. The faces are bound by straight or linear edges 424 and comers or vertices 422. The body 410 defines a circumscribed sphere 402, which is the smallest sphere containing the polyhedron, and is shown with a dashed line about the outermost points 404 of the body 410.

[0053] The embodiment shown in FIGS. 7, 8 are included for illustrative purposes as examples of convex polyhedral that are not made from pentagons or hexagons. Other convex polyhedra not made up of pentagons or hexagons can also be used for the design of a golf ball. In another embodiment of the present invention a convex polyhedron is shown in FIG.

7. The polyhedron belongs to a class of solids called Geodesic polyhedron which are derived from an icosahedron by subdividing each face into smaller faces using a triangular grid, and then applying a canonicalization algorithm to make the result more spherical (see https://en.wikipedia.org/wiki/Geodesic_polyhedron). The vertices of the polyhedron are the only points on the polyhedron that he on a sphere. Any point along the edges or on the face of the triangles lies below the surface of a circumscribed sphere. The polyhedron in FIG. 7 is made from 320 triangles but any geodesic polyhedron with an arbitrary number of triangles can be used as the design of a golf ball.

[0054] In another embodiment of the present invention a convex polyhedron is shown in FIG. 8. The polyhedron belongs to a class of solids called Geodesic cubes which are made from rectangular faces. A geodesic cube is a polyhedron derived from a cube by subdividing each face into smaller faces using a square grid, and then applying a canonicalization algorithm to make the result more spherical (see http://dmccooey.com/polyhedra/GeodesicCubes.html). The vertices of the polyhedron are the only points on the polyhedron that he on a sphere. Any point along the edges or on the face of the rectangular faces lies below the surface of a circumscribed sphere. The polyhedron in FIG. 8 is made from 174 rectangular flat faces but any geodesic cube with an arbitrary number of faces can be used as the design of a golf ball.

[0055] It is important to note that the polyhedra described above and shown in FIGS. 2, 3, 7,

8 do not contain any dimples (i.e., curved or spherical depressions or indents), but instead have flat surfaces that he in a plane. However, the polyhedra provide enhanced aerodynamic characteristics, drag coefficient being one of them, that can help increase the carry distance of the golf ball.

[0056] However, golf ball geometries that are based on any convex polyhedra and either include or omit dimples are contemplated. In particular, for any convex polyhedra at least one of the faces may include one or more dimples. For example, FIG. 9 shows an

embodiment of a golf ball 500 that is based on a convex polyhedron with a plurality of polygonal faces 520. The convex polyhedron is identical to the one shown in FIG. 2, and includes a body 510 with an inner core and an outer shell with an outer surface 512. A plurality of faces 520 are formed in the outer surface, creating a pattern 516. All the faces are formed in the outer surface 512 and each of such faces is flat and lies in a plane. The faces are bound by straight or linear edges 524 and comers or vertices 522. The body 510 defines a circumscribed sphere 502, which is the smallest sphere containing the polyhedron, and is shown with a dashed line about the outermost points 504 of the body 510.

[0057] However, in this embodiment each face 520 of the polyhedron contains one dimple 560. The dimples 560 have a substantially spherical shape and are created by subtracting spheres 570 from each of the faces 510 of the polyhedron 500. Table 3 lists the coordinates x, y and z of the center of the spheres 570 along with the diameter d of the spheres 570. The coordinates x, y, z and the diameter of the spheres d are normalized by the diameter D of the circumscribed sphere 504. The coordinates x, y, and z are with respect to the center of the circumscribed sphere. The total dimple volume ratio, defined as the aggregate volume removed from the surface of the polyhedron divided by the volume of the polyhedron, is 0.317%.

[0058] As shown, the dimples 560 are each located at the center of each 510. However, the dimples 560 can be positioned at another location such as offset within each face 510, or overlapping two or more faces 510. In addition, while spherical dimples 560 are shown, the dimples 560 can have any suitable size and shape. For example, non-spherical depressions can be utilized, such as triangles, hexagons, pentagons. And the dimples need not all have the same size and shape, for example there can be dimples with more than one size and more than one shape.

[0059] The effect that the addition of dimples has on the drag coefficient is now discussed.

A comparison of the drag curve of this embodiment against that of golf ball 100 is shown in FIG. 10. There are two important observations. First when dimples are added to the faces of a polyhedron, the drag crisis occurs at a lower Reynolds number range. That is, the drag coefficient starts to drop at a lower Reynolds number. Second, the drag coefficient in the post-critical regime increases. This effect may be desirable when designing a golf ball for players with lower swing speeds such as an amateur golf player where the range of Reynolds number that the golf ball experiences during a driver shot is reduced. As the total dimple volume approaches zero the drag curve of golf ball 500 would approach that of golf ball 100. Therefore, one can fine tune the exact behavior of the drag curve by adjusting the total dimple volume. Each dimple is formed by subtracting a sphere for the face of the polyhedron as explained above. The dimple volume is the amount of volume that each sphere subtracts from the volume of the polyhedron.

[0060] FIG. 11 compares the drag curve of the golf ball 500 shown in FIG. 9 against a commercial golf ball, namely the Callaway SuperHot, which is a dimpled ball marketed as having a low drag coefficient and shown in U.S. Patent No. 6,290,615. The drag coefficient for both balls were obtained by performing wind tunnel experiments using the actual commercial golf ball and a 3D printed prototype of the golf ball 500. The drag crisis for the golf ball 500 happens earlier, that is the drag coefficient starts to drop at lower Reynolds number. Clearly CD for golf ball 500 is consistently lower than that of the commercial golf ball for the range of Reynolds numbers 60,000-160,000. A lower drag coefficient can help a golf ball achieve longer carry distances. The golf ball 500 is the exact same polyhedron of golf ball 100, however the golf ball 500 has a dimple on each face. The polyhedron which golf ball 500 is based on is identical to the polyhedron of golf ball 100 in FIG. 2.

[0061] FIG. 12 shows an example of a golf ball based on a convex polyhedron with dimples in accordance with another embodiment of the invention. The golf ball 600 has a body 610 with an inner core and an outer shell with an outer surface 612. A plurality of faces 620 are formed in the outer surface, creating a pattern 616. All the faces are formed in the outer surface and each of such faces is flat and lies in a plane. The faces are bound by straight or linear edges 624. Here, the golf ball 600 is based on a polyhedron with 312 polygons.

[0062] The body 610 defines a circumscribed sphere 602, which is the smallest sphere containing the polyhedron, and is shown with a dashed line about the outermost points 604 of the body 610. The vertices 622a, 622b of the polyhedron are the only points 604 on the polyhedron that he on the sphere. Any point along the edges 624 or on the face of the polygons lies below the surface of the circumscribed sphere.

[0063] The golf ball body 610 is a polyhedron that is made from first faces 620a and second faces 620b. As shown, the first faces 620a have a first shape, namely pentagons, and the second faces 620b have a second shape different than the first shape, namely hexagons. All of the faces are flat and each such face forms a single plane. In one embodiment, there are 12 pentagons 620a and 300 hexagons 620b (a hexagon-to-pentagon ratio of 25: 1), each having comers or points 622a, 622b connected by boundaries such as straight lines or edges 624a, 624b. The first and second faces 620a, 620b form the pattern 616. The edges 624 are sharp, in that the radius of curvature of an edge is less than 0.001D, where D is the diameter of the circumscribed sphere. Ideally all edges should be the sharpest possible. A radius of curvature of 0 is the case with the sharpest possible edges, though golf ball edges might become a little rounded during manufacture or follow use.

[0064] The geometric shape of the embodiment illustrated in FIG. 12 falls into the class of convex polyhedral where the angle between any pair of adjacent faces is always less than 180 degrees. In this embodiment, the angle between a pentagon face 620a and an adjacent hexagonal face 620b is 170.2 degrees. The angle between two adjacent hexagon faces 620b varies from 167.1 degrees to 168.2 degrees.

[0065] Each face 620 is immediately adjacent and touching a neighboring face 620, such that each edge 624 forms a border between two neighboring faces 620 and each point 622 is at the intersection of three neighboring faces 620. And, each point 622 is at an opposite end of each linear edge 624 and is at the intersection of three linear edges 624. Accordingly, there is no gap or space between adjacent neighboring faces 620, and the faces 620 are contiguous and form a single integral, continuous outer surface 612 of the ball 600.

[0066] The pattern is based on an icosahedron shown in FIG. 2(c). In the icosahedron 170 five equilateral triangles 180 meet at each of its twelve vertices 182. The 12 pentagons 620a of the golf ball 600 shown in FIG. 12 are centered on the vertices of an icosahedron.

Therefore, any pair of 3 pentagons 620a form an equilateral triangle 680 shown with dashed line in FIG. 12. The pentagons 620a are all equilateral, that is the 5 edges 624a all have the same length equal to 0.102D, where D is the diameter of the circumscribed sphere. The hexagons 620b are not equilateral and the lengths of the edges 624b vary from 0.102D to 0.132D.

[0067] Table 4 lists the coordinates of all the vertices and how each face is formed, which defines the geometry of the polyhedron. Table 4 lists the coordinates x, y, and z of all of the vertices 622 of polyhedron of golf ball 600. Faces are constructed by connecting the group of vertices with straight lines. The coordinates x, y, z are normalized by the diameter D of the circumscribed sphere 604 and are with respect to the center of the sphere. The golf ball 600 contains 620 vertices 622, 930 straight edges 624 and 312 polygonal faces 620.

[0068] Each of the face 620 of the polyhedron contains one dimple 690. The dimples 690 have a substantially spherical shape and are created by subtracting spheres 670 from the face 620 of the polyhedron. Table 5 lists the coordinates x, y and z of the center of the spheres 670 along with the diameter d of the spheres 670. The coordinates x, y, z and the diameter of the spheres d are normalized by the diameter D of the circumscribed sphere 604. The coordinates x, y, and z are with respect to the center of the circumscribed sphere. [0069] A comparison of the drag curve of this embodiment against that of a commercial golf ball, namely the Bridgestone Tour is shown in FIG. 13, which is a dimpled ball marketed as having a low drag so that the ball travels further, and described in U.S. Patent No. 7,503,857. The graphs shows the invention having a lower drag coefficient. The drag coefficient for both balls were obtained by performing wind tunnel experiments using the actual commercial golf ball and a 3D printed prototype of the embodiment. The drag crisis for both golf balls occurs at approximately the same range of Reynolds number, namely from Re=50,000- 80,000. At Reynolds number of 100,000 CD for the Bridgestone Tour ball is 0.195 while CD for the current embodiment is 0.162, a drag reduction of 17%. Overall CD for the current embodiment is consistently lower than that of the commercial golf ball for the range of Reynolds numbers 80,000-160,000. A lower drag coefficient can help a golf ball achieve longer carry distances.

[0070] Other sizes and shapes of dimples are contemplated by the present invention, and any suitable size and shape dimple can be utilized in the golf balls of FIGS. 2-3, 7-9, 12. While FIG. 2 employs a convex polyhedron that belongs to a class of Goldberg polyhedral, FIGS. 9, 12 are revisions to the Goldberg principle in that each face of the polyhedron contains a dimple.

[0071] The following documents are incorporated herein by reference. Achenbach, E.

(1972). Experiments on the flow past spheres at high Reynolds numbers. Journal of Fluid Mechanics. Harvey, P. W. (1976). Golf ball aerodynamics. Aeronautical Quarterly. J. Choi, W. J. (2006). Mechanism of drag reduction by dimples on a sphere. Physics of Fluids, 149- 167. Ogg, S. S. (2001).

[0072] It is further noted that the description and claims use several positional, geometric or relational terms, such as neighboring, circular, spherical, round, and flat. Those terms are merely for convenience to facilitate the description based on the embodiments shown in the figures. Those terms are not intended to limit the invention. Thus, it should be recognized that the invention can be described in other ways without those geometric, relational, directional or positioning terms. In addition, the geometric or relational terms may not be exact. For instance, edges may not be exactly linear, hexagonal, pentagonal or spherical, but still be considered to be substantially linear, hexagonal, pentagonal or spherical, and faces may not be exactly flat or planar but still be considered to be substantially flat or planar because of, for example, roughness of surfaces, tolerances allowed in manufacturing, etc.

And, other suitable geometries and relationships can be provided without departing from the spirit and scope of the invention.

[0073] In addition, while the invention has been shown and described with boundaries formed by straight edges 124, 224, 324, 424, 524, 624 and points 122, 222, 322, 422, 522, other suitable boundaries can be provided such as curves with or without rounded points.

And the boundaries need not be sharped, but can be curved. And other suitable shapes can be utilized for the faces. And, while the invention has been described with a certain number of faces and/or dimples, other suitable numbers of faces and/or dimples, more or fewer, can also be provided within the spirit and scope of the invention. Within this specification, the various sizes, shapes and dimensions are approximate and exemplary to illustrate the scope of the invention and are not limiting. The faces need not all be the same shape and/or size, and there can be multiple sizes and shapes of faces.

[0074] The sizes and the terms“substantially” and“about” mean plus or minus 15-20%, and in one embodiment plus or minus 10%, and in other embodiments plus or minus 5%, and plus or minus 1-2%. In addition, while specific dimensions, sizes and shapes may be provided in certain embodiments of the invention, those are simply to illustrate the scope of the invention and are not limiting. Thus, other dimensions, sizes and/or shapes can be utilized without departing from the spirit and scope of the invention. [0075] Use of the term optional or alternative with respect to any element of a claim means that the element is required, or alternatively, the element is not required, both alternatives being within the scope of the claim. Use of broader terms such as comprises, includes, and having may be understood to provide support for narrower terms such as consisting of, consisting essentially of, and comprised substantially of. Accordingly, the scope of protection is not limited by the description set out above but is defined by the claims that follow, that scope including all equivalents of the subject matter of the claims. Each and every claim is incorporated as further disclosure into the specification and the claims are embodiment(s) of the present disclosure.

TABLE 1

TABLE 2

TABLE 3

TABLE 4

TABLE 5

[0076] The foregoing description and drawings should be considered as illustrative only of the principles of the invention. The invention may be configured in a variety of shapes and sizes and is not intended to be limited by the embodiment shown. In addition, the statements made with respect to one embodiment apply to the other embodiments, unless otherwise specifically noted. For example, the statements regarding FIG. 2(a) with respect to size, shape and geometry apply equally to the embodiments of FIGS. 3, 7-9, 12. It is further understood that the description and scope of invention apply equally (though the descriptions have not been repeated) for each structure that is the same or similar between each of the various embodiment, and whether or not those structures have been assigned a similar reference numeral .

[0077] Numerous applications of the invention will readily occur to those skilled in the art. Therefore, it is not desired to limit the invention to the specific examples disclosed or the exact construction and operation shown and described. Rather, all suitable modifications and equivalents may be resorted to, falling within the scope of the invention.

Claims

CLAIMS:

1. A golf ball comprising:

a body having an outer shell with an outer surface; and

a pattern formed in the outer surface of said body, the pattern comprising a polyhedron having a plurality of flat faces, each of said plurality of flat faces having one or more sharp edges.

2. The golf ball of claim 1, wherein said plurality of faces are circumscribed in a sphere, wherein only the sharp comers forming vertices of the polyhedron lie on the sphere.

3. The golf ball of claim 2, said sphere having a diameter of at least l.68in.

4. The golf ball of any of claims 1-3, wherein at least one of the plurality of faces of the polyhedron contains one or more dimples.

5. The golf ball of any of claims 1-4, wherein said plurality of faces are each in a plane.

6. The golf ball of any of claims 1-5, wherein said plurality of faces are contiguous to touch one another and form a single continuous outer surface of said body.

7. The golf ball of any of claims 1-6, wherein said plurality of faces are at an angle with respect to one another to define said one or more sharp edges and said one or more sharp comers.

8. The golf ball of any of claims 1-7, wherein said pattern comprises a Goldberg polyhedron.

9. The golf ball of any of claims 1-8, wherein said plurality of faces comprise a plurality of first faces having a first shape and a plurality of second faces having a second shape.

10. The golf ball of claim 9, wherein said first shape comprises a pentagon and said second shape comprises a hexagon.

11. The golf ball of claim 9 or 10, wherein said plurality of first faces comprise twelve and said plurality of second faces comprise 150.

12. The golf ball of claim 9 or 10, wherein a ration of said plurality of first faces to said plurality of second faces comprises 12.5: 1.

13. The golf ball of any of claims 1-12, wherein said plurality of flat faces having one or more sharp comers.

14. The golf ball of any of claims 1-13, wherein said edges are linear.

15. The golf ball of any of claims 1-14, wherein two neighboring flat faces form an angle substantially less than 180 degrees.

16. The golf ball of any of claims 1-15, wherein said sharp edges have a radius of curvature that is less than 0.001D, where D is the diameter of a circumscribed sphere of said golf ball.

17. A method of forming a golf ball, comprising: forming an outer surface; and, forming a pattern in the outer surface, the pattern having a plurality of flat surfaces defining sharp edges and points therebetween.