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PHYLLOTAXIS-BASED DIMPLE PATTERNS
FIELD OF THE INVENTION
The present invention is directed to golf balls. More particularly, the present invention is directed to a novel dimple packing method and novel dimple patterns. Still more particularly, the present invention is directed to a novel method of packing dimples using phyllotaxis and novel dimple patterns based on phyllotactic patterns.
Dimples are used on golf balls to control and improve the flight of the golf ball. The United States Golf Association (U.S.G.A.) requires that golf balls have aerodynamic sym- 15 metry. Aerodynamic symmetry allows the ball to fly with little variation no matter how the golf ball is placed on the tee or ground. Preferably, dimples cover the maximum surface area of the golf ball without detrimentally affecting the aerodynamic symmetry of the golf ball. 20
Most successful dimple patterns are based in general on three of five existing Platonic Solids: Icosahedron, Dodecahedron or Octahedron. Because the number of symmetric solid plane systems is limited, it is difficult to devise new symmetric patterns. 25
There are numerous prior art golf balls with different types of dimples or surface textures. The surface textures or dimples of these balls and the patterns in which they are arranged are usually defined by Euclidean geometry.
For example, U.S. Pat. No. 4,960,283 to Gobush discloses a golf ball with multiple dimples having dimensions defined by Euclidean geometry. The perimeters of the dimples disclosed in this reference are defined by Euclidean geometric shapes including circles, equilateral triangles, isos- 3J celes triangles, and scalene triangles. The cross-sectional shapes of the dimples are also Euclidean geometric shapes such as partial spheres.
U.S. Pat. No. 5,842,937 to Dalton et al. discloses a golf ball having a surface texture defined by fractal geometry and 40 golf balls having indents whose orientation is defined by fractal geometry. The indents are of varying depths and may be bordered by other indents or smooth portions of the golf ball surface. The surface textures are defined by a variety of fractals including two-dimensional or three-dimensional 45 fractal shapes and objects in both complete or partial forms.
As discussed in Mandelbrot's treatise The Fractal Geometry of Nature, many forms in nature are so irregular and fragmented that Euclidean geometry is not adequate to represent them. In his treatise, Mandelbrot identified a 50 family of shapes, which described the irregular and fragmented shapes in nature, and called them fractals. A fractal is defined by its topological dimension Dr and its Hausdorf dimension D. Dr is always an integer, D need not be an integer, and D=Dr. (See p. 15 of Mandelbrot's The Fractal 55 Geometry of Nature). Fractals may be represented by twodimensional shapes and three-dimensional objects. In addition, fractals possess self-similarity in that they have the same shapes or structures on both small and large scales. U.S. Pat. No. 5,842,937 uses fractal geometry to define the go surface texture of golf balls.
Phyllotaxis is a manner of generating symmetrical patterns or arrangements. Phyllotaxis is defined as the study of the symmetrical pattern and arrangement of leaves, branches, seeds, and petals of plants. See Phyllotaxis A 65 Systemic Study in Plant Morphogenesis by Peter V. Jean, p. 11-12. These symmetric, spiral-shaped patterns are known
as phyllatactic patterns. Id. at 11. Several species of plants such as the seeds of sunflowers, pine cones, and raspberries exhibit this type of pattern. Id. at 14-16.
Some phyllotactic patterns have multiple spirals on the surface of an object called parastichies. The spirals have their origin at the center of the surface and travel outward, other spirals originate to fill in the gaps left by the inner spirals. Frequently, the spiral-patterned arrangements can be viewed as radiating outward in both the clockwise and counterclockwise directions. These type of patterns are said to have visibly opposed parastichy pairs denoted by (m, n) where the number of spirals at a distance from the center of the object radiating in the clockwise direction is m and the number of spirals radiating in the counterclockwise direction is n. The angle between two consecutive spirals at their center C is called the divergence angle d. Id. at 16-22.
The Fibonnaci-type of integer sequences, where every term is a sum of the previous other two terms, appear in several phyllotactic patterns that occur in nature. The parastichy pairs, both m and n, of a pattern increase in number from the center outward by a Fibonacci-type series. Also, the divergence angle d of the pattern can be calculated from the series. Id.
When modeling a phyllotactic pattern such as with sunflower seeds, consideration for the size, placement and orientation of the seeds must be made. Various theories have been proposed to model a wide variety of plants. These theories have not been used to create new dimple patterns for golf balls using the science of phyllotaxis.
SUMMARY OF THE INVENTION
The present invention provides a method of packing dimples using phyllotaxis and provides a golf ball whose surface textures or dimensions correspond with naturally occurring phenomena such as phyllotaxis to produce enhanced and predictable golf ball flight. The present invention replaces conventional dimples with a surface texture defined by phyllotactic patterns. The present invention may also supplement dimple patterns defined by Euclidean geometry with parts of patterns defined by phyllotaxis.
Models of phyllotactic patterns are used to create new dimple patterns or surface textures. For golf ball dimple patterns, careful consideration is given to the placement and packing of dimples or indents. The placement of dimples on the ball using the phyllotactic pattern are preferably made with respect to a minimum distance criterion so that no two dimples will intersect. This criterion also ensures that the dimples will be packed as closely as possible.
BRIEF DESCRIPTION OF THE DRAWINGS
Reference is next made to a brief description of the drawings, which are intended to illustrate a first embodiment and a number of alternative embodiments of the golf ball according to the present invention.
FIG. 1A is a front view of a phyllotactic pattern;
FIG. IB is a detail of the center of the view of the phyllotactic pattern of FIG. 1A;
FIG. 1C is a graph illustrating the coordinate system in a phyllotactic pattern;
FIG. ID is a top view of two dimples according to the present invention;
FIG. 2 is a chart depicting the method of packing dimples according to a first embodiment of the present invention;
FIG. 3 is a chart depicting the method of packing dimples according to a second embodiment of the present invention;
FIG. 4 is a two-dimensional graph illustrating a dimple pattern based on the present invention;
FIG. 5 is a three-dimensional view of a golf ball having a dimple pattern defined by a phyllotactic pattern according to the present invention;
FIG. 6 is a golf ball having a dimple pattern defined by a phyllotactic pattern according to the present invention; and
FIG. 7 is a golf ball having a dimple pattern defined by a phyllotactic pattern according to the present invention.
Phyllotaxis is the study of symmetrical patterns or arrangements. This is a naturally occurring phenomenon. Usually the patterns have arcs, spirals or whorls. Some phyllotactic patterns have multiple spirals or arcs on the surface of an object called parastichies. As shown in FIG. 1A, the spirals have their origin at the center C of the surface and travel outward, other spirals originate to fill in the gaps left by the inner spirals. See Jem'Phyllotaxis A Systemic Study in Plant Morphogenesis at p.17. Frequently, the spiralpatterned arrangements can be viewed as radiating outward in both the clockwise and counterclockwise directions. As shown in FIG. IB, these type of patterns have visibly opposed parastichy pairs denoted by (m, n) where the number of spirals or arcs at a distance from the center of the object radiating in the clockwise direction is m and the number of spirals or arcs radiating in the counterclockwise direction is n. See Id. Further, the angle between two consecutive spirals or arcs at their center is called the divergence angle d. Preferably, the divergence angle is less than 180°.
The Fibonnaci-type of integer sequences, where every term is a sum of the previous two terms, appear in several phyllotactic patterns that occur in nature. The parastichy pairs, both m and n, of a pattern increase in number from the center outward by a Fibonacci-type series. Also, the divergence angle d of the pattern can be calculated from the series. The Fibonacci-type of integer sequences are useful in creating new dimple patterns or surface texture.
Important aspects of a dimple design include the percent coverage and the number of dimples or indents. The divergence angle d, the dimple diameter or other dimple measurement, the dimple edge gap, and the seam gap all effect the percent coverage and the number of dimples. In order to increase the percent coverage and the number of dimples, the dimple diameter, the dimple edge gap, and the seam gap can be decreased. The divergence angle d can also affect how dimples are placed. The divergence angle is related to the Fibonacci-type of series. A preferred relationship for the divergence angle d in degrees is:
where F1 and F2 are the first and second terms in a Fibonacci-type of series, respectively.
Near the equator of the golf ball, it is important to have as many dimples or indents as possible to achieve a high percentage of dimple coverage. Some divergence angles d are more suited to yielding more dimples near the equator than other angles. Particular attention must be paid to the number of dimples so that the result is not too high or too low. Preferably, the pattern includes between about 300 to about 500 dimples. Multiple dimple sizes can be used to affect the percentage coverage and the number of dimples; however, careful attention must be given to the overall symmetry of the dimple pattern. The dimples or indents can be of a variety of shapes, sizes and depths. For example, the indents can be circular, square, triangular, or hexagonal. The dimples can feature different edges or sides including ones that are straight or sloped. In sum, any type of dimple known to those skilled in the art could be used with the present invention.
The coordinate system used to model phyllotactic patterns is shown in FIG. 1C. The XY plane is the equator of the ball while the Z direction goes through the pole of the ball. Preferably, the dimple pattern is generated from the equator of the golf ball, the XY plane, to the pole of the golf ball, the Z direction. The angle <|) is the azimuth angle while 9 is the angle from the pole of the ball similar to that of spherical coordinates. The radius of the ball is R while p is the distance of the dimple from the polar axis and h is the distance in the Z direction from the XY plane. Some useful relationships are:
where i is the index number of the dimple.
Another consideration is how to model the top and bottom hemispheres such that the spiral pattern is substantially continuous. If the initial angle <|) is 0° and the divergence angle is d for the top hemisphere, the bottom hemisphere can start at -d where:
This will provide a ball where the pattern is substantially continuous.
When modeling a phyllotactic pattern such as with sunflower seeds, consideration for the size, placement and orientation of the seeds must be made. Similarly, several special considerations have to be made in designing or modeling a phyllotactic pattern for use as a golf ball dimple