Hyperbolic geometry model

A hyperbolic geometry model formed by several pieces of flexible material joined along their edges in a predetermined manner. In one embodiment, the pieces are polygons arranged so that the sum of the angles about each vertex or corner exceeds 360.degree.. In a second embodiment the pieces are elongated strips joined edge-to-edge about a center line with the inner strips being comparatively straight and the strips away from the center line having a progressively decreasing radius of curvature. In a final embodiment the pieces are arcuate strips joined edge-to-edge about a center point with the radius of curvature of the strips progressively increasing to a finite limit away from the center point. The resulting hyperbolic geometry model is a sheet of flexible material which, because of its increasing fullness away from the center of the model, exhibits geometric properties which illustrate the characteristics of a hyperbolic plane.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
This invention relates to geometric surfaces, and more particularly to a 
model for illustrating the geometric properties of the hyperbolic plane 
and to a structure for providing a sheet of thin, flexible material which 
cannot be flattened and which has a large surface area in relation to its 
volume. 
2. Description of the Prior Art 
Three basic geometrical surfaces are the Euclidean plane, the spherical 
plane and the hyperbolic plane. Each of these planes have their own 
geometric properties. For example, in the familiar Euclidean plane a pair 
of lines intersecting a third line at right angles are parallel to each 
other. In a spherical plane, a pair of lines intersecting a third line at 
right angles (e.g. a pair of meridians intersecting the equator) converge 
toward each other and eventually overlap. In the hyperbolic plane, a pair 
of lines intersecting a third line at right angles continuously diverge 
away from each other. 
The geometric properties of the Euclidean plane can be easily demonstrated 
on a flat, planar surface. Similarly, a spherical geometry model for 
illustrating the properties of spherical geometry is formed by a simple 
sphere. However, it has not heretofore been possible to produce a physical 
model for demonstrating the properties of hyperbolic geometry, except on 
patches of small intrinsic size. 
Although physical, hyperbolic geometry models have not been produced, the 
characteristics of the hyperbolic plane have been extensively studied and 
are described in detail in Coxeter, H. S. M., Non-Euclidean Geometry, 
pages 147-167, University of Toronto Press, Toronto, Canada 1942 and 
Coxeter, H. S. M., Twelve Geometric Essays, pages 199-214). Southern 
Illinois University Press, Carbondale, Illinois 1968. These references 
decompose the hyperbolic plane into "tessellations" or "tilings"; that is, 
an arrangement of points, line-segments and simple polygons (called 
vertices, edges and faces, respectively) such that every edge joins two of 
the vertices and it is a common side of two of the faces. Although the 
hyperbolic plane can be described as being composed of these 
tessellations, the tessellations disclosed therein are distorted so that 
they can be illustrated in a Euclidean plane. This distortion is necessary 
since the sum of the angles of the corresponding Euclidean polygons about 
each vertex exceeds 360.degree., and it is not possible to accurately 
portray a collection of polygons having this property in the Euclidean 
plane. 
SUMMARY OF THE INVENTION 
The primary object of the invention is to produce a physical embodiment of 
a hyperbolic geometry model for illustrating the properties of hyperbolic 
geometry. 
It is another object of the invention to fabricate hyperbolic geometry 
models utilizing a variety of techniques. 
These and other objects of the invention are accomplished by simulating a 
hyperbolic plane with a replica which, while resembling a tessellation of 
a hyperbolic plane, may, in actuality, be a properly interconnected 
tessellation composed of Euclidean polygons or strips. Although the 
possibility of using truly hyperbolic, or "saddle-shaped", polygons or 
strips as components of the tessellation for especially accurate models is 
not excluded, it is considered sufficient for most purposes to use 
Euclidean polygons of small intrinsic area or strips of small intrinsic 
width, which are only slightly different from their hyperbolic 
counterparts. Here "intrinsic area" is equal to: (area).times. (absolute 
value of Gaussian curvature), and corresponds to "solid angle" equal to: 
(area)/(radius).sup.2 on the surface of a sphere. "Intrinsic width" is 
equal to: (width).times.(square root of absolute value of Gaussian 
curvature). The physical dimension of Gaussian curvature is 
(Length).sup.-2 and a hyperbolic Gaussian curvature is always negative. 
The physical model of the hyperbolic plane is constructed by joining 
several flexible embodiments of these polygons or strips to each other 
along their edges in a predetermined pattern. Each replica may be composed 
of Euclidean polygons or strips, but they are joined with others in a 
manner that near each point approximates the shape of a local segment of 
the hyperbolic plane. Consequently, the model closely approximates a 
relatively large segment of the hyperbolic plane. The replicas may be 
either tessellations with the Euclidean angles about each vertex exceeding 
360.degree., or strips joined edge-to-edge about a center line with the 
radius of curvature of the strips progressively decreasing to a finite 
limit away from the center line, or arcuate strips joined edge-to-edge 
about a center point with the radius of curvature of the strips 
progressively increasing toward a finite limit away from the center point.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
As best illustrated in FIG. 1, one embodiment of a hyperbolic geometry 
model 10 utilizes a plurality of Euclidean triangles 12 which, when joined 
to each other, resemble a triangular tessellation of the hyperbolic plane. 
Each inside angle of the Euclidean triangles is 60.degree., but, since 7 
triangles adjoin each other at each vertex, the sum of the angles about 
each vertex is 420.degree. (60.degree..times.7). Exact congruence with 
hyperbolic geometry is not required within each polygon (i.e. the polygon 
in the Euclidean plane does not correspond exactly to the polygon in the 
hyperbolic plane). However, the shape of the replica is still limited by 
certain factors, in order for each portion of the replica to accurately 
resemble a particular localized area in the hyperbolic plane, so that a 
collection of such portions represents a relatively large section of the 
hyperbolic plane. The first requirement is that the length of any two 
sides of polygons in the replica must have the same ratio as in the 
abstract tessellation. Secondly, the magnitudes of any two angles in the 
replica must have the same relationship as in the abstract tessellation so 
that if one angle is larger than another in the tessellation, that same 
angle must be larger than the other in the replica. Similarly, if two 
angles are equal in the tessellation, they must be equal in the replica. 
Finally, if the sides are straight or of constant curvature in the 
replica, they must be straight or of constant curvature in the abstract 
tessellation. Although the term "polygon" is sometimes used solely to 
designate a plane figure having a large number of sides, the more general 
mathematical definition, namely, a surface figure having three or more 
geodesic sides, or a physical object such as a fabric or lattice section 
having the shape of such a figure, is to be used herein. 
Another embodiment of a hyperbolic geometry model 14 utilizing octagons 16 
in the replica, is illustrated in FIG. 2. In a Euclidean octagon the 
inside angle between each edge is 135.degree.. As illustrated in FIG. 2, 
each vertex of the octagonal replica is a common vertex of three octagons 
so that the sum of the angles about each vertex is 
405.degree.(135.degree..times.3). 
Specific arrangements of "regular" polygons, i.e. where all of the sides 
and angles of the polygons are identical, can be denoted [n,m] indicating 
that the polygons are n-sided figures repeating m to a corner. In the 
hyperbolic plane n.times.m-2m-2n is greater than zero which is equivalent 
to requiring that the total angle of m Euclidean regular polygons of n 
sides at one corner is greater than 360.degree.. For a sum of the angles 
of m Euclidean regular polygons of n sides at one corner equal to 
360.degree. produces Euclidean tessellations and if the sum of the angles 
is less than 360.degree. spherical tessellations (Platonic solids) are 
produced. 
The replicas may be arranged in four ways: (a) point centered--where the 
center of the model is m polygons meeting at the point as in FIG. 1, (b) 
face centered--where the center of the model is the center of a polygon as 
in FIG. 2, (c) line centered--where the polygons 18 are arranged about a 
straight line 20 running between parallel rows of polygons as illustrated 
in FIG. 3, and (d) strip centered--where a straight line 22 runs through 
the center of a row of polygons 24 as illustrated in FIG. 4. It should be 
noted that the tessellations of FIGS. 3 and 4 are Euclidean tessellations 
for illustrative purposes only. In practice, the polygons would be made to 
produce hyperbolic tessellations. 
A second embodiment of a hyperbolic geometry model as illustrated in FIG. 5 
utilizes a plurality of arcuate strips 26 centered about a point 28. The 
strips 26 each have a radius of curvature which progressively increases 
away from the point 28. Each of the arcuate strips is a replica of an 
annulus of the hyperbolic plane. In order for the replicas in Euclidean 
geometry to closely approximate the annuli in the hyperbolic plane they 
must have a constant width, their inner and outer edges must have a 
constant curvature, and each spiral must form an annulus whose inner and 
outer edges are the same length of the inner and outer edges of the 
annulus it is modeling on the abstract hyperbolic plane. This means that 
the ratio between the lengths of the inner and outer edges of each segment 
must always be the same. 
The circumference of a circle in the hyperbolic plane is given by the 
formula: 
EQU C=2.pi.sinh r 
where r is the radius and C is the circumference of the circle. This 
formula can be utilized to drive the formula: 
##EQU1## 
where r is the radius of the inner circle in the hyperbolic plane, and w 
is the width of the strip. Here r and w are "intrinsic" widths, as defined 
in the Summary of the Invention. 
As illustrated in FIG. 6, a pair of lines 30, 32 intersecting a third line 
34 at 90.degree.. The lines 30, 32 are spaced apart from each other along 
the line 34 by a distance d, and a fourth line 36 intersects the lines 30, 
32 at 90.degree. and is spaced apart from the line 34 along the lines 30, 
32 by a distance r. The length x of the line 36 between the lines 30, 32 
can be calculated according to the formula x=d cosh r. This property of 
the hyperbolic plane can be utilized to derive the hyperbolic geometry 
model illustrated in FIG. 7 in which a plurality of strips 38 are arranged 
about a center line 40 with the radius of curvature of the strips 
progressively decreasing away from the center line. The above formula can 
be utilized to derive the following formula which can be utilized to 
calculate the dimensions of the strips 38 illustrated in FIG. 7: 
##EQU2## 
where r is the intrinsic distance from the inner edge of the strip 38 to 
the center line, and w is the intrinsic width of the strip. 
The resulting embodiments of the hyperbolic geometry model closely 
approximate the hyperbolic plane and exhibit such properties of the 
hyperbolic plane as an increasing fullness away from a center point or 
center line. The models may be formed by edge joining replicas made from a 
variety of flexible materials such as, for example, cloth, and the edge 
joining can be accomplished through a variety of techniques such as by 
sewing.