Hyperbolic Immersion

Hyperbolic Immersion is an attempt to capture some of the beauty in geometry in a form that is tangible, durable and without the false precision of a computer generated model. The hope is to provide people of all levels of mathematical interest, training and ability with something to think about. It was constructed for Burning Man 2003, where it was displayed at about 11:37 (a location, not a time), between David Best's Temple of Honor and Zachary Coffin's Gravity Rocks. It is currently making friends with cats in semi-rural Washington.

This sculpture represents an isometric immersion of a subset of the hyperbolic plane into Euclidean 3 space. David Hilbert proved in 1901 the impossibility of smoothly extending to an immersion of the complete plane. For the specific construction method employed in this sculpture, a much simpler calculation shows that the construction is necessarily finite.

The piece is made from a few hundred translucent green triangles, cut from sheets of polycarbonate. The triangles are linked together using scraps of bike tire, attached with machine screws. The triangles meet seven at a vertex, giving an approximation of a tessellated hyperbolic plane exhibiting the group of discrete symmetries with the smallest covolume. The surface is suspended by a steel frame at about 5' above ground level, and the model extends about 7' in its largest dimension.

Shadows

Suspended

Dust Storm

Hilbert's Theorem

The Poincaré Disk

$\begin{displaymath}\begin{array}{rcl} t_n &=& t_{n-1}+2u_{n-1}+v_{n-1}\\ u_n &=& 2u_{n-1}+v_{n-1}\\ v_n &=& u_{n-1}+v_{n-1} \end{array}\end{displaymath}$

$\begin{displaymath}t_0=0,\:u_0=7,\:v_0=0\end{displaymath}$

$\begin{displaymath} M=\threebythree{1}{2}{1}{0}{2}{1}{0}{1}{1} \end{displaymath}$

$\begin{displaymath}\det(M-\zeta I)=-\zeta^3+4\zeta^2-4\zeta+1\end{displaymath}$

$\begin{displaymath}\zeta_1=1,\:\zeta_{2,3}=\frac{3\pm\sqrt5}{2}\end{displaymath}$

$\begin{displaymath}\xi_1=\vectxyz100,\:\xi_2=\vectxyz{-\sqrt5}{\fee}{1},\: \xi_3=\vectxyz{\sqrt5}{1/\fee}{1}\end{displaymath}$

$\begin{displaymath}\begin{array}{rcl} t_n &=& (1\;0\;0)\;M^n\left(\begin{array}... ... & \\ &>& \frac1V\cdot\frac43\pi n^3, \:\: (n>>0) \end{array}\end{displaymath}$

Calculation
The function t(n) is the number of triangles within a combinatorial radius of n triangle edges from a central vertex. This number grows exponentially, but the volume of the sphere they must all fit inside only grows as the third power of n. I conjecture that this upper bound on the size that the pattern can symmetrically extend whenever it is physically manifested is a reflection of a mathematical bound on the combinatorial radius of this pattern embedded or even immersed into euclidean 3-space, independent of the triangles having any volume. I'm guessing that this mathematical bound is less than n=15.

Nebula and the Hyperbolic Plane

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