Geom-E-Tree

Geom-E-Trees

The first part of this page consists of new renditions of graphics done between 1972 and 1980 on pen plotters and TEKscope CRT's. The GIF images were generated by linking into the gifdraw library by Quest Protein Database Center, Cold Spring Harbor Labs. (Actually, I call the routines from my own set of routines which can produce PostScript, Tek4010, or GIF output).

At the end of this page, I give some references to tree-related topics.

Trees

I am interested in tree structures of all kinds, from simple geometric abstractions, to the fate map of cells during embryogenesis.

Filling space by Hand

This Pixel Tree was done with MacPaint in the late 1980's!

Trees and the "perfect reduction factor"

The number given is the angle between branches. I have been able to compute the perfect reduction factor to use in each case. A page on that calculation is in order.

Binary 10 15 30 45 60 72 120 150 175 180
Ternary 30 45 60 72 90 120
Quaternary 45 60 72 90
5-ary 30 45 60 72
6-ary 30 45 60
Geom-E-Tree Animation (1300+) trees recorded frame-by-frame on 8mm film.

I'd like to write a Java program that could have your browser draw the trees for you, or go through an animated sequence. Dave Cassard produced a first attempt at this while enrolled in a Java programming class.

Radial Trees

These trees are drawn on concentric circles using two different radius-functions. I.E., a reduction factor (Rf) is not applied to the length of the branches, but there is a relation between successive radii.

Binary Radial Rf#1
Binary Radial Rf#2
There are others of higher degree.

Nested Polygon Sequences

Done for John K. Richards in 1973, using a CalComp plotter. Will write story behind it some day. Copyright registered in 1970's.

Notation n[i|o][+|-]c)
n is number of sides to begin with,
i|o is direction (inward or outward),
c is the +/- integral change in then as you go inward or outward.

3o+1, Whole Mandala, actually drawn 30i-1.
4o+2, Even Mandala, 30i-2
3o+2, Odd Mandala, 29-2.
3i+1, triangle going in, to increasing sides
4i+2, square going in, evenly
3i+2. trianle going in, by twos.
The are others.

Note that while the figure is named 3o+1, it was drawn as 30i-1... I started with a 30-gon of a given radius, and went inward (inscribed) from there.

Polygonal Billiards

Paths of particle reflections ("bouncing around ") inside regular concave polygons. Originally conceived in 3D to explore internal reflection dynamics of pyramid structures. When that proved difficult, I dropped to 2D to see if any interesting things happen. First program for Triangles by Corey Hirsh. Generalized polygon program by myself. Debugged by Greg Davis!

I recently discovered that others have done math research on this exact subject.

Sorry for the bright colors.. will change background or something ASAP.

The number given is the angle of the initial ray, beginning at the center of the polygon.

Triangles 1, 2, 3, 5, 8, 9, 10, 11, 12, 13, 14, 30.5, 45.
Squares 13, 21, 29, 31, 32.
Pentagons 18, 36, 4.
Hexagons 3, 13, 16, 17, 20, 22, 24, 27, 29.
Octagons 1.

Dendrimer Molecules

See May 1995 Scientific American. This article describes the contruction of tree-like polymers. It has a diagram similar to the radial trees above. The possibility of such molecules had occurred to me as well when I was drawing all these trees.

See also February 23, 1996 Science: Self-Assembling Dendrimers, p 1095; Molecular Trees: A New Branch of Chemistry, p 1077. Both articles contain bibliographies.

Tree-Structured Robot

See October 1994 Scientific American, Page 112. Hans P. Moravec at CMU has designed a binary tree shaped robot - trunk, two arms with two limbs each, etc, down to many tiny fingers.

Back to John Miller's Home Page

Created By: miller@lclark.edu