Geom-e-Trees and Poly-Gones

New! April 2014: The Making of TERNARY TREE STAR (version 1.618), a 2012 artwork of mine.

History of my work on trees, polygon billiards, et cetera

See also the histories of my iOS apps: Geom-e-Tree and PolygonFlux.

The first part of this page consists of new renditions of graphics done between 1972 and 1980 at Lewis & Clark College 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 called the routines from my own set of routines which can produce PostScript, Tek4010, or GIF output). 2014 update: Really, I should generate SVG! That's next.

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

Here is 'Pixel Tree', drawn using 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.

• Binary 10 15 30 45 60 72 120 150 175 180
• Ternary 30 45 60 72 90 120
• 45 60 72 90
• 5-ary 30 45 60 72
• 6-ary 30 45 60

Geom-e-Tree Animations (1980, 2012)

In Fall of 1979, we recorded 1300+ trees frame-by-frame on Super8mm film. This was a fun Independent Study project done with student David C. King. See YouTube for the one of the original films Geom-e-Tree (1980) and a digital replica using Processing Geom-e-Tree (2012).

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 recurrence relation between successive radii.

Recurrence Relation Method #1 assures that the areas of the concentric shells are the same.

   Input R from keyboard;
   R = R * (1 + pi/n);

	
	



 Recurrence Relation Method #2 is left as an exercise the reader. LOL


   Input R from keyboard;
   A = R^2;
   R = SQRT( A + R^2 );



	
	



There are others of higher degree.
These are very similar in shape to Dendrimers, or molecular trees.
See below.



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.



Now Wikipedia has a good article on Dendrimers



Tree-Structured Robot


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

Nested Polygon Sequences

Not Tree-related, but included here when I first made this web page in 1990's.

Done for John K. Richards in 1973, using a CalComp plotter. Will write story behind it some day. Visual Arts (VA) copyright registered in April 1978 under title Synergy.

I made a series of plots varying 3 paraemters:

Notation n[i|o][+|-]c)

n   is the number of sides for initial polygon
i|o is the direction (inward or outward)
+|- is whether to increment n or decrement n
c   is the integral change for 'n'

• 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, increasing sides
• 4i+2, square going in, evenly
• 3i+2. triangle 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.

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.

In 2011 I wrote PolygonFlux, an immersive iOS app, that allows one to explore this space, and later, PolygonJazz, combining sound with geometric flux. For a complete history of all this, see [LINK].