Understanding the Vegreville Pysanka
This egg was designed by Ron Resch while he was
Professor of Computer Science at the University
of Utah. In the concept and planning of the Egg,
9 "firsts" were achieved in mathematics,
engineering and architecture. The design involved cutting edge computer
graphics and students like Jim Blinn and Robert McDermott. I found 4 books that
describe this project:
(1)Paul Hoffman's book "Archimede's Revenge", chapter 5
"Adventures of an Egg Man".
(2)
Jim Blinn's book "Dirty Pixels" , chapter 1 "The World's Largest Easter Egg".
(3) Jay Kappraff "Connections", page 336 Figure 9.13, Chapter 9
"Transformations of the Platonic Solids",
(4)Theoni Pappas "The Magic of Mathematics", page 73 "Laying an Egg Mathematically".
Because the egg shape is axisymmetrical, the analogy with the earth is convenient. The pointed end on top is the North Pole (the POINT) and the fat end on the bottom is the South Pole (the END). The axis of symmetry is tilted 60 degrees from vertical. The area near the equator is a barrel shape. The poles are made of computer simulated caps. At the poles are 6-point stars, which is a common motif in Canada. At the equator the stars are made out of 2 colors (gold and silver) in the Mitsubishi arrangement. The stars are placed at the middle of a rhombus shape. The barrel of rhombi is bound
by the silver bands. Outside of the silver bands, there are rows of 3-point stars.
To simplify the fabrication, Prof. Ron Resch chose to use the
maximum number of 12-inch equilateral triangles. There are 2208
pieces of equilateral triangles and cover nearly half of the
surface. To make this surface follow the egg smoothly, the
equilateral triangles are interlaced with isosceles triangles
or 3-point stars. Symmetry helps to minimize the number of
different tiles. The two caps each have a 6-point star.
The sector(half-slice of a point) is repeated 12 times. The
barrel has 9 rhombi. Each rhombus is symmetrical about the
y axis. Only 1/18 of the barrel surface needs to be modeled.
If color is ignored, the number of different shapes is again
reduced in half. The barrel has 36 identical staves that
cover 10 degrees of longitude. The staves are made of
alternating columns of triangles and stars.
Within the barrel, the circumference is largest at the equator.
The circumference shrinks as we move towards the silver band.
This shrinkage is minimal around the equator and all the tiles
are nearly triangular. Outside of the silver bands, the gaps
between the equilateral triangles turn into 3-point stars. The
stars are hexagons with sides of 12 inches. The stars get
progressively thinner as we move towards the poles. The top
row (near the North Pole) of stars are more narrow than the
bottom row (near the South Pole) because the shrinkage is
more severe near the North Pole. The barrel extends from
the 5 bands of stars at the top to the 4 bands of stars at
the bottom. All the stars point down. The columns of stars
are separated by columns of fix-size triangles pointing up.
Between the barrel and the caps are the dark area (bronze-brown)
which hides the irregular shapes. The tiling in dark color
merges the 6-fold rotation symmetry of pole design with the
9-fold symmetry of the barrel. This is also the area where
the curvatures of the barrel are matched closely to the
curvatures of the two ends.
To state the problem in another way, the egg surface is
covered by 3 piece of continuous triangulation: the BARREL,
the END and the POINT. The triangulation consists of a
fix-size 12-inch equilateral triangle and a shrinkable
3-point star. The area of this star is 4X the fix-size
triangle at the equator
(24 inch equilateral triangle). The star area can be
compressed down to zero. In practice, the barrel is truncated
before the 3-point stars get too small. The 3-point stars are
colored silver and gold in 9 bands at the top and the bottom
of the barrel. The bronze-brown triangles in these bands
between the stars are the 12-inch equilateral triangles.
The triangles are flat aluminum plates (0.125 inch thick)
and structurally connected to the axis of symmetry with spokes.
The 3-point stars are aluminum plates (0.062 inch thick) bend
into 4 planes. The stars are bolted to 6 equilateral triangles
on its boundary and not connected to the core.
The size of the egg can be measured by the number of triangles. The diamond shape(rhombus) on the barrel has 6 triangles across the equator. So the shorter diagonal of the diamond is 6 feet long (roughly). The longer diagonal of the diamond is 10.63 feet long. About 10 adults holding hands can surround the equator of this giant egg.
If you compare the photos of the egg and the background, it is amazing how everything seems to change. The background is always different. The pysanka is kinetic and turns in the wind like a weathervane. This is a good way to minimize the loading from the 100-mile-per hour wind that sweeps across Vegreville. The coloring of the tiles has symbolic meaning. The color arrangement was a joint effort by Prof. Ron Resch and Paul Sembaliuk. Sembaliuk is an Albertan artist and authority on Ukrainian culture. Prof. Ron Resch has training in sculpture and industrial design. To make the egg a reality, he has also taken the task of fabrication of the tiles and designing the ingenious way the tiles are bolted together. The on-site installation is truly awesome.
The SYMBOLOGY is described by Annette Zoppo in a booklet: "By beginning at either end of the Egg in sequence, 5 distinct symbols are inherent in the design. The radiating gold star on
the end section, symbolizes Life and Good Fortune. This leads to the series of 3-pointed stars in alternating gold and silver, which symbolizes the Trinity, representative of the strong devotion to the faith of our ancestors. A band of silver, circumscribing the Egg, has no end and no beginning, symbolizes Eternity. On the central barrel section are located windmills, of 6 vanes and points in gold and silver, which symbolize the rich harvest. The silver wolves' teeth pointing to the center from the silver band, which is the most prominent motif of the design, symbolize the main message, of protection and security afforded our pioneers by the R.C.M.P. The 3 colors, bronze, silver and gold are rich color symbolizing Prosperity. Bronze (charcoal brown) is the basic color of the design and suggests the Good Earth, the land on which our forefathers struggled for
existence."
Vegreville is a farming community with the largest Ukrainian settlement in Canada. The idea of an egg monument came from Town Council member Mrs. Kay McKenzie. The timing and funding of the project were related to the 100th Anniversary of the Royal Canadian Mounted Police. The Egg was dedicated in the summer of 1976.
This may be a crazy idea for a science fair project. Start the form with a balloon. Add paper màche or plaster gauze on the surface to get a rigid form. The aspect ratio and the profile of the egg are important parameters. After the form is done, mark with a pencil the necessary lines for the tessellation. Finish the surface with color patches of origami paper. It is not as easy as it sounds, but you will learn a lot from doing this. First paint the background color (the 3-point stars or the dark-brown irregular shapes), then glue on the 2208 equilateral triangles made from color paper. Send me a photo when you are done !
Tessellation with JAVA programs
Examples of Math/Art
M.C. Escher interlocking shapes
Vegreville, Canada World's largest Easter Egg
University of Saskatchewan Debate Team 2000
Vegreville Pysanka photo album
Penrose tile in Architecture
Year of the Dragon design by Makoto Nakamura
Screenshots from the
Tessellations CD
I like to thank all the people who have posted the pysanka on
the WEB. I have spent a few hours thinking about the wonder of this egg. Please send me more photos. If you see any analysis or description of the egg construction, please send me a copy or a reference!
