MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

128 Mathematical Recreations
Recreation 29 (Spidrons [242]). To create a spidron, subdivide an equilateral
triangle by connecting its center to its vertices and reflect one of the
three resulting 30 ◦ − 30 ◦ − 120 ◦ isosceles triangles (with area one-third of the
original equilateral triangle) about one of its shorter sides. Now, construct an
equilateral triangle on one of the shorter sides (also with area one-third of the
original equilateral triangle) and repeat the process of subdivision-reflectionconstruction.
This will produce a spiraling structure with increasingly small
components. By deleting the original triangle we arrive at a semi-spidron and
joining two of them together results in a spidron which is the seahorse shape
of Figure 4.34(a).
Note that, since 2 2 2 + + + · · · = 2 · = 1, the sum of the areas
3 9 27 1−1/3
of the sequence of triangles following an equilateral triangle in a spidron is
equal to the area of the equilateral triangle itself. In other words, an entire
semi-spidron was lurking within the original equilateral triangle, waiting to be
released! Systems of such spidrons are notable for their ability to generate
beautiful tiling patterns in two dimensions (Figure 4.34(b)) and, when folded,
splendidly complex polyhedral shapes in three dimensions (Figure 4.34(c)).
They were invented in 1979 by graphic artist Dániel Erdély as part of a homework
assignment for Ernö Rubik’s (of Rubik’s Cube fame) Theory of Form class
at Budapest University of Art and Design. Possible practical applications of
spidrons include acoustic tiles and shock absorbers for machinery. [242].
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