![]() ![]() Rigby, Napoleon, Escher, and Tessellations, Mathematics Magazine, Vol. (The paper also included the tri-bar or Penrose triangle, which is constructed impossibly from three 90-degree angles: in 1961 Escher built his never-ending Waterfall using three of them. The third part is the content of Kiepert's theorem. The basis of these tessellations will be the regular polygons. (To see that in the applet, check both Show tessellation and Hint boxes.) We will use some of the same techniques used by Escher. The tessellation can be obtained from the Napoleon's tessellation we discussed elsewhere bu joining vertices of the equlateral triangles to their centers. The applet serves to illustrate the second part of the claim. A' and B' are two of the vertices of that equilateral triangle and since ΔA'B'C' is equilateral, C' is the third vertex. By Napoleon's theorem, their centers form an equilateral triangle. To prove the first part, consider Napoleon triangles ABC'', AB''C abd A''BC. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet. Example 1: Create a figure from a square that will tessellate a plane. Interesting tessellations may be formed beginning with a square or equilateral triangle. tessellation may be created using slides, flips, and turns. This applet requires Sun's Java VM 2 which your browser may perceive as a popup. tessellation A tessellation is a complete pattern of repeating shapes or figures that cover a plane leaving no spaces or gaps. (They are the same in the applet below.) In all likelihood, Escher had this in mind but did not mention in his Notes. For the theorem to work those orientations must be the same. Congruent copies of hexagon AC'BA'CB' can be used to tessellate the plane.Īs Rigby observed, the theorem was not stated accurately because no assumption had been made as to the orientation of the three 120° angles used in the construction nor the orientation of ΔA'B'C'.Let A be such that B'A = B'C and ∠CB'A = 120°. Let C be the point such that A'B = A'C and ∠CA'B = 120°. Let A'B'C' be an equilateral triangle and B any point. ![]()
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