triangles, equilateral triangles, Persian Tile. Rigby, Napoleon, Escher, and Tessellations, Mathematics Magazine, Vol. A tessellation is created when a shape is repeated over and over again covering a. The third part is the content of Kiepert's theorem. (To see that in the applet, check both Show tessellation and Hint boxes.) 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. This applet requires Sun's Java VM 2 which your browser may perceive as a popup. (They are the same in the applet below.) In all likelihood, Escher had this in mind but did not mention in his Notes. In the picture on this page, the 'F's show that the shapes can be flipped and rotated without affecting the shapes' ability to tessellate. For the theorem to work those orientations must be the same. The 3 regular geometric shapes that tessellate are the equilateral triangle, square and hexagon. 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°. The bulk of Eschers tessellations are based on quadrilaterals, which the novice will find much easier to work with. Let A'B'C' be an equilateral triangle and B any point. Escher organizes his tessellations into two classes: systems based on quadrilaterals, and triangle systems built on the regular tessellation by equilateral triangles.
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