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By David Wells

A better half quantity to the author's "Dictionary of Curious and engaging Numbers", which specializes in mathematics and quantity thought. The entries during this booklet disguise curves, topology, tilings and all branches of aircraft and three-d geometry, from Euclid to fractals.

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They form a rectangle. IfP, Q, Rand S are the mid-points of the arcs AB, BC, CD and DA, then the sides of the rectangle are parallel to P Rand Q S, and P Rand Q S meet at the centre of the rectangle. If the excentres of the same four triangles are added, then together with the incentres, they form a rectangular 4 x 4 grid of 16 points. The centroids of the same four triangles form a quadrilateral similar to the original, as do their four nine-point centres. The four orthocentres form a quadrilateral congruent to the original.

It can be seen in many ways, for example as cross-pieces rotated about the vertices of a square grid, their free ends joined by short segments, or as two identical tessellations of elongated hexagons, overlapping at right angles. The latter suggests that the Cairo tile has many different forms, depending on the shape of the overlapping hexagons. Its dual tessellation, formed by joining the centre of each tile to the centre of every adjacent tile, is a semiregular tessellation of squares and equilateral triangles.

Continue the series to infinity and then add them all up. The result is the blancmange curve, which is continuous but does not have a tangent anywhere. The first four stages in its construction are shown below. In each figure but the first, the bold line is the sum of the previous stage and the new zigzag. BLASHKE'S THEOREM • 17 The fifth step shows the blancmange shape more prominently; by the eighteenth step it is difficult to distinguish the curve from its appearance after an infinite number of steps: The figure below shows another property of the blancmange curve.

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