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If so, it could be extended to more points via Helly's theorem. Andy Fingerhut asks, given a maximum (not minimum) matching of six points in the Euclidean plane, whether there is a center point close to all matched edges (within distance a constant times the length of the edge). Business card polyhedral origami.Ī factoid about similar triangles inspired by a trigonometric identity. Ronald Kyrmse investigates grid polygons in which all side lengths are one or sqrt(2). Is there a three-dimensional analogue? From MathSoft's favorite constants pages. , defining the unique non-equilateral triangle with three equally large inscribed squares.'The first six books of the Elements of Euclid, in which coloured diagrams and symbols are used instead of letters for the greater ease of learners.' The construction below achieves total length approximately 5.1547, but better bounds were previously known.

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What is the probability that a dropped needle lands on a crack on a hardwood floor? This is a set that intersects all lines through the unit disk. The truncated icosahedron recently acquired new fame and a new name when chemists discovered that Carbon forms molecules with its shape. However since each pair of links crosses four times, it can't be drawn with circles. Geometric sculpture by Stephen Luecking combining buckyball, hexagon, and amorphous shapes of carbon molecules.Ĭutting any one of five links allows the remaining four to be disconnected from each other, so this is in some sense a generalization of the Borromean rings. He asks if it has higher-dimensional generalizations. A 'Heron-type' formula for the maximum area of a quadrilateral, Col. The solution and proof are even prettier in four dimensions. What is the smallest cube that can be put inside another cube touching all its faces? There is a simple solution, but it seems difficult to prove its correctness. Pankaj Agarwal and Sandeep Sen ask for triangulations of convex polytopes in which the vertex or edge degree is bounded by a constant or polylog.

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Now included with Free 3 of 9 is a new version of the font called Free 3 of 9 Extended.įor more history and symbolism of the Borromean rings, see. Javan relates the history of the links and describes various generalizations with more than three rings. Dan Asimov discusses some related higher dimensional questions. Geoff Mess relates a proof that the Borromean ring configuration (in which three loops are tangled together but no pair is linked) can not be formed out of circles. Oklahoma State U.Ĭlass project displaying examples of the seven types of symmetry (frieze groups) possible for linear patterns in the plane. Kurchan asks, for each k, what is the smallest polyomino such that k copies can form a 'blocked' configuration in which no piece can be slid free of the others, but in which any subconfiguration is not blocked.

spirolateral gallery (requires JavaScript to view large images, and Java to view self-running demo or construct new spirolaterals).

Michael Trott animates morphs between a torus and a double-covered sphere, to illustrate their topological equivalence, together with several related animations. A recipe for making spiraling patterns in chemical reactions. Connelly had previously discovered non-convex polyhedra which are flexible (can move through a continuous family of shapes without bending or otherwise deforming any faces) these authors prove that in any such example, the volume remains constant throughout the flexing motion. Walz in Contributions to Algebra and Geometry, volume 38 (1997), No.1, 1-10. Rob Beezer makes truncated icosahedra out of old automobile tires. Durable bamboo models of the Platonic and Archimedean polyhedra, offered for sale.

Henry Baker's includes a, depicted below.

Primarily consists of Escher-like drawings but also includes an interesting section about Kepler's work on polyhedra. Vijay Raghavan points out an obscure reference to average case analysis of the Euclidean traveling salesman problem.

, Corinne Cerf extends previous lists of all small knots and links, to allow each component of the link to be marked by an orientation.

, structure formed for Expo 1958 in the form of nine spheres, representing an iron crystal.

3d ball-and-stick geometric construction kit.

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