Pentagonal Geometry and the Golden Ratio

• A Brunnian link. Cutting 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. However since each pair of links crosses four times, it can't be drawn with circles.

• Constructing a regular pentagon inscribed in a circle, by straightedge and compass. Scott Brodie.

• Cut-the-knot logo. With a proof of the origami-folklore that this folded-flat overhand knot forms a regular pentagon.

  

• The downstairs half bath. Bob Jenkins decorated his bathroom with ceramic and painted pentagonal tiles.

• Equilateral pentagons. Jorge Luis Mireles Jasso investigates these polygons and dissects various polyominos into them.

• Equilateral pentagons that tile the plane, Livio Zucca.

• The golden ratio in an equilateral triangle. If one inscribes a circle in an ideal hyperbolic triangle, its points of tangency form an equilateral triangle with side length 4 ln phi! One can then place horocycles centered on the ideal triangle's vertices and tangent to each side of the inner equilateral triangle. From the Cabri geometry site. (In French.)

• The golden section and geometry. Somehow leading to questions like how many stars there are on the US flag.

• Golden rectangles, Curtis McMullen.

• A golden sales pitch. Julie Rehmeyer dissects the myth of the golden ratio in classical art and describes some new uses for it in commerce.

• The golden section and Euclid's construction of the dodecahedron, and more on the dodecahedron and icosahedron, H. Serras, Ghent.

• Golden spiral flash animation, Christian Stadler.

• How many points can one find in three-dimensional space so that all triangles are equilateral or isosceles? One eight-point solution is formed by placing three points on the axis of a regular pentagon. This problem seems related to the fact that any planar point set forms O(n^7/3) isosceles triangles; in three dimensions, Theta(n³) are possible (by generalizing the pentagon solution above). From Stan Wagon's PotW archive.

• How to construct a golden rectangle, K. Wiedman.

• Isotiles, workbook on the shapes that can be formed by combining isosceles triangles with side lengths in the golden ratio.

• Jiang Zhe-Ming's geometry challenge. A pretty problem involving cocircularity of five points defined by circles around a pentagram.

• Largest 5-gon in a square, or more interestingly smallest equilateral pentagon inscribed in a square. Posting to sci.math by Rainer Rosenthal.

• Lattice pentagons. The vertices of a regular pentagon are not the subset of any lattice.

• Miquel's pentagram theorem on circles associated with a pentagon. With annoying music.

• Museum of harmony and golden section.

• Number patterns, curves, and topology, J. Britton. Includes sections on the golden ratio, conics, Moiré patterns, Reuleaux triangles, spirograph curves, fractals, and flexagons.

• Parallel pentagons. Thomas Feng defines these as pentagons in which each diagonal is parallel to its opposite side, and asks for a clean construction of a parallel pentagon through three given points. (He is aware of the obvious reduction via affine transformation to the construction of regular pentagons, but finds that non-elegant.)

• Penrose mandala and five-way Borromean rings.

• Penrose tilings. This five-fold-symmetric tiling by rhombs or kites and darts is probably the most well known aperiodic tiling.

• Perplexing pentagons, Doris Schattschneider, from the Discovering Geometry Newsletter. A brief introduction to the problem of tiling the plane by pentagons.

• Pentagon packing on a circle and on a sphere, T. Tamai.

• Pentagonal Tessellations. John Savard experiments with substitution systems to produce tilings resembling Kepler's.

• Pentagons that tile the plane, Bob Jenkins. See also Ed Pegg's page on pentagon tiles.

• The pentagram and the golden ratio. Thomas Green, Contra Costa College.

• Russian math olympiad problem on lattice points. Proof that, for any five lattice points in convex position, another lattice point is on or inside the inner pentagon of the five-point star they form.

  

• A Venn diagram made from five congruent ellipses. From F. Ruskey's Combinatorial Object Server.

  

• Wonders of Ancient Greek Mathematics, T. Reluga. This term paper for a course on Greek science includes sections on the three classical problems, the Pythagorean theorem, the golden ratio, and the Archimedean spiral.