The sides of a pentagon, hexagon, and decagon, inscribed in congruent circles, form a right triangle.

## The Arithmetic-Geometric Mean Inequality

A visual intuitive proof that **√ab** cannot be larger than **(a+b)/2**, where a, b ∈ R*+

## When a plane intersects a dodecahedron

A cross-section of the **dodecahedron** can be an equilateral triangle, a square, a regular pentagon, a regular hexagon (two ways), or a regular decagon.

## Target 10

Here is a little puzzle of our creation you can make with your kids or in class…

## Pascal’s Theorem

## Sangaku: Semicircle inscribed in a right triangle

Find the radius* r* of the semicircle inscribed in the right triangle below:

show solution

## Fibonacci Spiral Jigsaw Puzzle

Each piece of this puzzle is similar (the same shape at a different size). The placement of the pieces is based on the golden angle (≈137.5º), and results in a pattern frequently found in nature (phyllotaxis), for instance on sunflowers. The puzzle features 8 spirals in one direction, and 13 in the other. You can build your own Fibonacci spiral puzzle by following John Edmark’s tutorial.

## Transform a Ball with 2 Holes into a CD

Topology is a fascinating branch of mathematics that describes the properties of an object that remain unchanged under “smooth” deformations. If we imagine objects to be made of clay, a smooth deformation is any deformation that does not require the discontinuous action of a tear or the punching of a hole, such as bending, squeezing and shaping. These deformations are called “continuous deformations“. Continue reading “Transform a Ball with 2 Holes into a CD”

## Möbius Curiosities

An intriguing Möbius maze created by Dave Phillips. Did you notice that the arrows, which should indicate the direction to follow, are wrongly placed?

## Intriguing Geometric Dissections

Any two polygons with equal area can be dissected into a finite number of pieces to form each other. One of the most interesting dissection or geometric *equidecomposition* puzzles is that discovered by Harry Lindgren in 1951 (see Fig. 2 further below). As you know, in this kind of puzzles, the *geometric invariant* is the area, since when a polygon is cut and its pieces are distributed differently, the overall area doesn’t change.

Lindgren was the first to discover how to cut a dodecagon into a minimal number of pieces that could pave a square, when rearranged differently. His solution is very elegant, he first built a regular Euclidean pavement by cutting a dodecagon as shown in fig. 1.a, then arranging the four pieces symmetrically on the plane (fig 1.b). The tessellation achieved with these pieces corresponds, by superposition, to a regular paving of squares. The example in fig. 2 shows how the puzzle appears once finished.