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.
