No matter how you choose to place the red points on the circumference, the three blue points will lie on a straight line (shown in red). This works for any conic (which may be a circle, ellipse, parabola or hyperbola).

## 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.

## The Infinite Chocolate Bar

Although the principle of these kinds of “vanish puzzles” is really quite simple, they still confound countless numbers of puzzle enthusiasts!

**Related puzzles**

· The Area Paradox

· Math-Magic Vanishing Space

## Rolling Polygons

Imagine the wheels of your bike are polygons. Then, to ride smoothly the road should be made of ‘catenaries‘ (yes, those bumpy things).

It is the math professor Stan Wagon who first demonstrated this concept with a real square-wheel bike at Macalester College in St. Paul, Minnesota. Continue reading “Rolling Polygons”

## Linear to rotational motion

Intriguing linear motion perceived as circular motion! Watch as the black balls rotate in a circle, then focus on one ball at a time and you will notice that it follows a straight line. Also, watch at the moment when there are only four balls moving, it forms a rotating square between the four balls. This is just neat example of looking deeper into something so simple and discovering a hidden pattern.

**More**

Pattern with Arabesque paths moving in a linear fashion induces rotational motion to a hexagonal device.

## Haruki’s Theorem

Given 3 circles, each intersecting the other two in two points, the line segments connecting their points of intersection satisfy: ace/bdf = 1