Geometric shapes are not limited only to the figurative aspect, they can also play active roles, for instance, serving in microelectronics to build operational printed circuits such as: small inductors (magnified, fig. a below), resistors (fig. b) and capacitors (fig. c). (image taken from my book “Almanach du Mathématicien en Herbe“)
A math-magic article I wrote for the German magazine Zeit Wissen: with the 13 triangular and square pieces (fig. 1) it is possible to form two large squares shown in fig. 2. Though the second large square has an extra piece the dimensions of the squares seem to be the same! Can you explain why this is possible?
This puzzle is available as greeting cards from my online store.
This is one of my earliest color optical illusions. There is no yellow or green in the diamond shapes, just vertical black lines! (If you don’t believe it, use a eyedropper tool to check it.)
The simplest right triangle with rational sides (the longest side has a denominator of 45 digits!) and area 157, was found by Don Zagier in 1993.
Here is another geometrical Op Art of my creation: “Deep Blue” (2001). The yellowish scintillating rays you see in this picture are a construct of your brain. This work is available as prints from Saatchi Art gallery.
If a cyclic quadrilateral ( = with vertices lying on a common circle) has diagonals which are perpendicular, then the perpendicular to a side from the point of intersection of the diagonals will bisect the opposite side (AF = FD).
Here is an intriguing Roman crystal 20-sided die (icosahedron), used in fortune-telling, ca. 1st century AD.