Category Archives: geometry

Area paradox unmasked

In my last post I presented a paradox, where a set of four pieces forming an 8×8 square could apparently be rearranged to form a 5×13 rectangle, summoning an extra unit of area out of thin air. Quite a few … Continue reading

Posted in geometry, pictures, puzzles, solutions | Tagged , , , | 1 Comment

An area paradox

Here’s a fun paradox which has been around for quite a while and was apparently a favorite of Lewis Carroll. As you can verify for yourself, the two figures above are composed of two different rearrangements of the same four … Continue reading

Posted in challenges, geometry, paradox, pictures | Tagged , , | 18 Comments

Penn Alexander math club: map coloring

Today in math club I had the students explore map coloring. I tried to leave it as open-ended as possible to start—I just said that we were going to draw maps with countries, and try to give each country a … Continue reading

Posted in geometry, pattern, puzzles, teaching | Tagged , , , | 4 Comments


Just a link today: Mike Croucher over at Walking Randomly has some gorgeous pictures of fractal constructions called “pentaflakes”, made by recursively gluing pentagons together in various ways. He’s also made a Mathematica demonstration for playing around with various sorts … Continue reading

Posted in fractals, geometry, links | Tagged , , , | 1 Comment

Sine of an inscribed angle

Did you know that the ratio between the side of any triangle and the sine of the opposite angle is equal to the diameter of the triangle’s circumcircle? I didn’t! I just learned it today when researching the law of … Continue reading

Posted in geometry, proof | Tagged , , , , | 12 Comments

Carnival of Mathematics #23: Haiku Edition

Welcome to the 23rd Carnival of Mathematics: Haiku Edition! First, I must apologize for the delay: I usually have very little trouble with my hosting provider, but of course it went down just when the CoM was supposed to be … Continue reading

Posted in algebra, books, calculus, challenges, counting, fractals, geometry, infinity, links, meta, number theory, pascal's triangle, pattern, people, sequences, trig, video | 17 Comments

Video: The Story of Pi

A fun video about , including a nice visual explanation of the formula for the area of a circle, and some interesting places that shows up where you might not expect it to. The video was made by Tom M. … Continue reading

Posted in famous numbers, geometry, links, video | 5 Comments

Video: Möbius transformations revealed

For your viewing pleasure, a fantastically beautiful video about Möbius transformations, which are functions of the form where z, a, b, c, and d are complex numbers, and . For example, is a Möbius transformation with b=2, c=1, and a=d=0. … Continue reading

Posted in complex numbers, geometry, video

Geometric multiplication: an explanation

Now for an explanation/proof of that weird method of multiplication that I talked about in a previous post. You’ll recall that it’s really quite simple: to multiply a and b, draw a line from to and see where it crosses … Continue reading

Posted in algebra, geometry, proof

Geometric multiplication

I’ve just learned about a fairly useless, yet utterly beguiling method for performing multiplication, and I’d like to share it! Suppose you have two numbers you wish to multiply, call them a and b. Your first instinct is probably to … Continue reading

Posted in algebra, geometry, links, people | 4 Comments