A sequence of integers is called graphical if there exists a graph with it as its degree sequence. A theorem by Erdős and Gallai characterizes which sequences are graphical, but gives no algorithm to explicitly construct such a graph. Can you construct it?
The folks at Pixar are widely known as some of the world's best storytellers and animators. They are perhaps less recognized as some of the most innovative math whizzes around. Pixar Research Lead Tony DeRose delves into the math behind the animations, explaining how arithmetic, trigonometry and geometry help bring Woody and the rest of your favorite characters to life.
With your mouse, drag data points and their error bars, and watch the best-fit polynomial curve update instantly. You choose the type of fit: linear, quadratic, cubic, or quartic. The reduced chi-square statistic shows you when the fit is good. Or you can try to find the best fit by manually adjusting fit parameters.
This virtual exhibition shows more than 200 mathematical situations. They give students opportunities to experiment, try out, make hypotheses, probe them, try to validate them but also the possibility to try to demonstrate and debate about mathematical properties.
Take a cube. Chop it into 3×3×3 = 27 small cubes. Poke holes through it, removing 7 of these small cubes. Repeat this process for each remaining small cube. Do this forever! The result is the Menger sponge.
What’s the volume of the Menger sponge? At each stage we remove 7/27 of the volume, so only 20/27 of the volume is left. As we repeat this forever, the volume drops to zero. So, the final volume is zero!
What’s the surface area of the Menger sponge? That’s a bit harder to work out, but it’s infinite!
What’s the dimension of the Menger sponge? That depends on what you mean by ‘dimension’.