Some Strange Sums
If someone told you to compute the value of the infinite sum \(Z = 1+2+3+4+\cdots\), you would tell them that it simply does not converge. But what if they insisted that you assign a finite value to...
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Archimedes’ Circular Reasoning
Every geometry textbook has formulas for the circumference (\(C = 2 \pi r\)) and area (\(A = \pi r^2\)) of a circle. But where do these come from? How can we prove them? Well, the first is more a...
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Slicing Spheres
Last week we saw how to compute the area of a circle from first principles. What about spheres? To compute the volume of a sphere, let’s show that a hemisphere (with radius \(r\)) has the same volume...
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Spherical Surfaces and Hat Boxes
To round off our series on round objects (see the first and second posts), let’s compute the sphere’s surface area. We can compute this in the same way we related the area and circumference of a circle...
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A Slice of Interdimensional Sponge Cake
The Menger sponge is a delightfully pathological shape to soak up. Start with a cube of side-length 1, and “hollow it out” by slicing it into 27 sub-cubes (like a Rubik’s Cube) and removing the 7...
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Seeing Stars
Let us now take a closer look at the hex-fractal we sliced last week. Chopping a level 0, 1, 2, and 3 Menger sponge through our slanted plane gives the following: Studying the slicings of various...
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