Did you know that every sufficiently smooth function on an interval can be expressed as an infinite sum of sines...
Continue reading...analysis
Volume of a Cone in N Dimensions
One of the first geometric formulas we learn in plane geometry is that the area of a triangle is: Area of a...
Continue reading...Brouwer Fixed Point Theorem
One of the most useful theorems in mathematics is an amazing topological result known as the Brouwer Fixed Point Theorem....
Continue reading...Seven Shuffles
How many shuffles does it take to randomize a deck of cards? The answer, of course, depends on what kind...
Continue reading...Gamma Function
The Gamma function is an amazing integral: Gamma(x) = INTEGRALt=0 to infinity tx-1e-t dt . Using integration by parts, you can show...
Continue reading...Arclength Surprise
Consider a unit circle, and any arc S on the unit circle in the first quadrant. No matter where S is...
Continue reading...Rising Sequences in Card Shuffling
In Seven Shuffles we saw that it takes about 7 random riffle shuffles to randomize a deck of 52 cards. This means...
Continue reading...Gabriel’s Trumpet
The surface of revolution of y = 1/xfor x greater than 1, has finite volume but infinite surface area! Put another way, this...
Continue reading...e is irrational
If e were rational, then e = n/m for some integers m, n. So then 1/e = m/n. But the series expansion for 1/e is 1/e =...
Continue reading...Polyhedral Face Vectors
A polyhedron is any closed region of 3-space cut out by a finite set of planes. Take any polyhedron and do the...
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