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# Category Archives: complex numbers

## The Möbius function proof, part 2 (the subset parity lemma)

Continuing from my previous post, we are in the middle of proving that satisfies the same equation as , that is, and that therefore for all , that is, is the sum of all the th primitive roots of unity. … Continue reading

Posted in arithmetic, combinatorics, complex numbers, primes, proof
Tagged circle, complex, moebius, mu, primitive, proof, roots, sum, unit, unity
2 Comments

## Mystery curve, animated

As a follow-on to my previous post, here’s an animation (17MB) showing how the “mystery curve” arises as a sum of circular motions: Recall that the equation for the curve is . The big blue circle corresponds to the term—it … Continue reading

Posted in complex numbers, geometry, programming
Tagged animation, circles, complex, curve, graph, parametric, random, symmetry
6 Comments

## Random cyclic curves

Princeton Press just sent me a review copy of a new book by Frank Farris called Creating Symmetry: The Artful Mathematics of Wallpaper Patterns. It looks amazing and I’m super excited to read it. Apparently John Cook has been reading … Continue reading

Posted in complex numbers, geometry, programming
Tagged complex, curve, graph, parametric, random, symmetry
24 Comments

## Monday Math Madness #31

This week’s Monday Math Madness is a nice little problem involving complex exponentiation. Go check it out, and maybe win a prize!

Posted in challenges, complex numbers, links
Tagged complex, exponentiation, madness, math, monday

## 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

## Nuclear Pennies Game: Analysis

And now, for the promised analysis of the Nuclear Pennies Game! First, recall the rules of the game: there is a semi-infinite (i.e. with a beginning but no end) strip of squares, each of which can contain a stack of … Continue reading

Posted in algebra, complex numbers, games, proof
3 Comments