February 2021 - mathfun@mailman.xmission.com

[math-fun] Fractions in primorial base
by Keith F. Lynch 02 Feb '21

02 Feb '21

I calculated fractions in the primorial base. Consecutive places after the point represent the product of the first n primes, i.e. 1/2, 1/6, 1/30, 1/210, 1/2310, etc. I only go through the 15th prime, 47, so that every number will fit in 64 bits. Fractions with a square-free denominator all terminate. All other fractions do not. The expansion of 1/4 appears to be A102781. Other expansions (e.g. 1/8, 1/9, 1/25) aren't in OEIS. As usual, A through Z represent 10 through 35. When I ran off the end of the alphabet, I started over with the lowercase alphabet; a through z represent 36 through 61. Here are the first few integer reciprocals. For more fractions, and the code that generated them, see http://KeithLynch.net/primorial.txt 1/2: 0.1 1/3: 0.02 1/4: 0.01235689BEFIKLN... 1/5: 0.011 1/6: 0.01 1/7: 0.0042 1/8: 0.003529CE5LN9AAZ... 1/9: 0.003238B67JACDSV... 1/10: 0.003 1/11: 0.00251 1/12: 0.00235689BEFIKLN... 1/13: 0.002219 1/14: 0.0021 1/15: 0.002 1/16: 0.001614EGEAR4PQf... 1/17: 0.00153B8 1/18: 0.0014745CF9KOREF... 1/19: 0.0014078I 1/20: 0.00135689BEFIKLN...

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[math-fun] Question about compact convex polyhedra in 3-space
by Dan Asimov 02 Feb '21

02 Feb '21

Suppose P is a compact convex polyhedron in 3-space, topologically equivalent to the sphere, with finitely many convex planar faces each bounded by a convex polygon with finitely many straight sides. Then every point x of P that is not a vertex has a neighborhood that is isometric to a neighborhood in the plane. And each vertex is locally isometric to a neighborhood of the vertex of a cone, with total angle around the vertex* equal to some theta in the open interval (0, pi). For the jth vertex, call this number theta_j. If this cone is approximated by a smooth surface then its unit normals translated to the origin will cover an area of the unit sphere equal to 2pi - theta. So if we assign the number 2pi - theta_j to the jth vertex, the Gauss-Bonnet theorem tells us that Sum (2pi - theta_j) = 4pi or equivalently Sum theta_j = 2pi(V-2) if V is the number of vertices.** Now suppose we have a finite number of regions P_j in the plane, each bounded by a polygon with straight sides, and that we are given a pairing of all the edges with edges of equal length so that, if the resulting abstract surface has V vertices, E edges, and F faces, the following two conditions are satisfied: 1) V - E + F = 2 (The Euler characteristic of the resulting surface is 2, from which we know that it must be a sphere.) and 2) The total angle about vertex j is theta_j in the interval (0, pi), such that Sum theta_j = 2pi(V-2). Question: --------- Does there then necessarily exist a convex polyhedron in 3-space that is isometric to the resulting abstract surface? —Dan ————— * The total angle of a cone can be seen if you slit the cone from the base to the vertex and flatten it out, it will fill a certain fraction f of a disk. Then its total angle is 2pi*f. ** E.g., for a cube V = 8 and each theta_j = 3pi/2, and indeed 8(3pi/2) = 2pi(8-2).

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[math-fun] The only explicit formula for the Heighway Dragon function,
by Bill Gosper 01 Feb '21

01 Feb '21

as far as I know, is my old Fourier approximation <http://gosper.org/explicittwindragon.pdf>, The formula approaches exactitude as the summation ranges over integers ever farther from 0 . I wrote this as a piece of evidence supporting my pitch to Wolfram Inc to at long last support (optionally infinite) MatrixProducts. (ContinuedFactionK would become a special case.) Product[m_?MatrixQ,...] unfortunately does elementwise rather than matrix multiplication. As I incessantly remind, Path Invariance <http://gosper.org/pathi.pdf> is a nice introduction to (2⨉2) iterated matrix products. —rwg

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[math-fun] Reciprocals of integers in binary
by Keith F. Lynch 01 Feb '21

01 Feb '21

I wondered what reciprocals of integers looked like in binary, so I computed them. I noticed that lots of them had first and second halves of the repeating section that were complements of each other. For instance 1/97 is 0.000000101010001110100000111111010101110001011111.... That's 0.000000101010001110100000 111111010101110001011111.... For which numbers is this the case? It turns out to be precisely those in http://oeis.org/A014657 (except the first two) times any power of two (including zero). And I see why.

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