April 2006 - mathfun@mailman.xmission.com

Re: [math-fun] favorite theorem
by Henry Baker 28 Apr '06

28 Apr '06

Some suggestions: Hamilton: ii=jj=kk=ijk=-1 (I guess he was left-handed...) Cantor: |R|>|Z| (diagonalization, which was later used by Godel) Polya's theory of counting Newton: solution of 2-body problem is a conic section Risch: integration in closed form Newton: "Newton's Method" sqrt, etc., widely applicable to lots of things. Church's Lambda Calculus (also "combinators") -- theory of substitution in trees. Tarski: decision procedure for real closed fields. At 09:08 AM 4/28/2006, Cordwell, William R wrote: >One of my students asked me what my favorite theorem was, a question that I found interesting and surprisingly difficult to answer. > >So...what's your favorite theorem or proof (and why)? > >--Bill C.

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[math-fun] Re: favorite theorem
by James Propp 28 Apr '06

28 Apr '06

Let me say, only half in jest, that my favorite theorem in math is the classification theorem for finite sets, i.e., the fact that finite sets are classified up to isomorphism by the counting numbers (including zero)! Jim Propp

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Re: [math-fun] Re: favorite theorem
by dasimov＠earthlink.net 28 Apr '06

28 Apr '06

Jim wroe: << Let me say, only half in jest, that my favorite theorem in math is the classification theorem for finite sets, i.e., the fact that finite sets are classified up to isomorphism by the counting numbers (including zero)! >> And I'll respond, half-seriously: How would you prove that? --Dan

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[math-fun] favorite theorem
by Cordwell, William R 28 Apr '06

28 Apr '06

One of my students asked me what my favorite theorem was, a question that I found interesting and surprisingly difficult to answer. So...what's your favorite theorem or proof (and why)? --Bill C.

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Re: [math-fun] favorite theorem
by dasimov＠earthlink.net 28 Apr '06

28 Apr '06

Bill Cordwell writes: << One of my students asked me what my favorite theorem was, a question that I found interesting and surprisingly difficult to answer. So...what's your favorite theorem or proof (and why) >> Great question. There's probably not one single favorite, but many I'm equally fond of. The question is slightly muddied for me by the fact that many theorems that were the cat's pajamas when I first learned them have become so familiar by now that their wonderfulness seems tarnished by time. But a few are * Real division algebras occur only in dimensions 1,2,4,8. * The only number fields Q(sqrt(-d)) whose ring of algebraic integers has unique factorization (for d > 0) are d = 1, 2, 3, 7, 11, 19, 43, 67, 163. * Bott periodicity, especially for O(oo), the limit of the isometry groups O(n) of R^n: (I.e., O(oo) = the union of all the O(n)'s with O(n) a subset of O(n+1).) Bott proved: The homotopy groups pi_k(O(oo)) depend only on k mod 8: k mod 8: 0 1 2 3 4 5 6 7 ____________________________________________________________ pi_k(O(oo)): Z_2 Z_2 0 Z 0 0 0 Z * Goedel incompleteness * Cantor: Infinitely many infinities * The 3-sphere S^3 admits infinitely-differentiable foliations by surfaces, but no real-analytic foliation. * The Chern-Gauss-Bonnet theorem: The (integral of the Gaussian curvature of a closed surface S) = 2pi * X(S) -- (where X(S) = the Euler characteristic of S) -- and its higher-dimensional analogues. * Classification of regular polytopes in all dimensions: dimension: 2 3 4 5 6 7 8 9 . . . ______________________________________________ # regular polytopes: oo 5 6 3 3 3 3 3 . . . ------------------------------------------------------------------------------ * Classification of surfaces under connected sum, as a commutative semigroup with 0: Two generators T, P with one relation: P + P + P = T + P. ------------------------------------------------------------------------------ * The kissing number in n dimensions (the maximum number of unit spheres that can all be simultaneously tangent to another one while all having disjoint interiors): Known cases to date: dimension: 1 2 3 4 8 24 _______________________________________________ kissing number: 2 6 12 24 240 196560 ------------------------------------------------------------------------------ --Dan

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Re: [math-fun] favorite theorem
by Jud McCranie 28 Apr '06

28 Apr '06

At 12:08 PM 4/28/2006, Cordwell, William R wrote: >One of my students asked me what my favorite theorem was, a question that >I found interesting and surprisingly difficult to answer. > >So...what's your favorite theorem or proof (and why)? Probably Godel's Incompleteness Theorem, because it is so interesting. (Not my favorite proof, though.)

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RE: [math-fun] Hero[n]'s formula
by Michael Reid 28 Apr '06

28 Apr '06

> If we have a right tetrahedron, where one of the corners > matches a cube's corner, there are three leg-faces and > one hypote-face. It turns out that the square of the > hypoteface area is equal to the sum of the squares of the > three legface areas. (I don't know why, or if it works in > more dimensions.) Any triangle can be a hypoteface, and > the lengths of the X,Y,Z legs are easy to calculate: > If the hypoteface edges are A,B,C, then X2+Y2 = A2, etc. > and X2 = (A2+B2-C2)/2. The legface areas are XY/2 etc., > the area squares are X2Y2/4, and it's not hard to check that > the sum of squares of the three legface areas matches Hero^2. it's already been pointed out that only non-obtuse triangles can occur as a "hypoteface". however, we can reduce to that case as follows. let K(a, b, c) = sqrt(s(s - a)(s - b)(s - c)) , which is heron's formula. then 16 K^2 = c^2 (2 a^2 + 2 b^2 - c^2) - (a^2 - b^2)^2 , which shows that K(a, b, c) = K(a, b, c') , where (c')^2 = 2 a^2 + 2 b^2 - c^2 . this corresponds to a simple geometric transformation: attach two (a, b, c) triangles along their c edge to make a parallelogram. then divide the parallelogram into two congruent (a, b, c') triangles using the other diagonal. the relationship c^2 + (c')^2 = 2 (a^2 + b^2) is the usual parallelogram law. it is easy to show that any obtuse triangle can be transformed into a non-obtuse triangle by a sequence of transformations of this ilk. (always attach along the edge opposite the obtuse angle; repeat as needed ... have patience ... you will get there!) mike

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RE: [math-fun] code
by Schroeppel, Richard 28 Apr '06

28 Apr '06

The third column looks proportional to the reciprocal of columns 1 (and 2). Roughly, 147000/column1. -- Rich -----Original Message----- From: math-fun-bounces+rschroe=sandia.gov(a)mailman.xmission.com [mailto:math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com] On Behalf Of Russ Cox Sent: Thursday, April 27, 2006 6:41 PM To: math-fun(a)mailman.xmission.com Subject: [math-fun] code (This is a serial dump from a USB measurement device.) <clip>

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[math-fun] code
by Russ Cox 27 Apr '06

27 Apr '06

(This is a serial dump from a USB measurement device.) I've been staring at this for a few hours. The first two columns are raw values (decimal and hex) and the third column is some kind of encoding of them. I wondered if maybe the third column was a 12-bit floating point format, but it's a little weird. Any ideas? Russ 61 0x3d 0x966 68 0x44 0x86c 98 0x62 0x5dc 99 0x63 0x5cc 100 0x64 0x5bd 101 0x65 0x5ad 105 0x69 0x57e 106 0x6a 0x56f 107 0x6b 0x55f 108 0x6c 0x54f 110 0x6e 0x540 112 0x70 0x521 114 0x72 0x511 115 0x73 0x501 118 0x76 0x4e2 119 0x77 0x4d2 123 0x7b 0x4b3 124 0x7c 0x4a4 126 0x7e 0x494 127 0x7f 0x484 129 0x81 0x475 131 0x83 0x465 137 0x89 0x436 139 0x8b 0x427 143 0x8f 0x407 150 0x96 0x3d8 152 0x98 0x3c9 157 0x9d 0x3aa 175 0xaf 0x34c 181 0xb5 0x32d 185 0xb9 0x31d 189 0xbd 0x30d 192 0xc0 0x2fe 196 0xc4 0x2ee 201 0xc9 0x2de 205 0xcd 0x2cf 210 0xd2 0x2bf 214 0xd6 0x2b0 219 0xdb 0x2a0 225 0xe1 0x290 230 0xe6 0x281 236 0xec 0x271 242 0xf2 0x261 248 0xf8 0x252 255 0xff 0x242 269 0x10d 0x223 278 0x116 0x213 286 0x11e 0x204 295 0x127 0x1f4 304 0x130 0x1e4 314 0x13a 0x1d5 325 0x145 0x1c5 336 0x150 0x1b6 349 0x15d 0x1a6 363 0x16b 0x196 377 0x179 0x187 1568 0x620 0x05e

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RE: [math-fun] A problem for the shower
by Cordwell, William R 27 Apr '06

27 Apr '06

Sorry, I was catching up on my EMail in chronological order, and didn't see that others had posted this. Bill C. -----Original Message----- From: math-fun-bounces+cordwell=sandia.gov(a)mailman.xmission.com [mailto:math-fun-bounces+cordwell=sandia.gov@mailman.xmission.com]On Behalf Of Cordwell, William R Sent: Thursday, April 27, 2006 12:52 PM To: math-fun Subject: RE: [math-fun] A problem for the shower I think that solid rocket motors try to achieve this, where the fuel has something of a star-shaped hole in the middle, and, as it burns, the amount of fuel exposed (surface area) stays roughly constant (so as to give a constant thrust). The hole gradually becomes "smoother", which keeps the surface area constant as the hole gets larger. Going backwards probably doesn't work, I would guess. Bill C. -----Original Message----- From: math-fun-bounces+cordwell=sandia.gov(a)mailman.xmission.com [mailto:math-fun-bounces+cordwell=sandia.gov@mailman.xmission.com]On Behalf Of David Wolfe Sent: Wednesday, April 26, 2006 10:27 AM To: math-fun Subject: [math-fun] A problem for the shower A question struck me in the shower this morning: Suppose you would like a piece of soap which dissolves at a uniform rate. I.e., as it gets smaller, the soap bar's surface area remains constant until it vanishes. I have convinced myself that one should be able to construct such a bar of soap with arbitrarily small holes. As the soap gets smaller, the holes are exposed to increase the surface area. Need it have holes? _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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