June 2020 - mathfun@mailman.xmission.com

[math-fun] Rolling polyhedra
by James Propp 26 Dec '20

26 Dec '20

I want to think about rolling polyhedra, and the polyhedra I want to think about rolling are polyhedra in which the sum of the angles at each vertex is a submultiple of 360 degrees. For instance, given any triangle T, we can create a tetrahedron whose four faces are all congruent to T, with angle-sum 180 degrees at each vertex (there's a name for such tetrahedra but I forget what it is). Are there other polyhedra in which the angles at each vertex sum to a submultiple of 360 degrees (not necessarily the same one at each vertex)? Jim Propp

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[math-fun] Chess variant
by David Wilson 13 Nov '20

13 Nov '20

In Paul Morphy's "Opera House" chess game, Morphy never moves a piece backward. I propose Opera House Chess, with initial position and moves as in chess, except pieces cannot move backward. Is it playable? With victory conditions modified?

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[math-fun] One parameter quintic solution
by Bill Gosper 01 Nov '20

01 Nov '20

This is probably ho-hum, but while trying to simplify a Chowla-surdberg, I found that 0 = a + 80 x + 20 x^3 + x^5 has solutions {((-a - Sqrt[4096 + a^2])^(1/5) E^((2 I \[Pi])/5))/2^( 1/5) + ((-a + Sqrt[4096 + a^2])^(1/5) E^((2 I \[Pi])/5))/2^( 1/5), ((-a + Sqrt[4096 + a^2])^(1/5) E^(-((2 I \[Pi])/5)))/2^( 1/5) + ((-a - Sqrt[4096 + a^2])^(1/5) E^(-((4 I \[Pi])/5)))/2^( 1/5), ((-a - Sqrt[4096 + a^2])^(1/5) E^(-((2 I \[Pi])/5)))/2^( 1/5) + ((-a + Sqrt[4096 + a^2])^(1/5) E^(-((4 I \[Pi])/5)))/2^( 1/5), (-a + Sqrt[4096 + a^2])^(1/5)/2^( 1/5) + ((-a - Sqrt[4096 + a^2])^(1/5) E^((4 I \[Pi])/5))/2^( 1/5), (-a - Sqrt[4096 + a^2])^(1/5)/2^( 1/5) + ((-a + Sqrt[4096 + a^2])^(1/5) E^((4 I \[Pi])/5))/2^(1/5)} Is it from a two-parameter family? C.f. https://en.wikipedia.org/wiki/Quintic_function#Other_solvable_quintics --rwg

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Re: [math-fun] FactorInteger
by Bill Gosper 20 Aug '20

20 Aug '20

Slight improvement: Mathematica's front end by default displays a centerdot for implicit multiplication. Strangely, this doesn't work in old the FI function below, which displays blanks. Blanks also (by default) separate digits in blocks of 3. So the display is sometimes ambiguous. But Wolfram CE Harry Calkins showed me this workaround: FI⨉[n : _Rational | _Integer] := Defer @@ {FactorInteger@n} /. {{a_Integer, 1} -> a, {a_Integer, b_} -> a^b} /. {a_} -> a /. List -> Times E.g. In[144]:= FI⨉@BernoulliB@50]@// tim During evaluation of In[144]:= 0.006522,1 (seconds) Out[144]= (5^2 417202699 47464429777438199)/(2 3 11) (The first Bernoulli number with a square factor!) Dirty trick. I used a tiny ⨉ symbol instead of x, so you need Front End completion to input FI⨉. You may want to edit the definition. Instead of little centerdots between the factors, there's a Preferences variable Appearances|Numbers|Multiplication|Automatically display multiplication symbol between multiplied numbers Use symbol: which you can bash to be a mini ⨉. Sorry, I don't know how to bash it under program control, as in an init file. Unfactor is way nicer than with FactorInteger: UnFI⨉@Defer[x_] := x In[153]:= UnFI⨉@%144 Out[153]= 495057205241079648212477525/66 —rwg On Wed, Jan 4, 2017 at 12:44 AM Bill Gosper <billgosper(a)gmail.com> wrote: > There are > > In[14]:= 52! > Out[14]= > 80658175170943878571660636856403766975289505440883277824000000000000 > > ways to shuffle a deck. > That's just a "little" over a googol ... > > In[366]:= Log[10.^100, %] > Out[366]= 0.679066483922048 > > to the 2/3 power. The square of the cube root. And vice versa. So if > anybody asks how many > ways to shuffle a deck, "Oh, about the cube root of a googol squared." > But this is actually low > by a fairly substantial factor of > In[372]:= (10.^100)^(.679 - 2/3) > > Out[372]= 17.1132830416181 > > The actual number of shufflings is enormously composite: > > In[15]:= FactorInteger@%14 > Out[15]= > {{2,49},{3,23},{5,12},{7,8},{11,4},{13,4},{17,3},{19,2},{23,2},{29,1},{31,1},{37,1},{41,1},{43,1},{47,1}} > > But this is ugly and hard to read. The doc suggests > > In[16]:= CenterDot@@(Superscript@@@%) > Out[16]= > 2^49\[CenterDot]3^23\[CenterDot]5^12\[CenterDot]7^8\[CenterDot]11^4\[CenterDot]13^4\[CenterDot]17^3\[CenterDot]19^2\[CenterDot]23^2\[CenterDot]29^1\[CenterDot]31^1\[CenterDot]37^1\[CenterDot]41^1\[CenterDot]43^1\[CenterDot]47^1 > > This actually looks neat. Except for the ^1s. Try it. But it's unclear > (actually, a nice little Mathematica exercise) how to get your original > number back. > So I wrote something called FI > > In[364]:= FI[52!] > Out[364]= 2^49 3^23 5^12 7^8 11^4 13^4 17^3 19^2 23^2 29 31 37 41 43 47 > > In[365]:= ReleaseHold@% > Out[365]= > 80658175170943878571660636856403766975289505440883277824000000000000 > > Here is the definition: > FI[n : _Rational | _Integer] := > Times @@ ((HoldForm @@ {#1[[1]]})^#1[[2]] &) /@ FactorInteger[n] > > And here's a spoiler for how to invert FactorInteger: > > In[367]:= Times @@ (Power @@@ FactorInteger[73 137]) > Out[367]= 10001 > > which also works on that CenterDot expression: > > In[368]:= Times @@ (Power @@@ %16) > > Out[368]= > 80658175170943878571660636856403766975289505440883277824000000000000 > > Finally, you might enjoy finding a fairly nice formula for how many 0s are > on the end of n!, and, > more generally, what are exponents of the various prime factors. > --Bill >

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[math-fun] How does a dull spoon manage to cut a stringy piece of meat
by Bill Gosper 14 Jul '20

14 Jul '20

while sliding back an forth against the inside of a slippery bowl? --rwg

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[math-fun] triadic dragon reminiscence
by Bill Gosper 11 Jul '20

11 Jul '20

Half a triadic dragon <http://gosper.org/semitriadic.png> self-similarly trisects into knobby canes (blue, orange+green, red+purple). But the whole triadic dragon self-similarly trisects, so two thirds of this half-dragon (blue+orange+green) is itself a triadic dragon. Deleting the blue cane leaves a (not self similar) congruently bisected backwards Z, orange+green, red+purple. But it is also in fact self-similarly trisected! (orange, green+red, purple). The Triadic Dragon joining 0 to 1 has bounding box West: -1/16, East: 17/16 (with index 239(!)/240), and N|S bounds ±9*i*/(16√3) (less certain here). Is there a plane figure that can be dissected into two, as well as three, but not six congruent pieces? —rwg

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[math-fun] Counting vs. cross-products vs. tree traversal
by Steve Witham 11 Jul '20

11 Jul '20

Counting in decimal is analogous to walking a tree.Â A tree of unbounded depth for that matter.Â "Carrying" is like popping the stack and going to another [[great...] grand] parent node.Â The higher digits are further *up* the tree and it just keeps going up... I'm getting hypoxia just thinking about it. Grade-school kids (me included) eventually learn to count without a lot of thinking. In contrast I find tree-traversals (usually implicit) one of the most awkward tasks in programming.Â I mean depth-first.Â Even when the number of levels is fixed, even when it's shallow enough to use recursion in a language with a limited stack.Â The bookkeeping, the program structure are just awkward. Even in Prolog (where depth-first tree search is everything) it might make a program a bit scattered-looking.Â Well let's try. (I'm making up a dialect of Prolog here. "(car : cdr)" shall mean cons.) Â Â Â (Decimal (d))Â ifÂ (Digit d). Â Â Â (Digit d)Â ifÂ (Element d (0 1 2 3 4 5 6 7 8 9)). Â Â Â (Decimal (d : n)) if (NZDigit d) and (ZDecimal n). Â Â Â (NZDigit d)Â ifÂ (Element d (1 2 3 4 5 6 7 8 9)). Â Â Â (ZDecimal (d))Â ifÂ (Digit d). Â Â Â (ZDecimal (d : n))Â ifÂ (Digit d) and (ZDecimal n). But all that aside, the digit-counting analogy sounds like a nice way to organize a tree search or a very-nested loop.Â Have an array of digit positions, and at each position a sub-array of possibilities.Â Then have a parallel array of indexes into those arrays.Â So, each time through the loop you increment the bottom digit with carry.Â Easy peazy, heh. (I'm currently writing a test for a module with a big cross-product of test cases.) Â --Steve

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[math-fun] A different kind of "square square"
by James Propp 03 Jul '20

03 Jul '20

The title of Eric's post gave me an idea for a different game to play with numbers. The prime p=101030101 could be called a "square prime" (in base ten) because the number of decimal digits of p is a perfect square AND if you write the digits of p in the form of a perfect square you get a pattern with all the symmetries of a square: 1 0 1 0 3 0 1 0 1 There are 44 square primes with 9 digits: 101030101, 101060101, 111181111, 111191111, 121272121, 121282121, 131333131, 141484141, 141494141, 151545151, 151585151, 161636161, 171767171, 181888181, 191939191, 191969191, 303050303, 313111313, 323222323, 323232323, 363646363, 363676363, 363686363, 373717373, 383828383, 383838383, 383888383, 727212727, 727272727, 727282727, 757545757, 757585757, 777767777, 797939797, 797949797, 909070909, 919141919, 919171919, 919191919, 929292929, 959555959, 979797979, 989868989, 989898989 On the other hand, there are only two 9-digit "square squares": 121242121 = 11011^2 and 404090404 = 20102^2. There are no 16-digit square squares (probably for congruential reasons but I don't see what they are; do you?). The are six 25-digit square squares: 1020100000204020000010201 = 1010000000101^2, 1020102020205020202010201 = 1010001000101^2, 1020104040208020404010201 = 1010002000101^2, 1232120202329232020212321 = 1110009100111^2, 1232122422348432242212321 = 1110010100111^2, and 1232124642369632464212321 = 1110011100111^2. (Note the one whose square root is not a palindrome.) Are there infinitely many square squares? Jim Propp

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Re: [math-fun] Solar panel calculations
by Henry Baker 01 Jul '20

01 Jul '20

Yes, I've been to that 'sunroof' web site, but it doesn't give me any details, so I haven't the slightest idea what calculations it is actually performing. Furthermore, it doesn't know the pitch of my roof, unless it is doing some clever image-matching from different satellite angles. At 08:16 AM 6/29/2020, Thomas Colthurst wrote: >I did this! Well, I and a small team. google.com/sunroof > >It even does the integral over every pixel of your roof, so >as to take into account shading from nearby trees and houses (as >well as non-flat roofs, horizon effects, chimneys and other roof >obstructions, ...). > >You might also like NREL's PVWatts calculator <https://pvwatts.nrel.gov/>, >which is less >accurate (because it doesn't do the full ray-tracing over the roof >thing) but provides more intermediate information. > >-Thomas C > >On Mon, Jun 29, 2020 at 10:52 AM Henry Baker <hbaker1(a)pipeline.com> wrote: >> I'm considering putting a solar PV system on my roof, and I >> thought that I should be able to calculate most of the numbers >> myself from first principles. I have searched for so-called >> online 'calculators', but found that most of them are aimed at >> idiots. >> >> Here's what I expected to find: >> >> A formula giving the exact sun position throughout the year, >> given latitude and longitude, as well as a factor which >> handles the distance from the sun w.r.t. time. >> >> Now this only provides the *input* energy. >> >> I now need to overlay this with average *weather* data >> on a per-hour basis to get the expected net energy. >> >> Finally, I need to *integrate* these data over an >> entire year to get the values for my own house. >> >> There may be such an online calculator, but I haven't >> been able to find it yet. >> >> It may be possible to factor the weather info out >> of the integration, so that I could 'correct' my >> location sunlight & weather data with my own lat/lon >> azimuth (angle of my house) and roof angle, but I'd >> need to see the equations to find out if this factoring >> is possible. >> >> I presume that someone on this list has already done >> these calculations for their own home?

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[math-fun] Prime squares and square squares
by Éric Angelini 30 Jun '20

30 Jun '20

Hello Math-Fun, today we'll look for « prime squares » and squares. Definition 1 A "prime square" is a prime which is the sum of 4 integers. Those integers must occupy the 4 vertices of a geometric square formed by 4 cell-centers properly chosen on an infinite grid A (the 4 integers hereunder in the upper left corner, for instance, sum up to 11, which is prime). The type-A infinite grid is an infinite sector of the plane where the cells are labeled by the successive anti-diagonals like below: +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | 1 | 3 | 6 | 10| 15| 21| 28| 36| 45| 55| 66| 78| 91|105|120| | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | 2 | 5 | 9 | 14| 20| 27| 35| 44| 54| 65| 77| 90|104|119| | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | 4 | 8 | 12| 19| 26| 34| 43| 53| 64| 76| 89|103|118| | | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | 7 | 12| 18| 25| 33| 42| 52| 63| 75| 88|102|117| | | | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | 11| 17| 24| 32| 41| 51| 62| 74| 87|101|116| | | | | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | 16| 23| 31| 40| 50| 61| 73| 86|100|115| | | | | | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | 22| 30| 39| 49| 60| 72| 85| 99|114| | | | | | | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | 29| 38| 48| 59| 71| 84| 98|113| | | | | | | | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | 37| 47| 58| 70| 83| 97|112| | | | | | | | | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | 46| 57| 69| 82| 96|111| | | | | | | | | | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | 56| 68| 81| 95|110| | | | | | | | | | | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | 67| 80| 94|109| | | | | | | | | | | | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | 79| 93|108| | | | | | | | | | | | | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | 92|107| | | | | | | | | | | | | | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ |106| | | | | | | | | | | | | | | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ Definition 2 A "square square" is a square which is the sum of 4 integers. Those integers occupy the 4 vertices of a geometric square formed by 4 cell-centers properly chosen on an infinite grid B (the four integers 6, 1, 4 and 5 hereunder sum up to 16, for instance, which is a square). The type-B grid is produced starting somewhere in the plane with 1 and labeling the successive cells clockwise — like the snail-path hereunder: +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | | | | | | | | | | | | | | | | | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | | |157|158|159|160|161|162|163|164|165|166|167|168|169| | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | | |156|111|112|113|114|115|116|117|118|119|120|121|122| | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | | |155|110| 73| 74| 75| 76| 77| 78| 79| 80| 81| 82|123| | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | | |154|109| 72| 43| 44| 45| 46| 47| 48| 49| 50| 83|124| | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | | |153|108| 71| 42| 21| 22| 23| 24| 25| 26| 51| 84|125| | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | | |152|107| 70| 41| 20| 7 | 8 | 9 | 10| 27| 52| 85|126| | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | | |151|106| 69| 40| 19| 6 | 1 | 2 | 11| 28| 53| 86|127| | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | | |150|105| 68| 39| 18| 5 | 4 | 3 | 12| 29| 54| 87|128| | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | | |149|104| 67| 38| 17| 16| 15| 14| 13| 30| 55| 88|129| | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | | |148|103| 66| 37| 36| 35| 34| 33| 32| 31| 56| 89|130| | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | | |147|102| 65| 64| 63| 62| 61| 60| 59| 58| 57| 90 131| | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | | |146|101|100| 99| 98| 97| 96| 95| 94| 93| 92| 91|132| | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | | |145|144|143|142|141|140|139|138|137|136|135|134|133| | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | | | | | | | | | | | | | | | | | | +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ (...snip -- see the rest here in full colors: https://bit.ly/2NGeAvU ) (1) Can you illustrate all geometric side-lengths of the "prime squares"? (2) Can you find all "square squares" constructible on a B grid? Best, É.

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