3464-Cube: Newsletter Challenge (March 2015)

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The answer depends on what you meant when you said 'generalized'.

In it's general form for higher dimensions Euler's Relation is written as V-E+F-C = 0, where V is the # of vertices, E is the # of edges, F is the # of faces, and C is the # of cells.

So in this case the answer is 0.

But if you meant 'the right side of the equation written as V-E+F = ?' (What is on the right hand side?) , then the answer is C, the number of cells.

The number of Cells (~= 3 dimensional cubes) is given by:

C = 2^(3464-3)[3464! / ((3464 - 3)! 3! )], where ! means factorial.

I can't calculate the exact value for C, since it exceeds 10³⁰⁷ ,which is the limit of what my electronics can compute. Let's just say there are a lot of cells.

2**3461 =73251242057016833184561729932351158717717202113918948613196577837026354009454257093494874698500631370420953025133511828815214216885473087979134165617315997488822605999290756925852594522105840603933315064656155558203499557160186049430537311287589564524381738848740911272278814005393524387191995726394644617460707542512180327052927724219040884985460886968460007925510274565857099584788471207601889254614717803825679164520928241381960010266912396656632930703687580528500937813981532024644024657487394345463636572758399105334182304325514047331974660322203648245155821913198871155356502472640694844666547647016863839520459663605704425867599681234890733555540063366112000175140029491117373566090393723739100953991871905262868463985212907477562753356495386007160436639964529335878202011562294712576813409415089087010300652683136390031943279480472045060604953964150915403728493932447914292019614638479598136237817008268159058785667207852624994258930258856169095854525842402431156799497690977168837931917288749623763052926411865497720953036913021952

3464!/(3464-3)!=41529570384

the remaining should be easy: I hope there's no misspelling!

or in scientific notation 3,85267650304*10**490

Can we form an orderly queue. I'd like to know how that was derived, not that I dispute it. Bcalc is a nice desktop utility (and free !!), but I forget how far it goes. That many digits, I'm thinking it maybe beem some home-spun coding.Total respect on doing the sum, but how did you do it ? A nasty sum like that can be broken down, but no way am I going there with so many clever sods around. Nasty big sums can often be broken down - I shall have to goad Roger Pink to refresh my mem. Honking big numbers, especially with expoments, can be cracked much easier than one would think. I'm not too fresh on numbers, but there is a clasical method of reducing them. He probably knows it off-hand, but if he doesn't then Roger will go mad over it. Gigantic sums can be reduced very quickly. This is so much fun, I'll dig out the book if need be.

there's a simple way to calculate it:

double 1 3461 times; but there's an other simple way: use a pocket calculator (please no name of manufacturer)

lol . Somebody OT'd you, so I negated the vote with a GA.

I believe that your formula of V-E+F-C = 0 is the case for 4 dimensions only. Higher dimensions introduce more terms in the equation , ie, at 6 dimensions you get V-E+F-C+D-G = 0 (I just made up the last two letters not sure what actual notation is for these). Regardless as long as it is an even number of terms in the equation the result is 0. Since the question refers to a dimension order that is even the Euler Characteristic is 0.

https://people.math.osu.edu/fiedorowicz.1/math655/HyperEuler.html

This link ^ implies that V-E+F-C = 0 is the correct formula for all higher dimensions.

The link gives that formula for the specific 4 dimensional case. The table at the bottom shows the case for higher dimensions and gives the characteristice as the sum from dimension 0 to dimension n-1 giving a result of 1 - (-1 to power of n) which is 0 for even n and 2 for odd n.

The thread's title uses 3463; the body uses 3464. Nor is it clear whether the generalization is limited to V-E+F, or goes on to include cells and other higher-dimensional subunits.

I agree that the title creates an ambiguity but to me it doesn't make sense to talk about higher dimensions and then only count the numbers of components for the first 3. An n dimensional object will have elements from dimension 0 (vertices) to dimension n-1. The question already gives examples of these where it gives cells (3 dimensional objects) for a 4 dimensional cube. The Euler characterisric when expressed as being the sum of the (even dimensioned elements) - (sum of odd dimension elements) for a multi dimensional cube gives an answer of 2 if it is an even dimensioned cube and 0 for an odd dimensioned cube.

The example of the 4 dimension cube is given in the question and shows the even dimensuioned elements as being 16 vertices (dimension 0 which considered even in this case) + 24 faces (dimension 2) =40. The odd dimsnioned elements are 32 edges (dimension 1) + 8 cells(dimension 3) = 40. So euler chracteristic is 0 for the 4 dimensional cube just as it was for the 2 dimensional case (square). A 5 dimensional cube has 32 vertices, 80 edges, 80 faces, 40 cells, and 10 4 dimensional elements. Doing the same calculation gives a Euler Characteristic of 2 just as in the case of the cube.

I agree, and also with the ordering of terms to produce alternating signs, as indicated in post 1 via the (-1)ⁱ factor.

There was an error in the title, which has since been corrected. The correct title is 3464-Cube.

V+F-E=2...

for 2 cube using the Eular unknown is Faces 4+f-4=2.. F=2.. check..

for 3 cube..unknown is Cell.. use formula again.. 8+6-12=2.. check

for 3 cube 16+24-32=8.. CHECK.. ALL VARIABLES WERE ALREADY GIVEN..

the right side is #cells..

the answer is CELLS..

keep in mind its only asking what the right side is equal to..

keeping it simple on this one!!!!!

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