Wikipedia:Reference desk/Archives/Mathematics/2023 October 23

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October 23

Venn diagram

If the Venn diagram of n sets can be drawn using n ellipses, what is the maximum value of n? 61.224.153.226 (talk) 04:25, 23 October 2023 (UTC)

If the ellipses are similar, the maximum value is 5 (Theorem 1 in reference [5] of the article Branko Grünbaum).  --Lambiam 08:33, 23 October 2023 (UTC)

The ellipses are not necessarily similar. 61.224.153.226 (talk) 08:35, 23 October 2023 (UTC)

I will need someone else to confirm, but based on the argument laid forth in this StackExchange answer on why you can't have a Venn diagram with four circles, and using the fact that two ellipses intersect in at most $4$ points, I believe that the maximum number of regions you can define with $n\geq 1$ ellipses is $2+\sum _{k=0}^{n-1}4k=2n^{2}-2n+2$. Naturally, the $2^{n}$ regions needed to define a Venn diagram on $n$ sets outpaces this amount after a certain point, which turns out to be $n\geq 6$, implying (hopefully) that $n=6$ is impossible. GalacticShoe (talk) 17:32, 23 October 2023 (UTC)

The proof of Theorem 1 in Grünbaum's article, while stated for a family of similar ellipses, actually does not use their similarity but merely that mutual intersections contain at most 4 points. If a family of $n$ curves whose intersectional cardinality is bounded by $j$ defines $2^{n}$ regions, the lemma on which the proof of his Theorem 1 is based states that

$j\geq {\frac {4(2^{n{-}1}\!-1)}{n(n-1)}}.$

To reach $n=6$, this requires $j\geq {\frac {124}{30}}>4.$  --Lambiam 07:56, 25 October 2023 (UTC)

Separated to n figures which are congruent

For which pairs of positive integers (m,n), a regular m-gon can be separated to n figures which are congruent?

Also, for which pairs of positive integer (m,n), the (2D) surface of the Platonic solid with m faces (m = 4, 6, 8, 12, 20) can be separated to n (2D) figures which are congruent? 61.224.153.226 (talk) 04:34, 23 October 2023 (UTC)

Any regular $m$-gon can be decomposed into $n$ congruent parts when $n\,{\mid }\,2m.$ It can also be decomposed into $2^{p}m$ congruent parts for any natural $p.$ The square can be decomposed into any number of congruent parts. I have not proved that this covers all cases.  --Lambiam 07:31, 23 October 2023 (UTC)

Are there any case known to NOT be able to decompose? 61.224.153.226 (talk) 08:37, 23 October 2023 (UTC)

This article by Michael Beeson shows that there are many more possibilities than those I mentioned above. Specifically, a regular $m$-gon can be decomposed into $m$ or $2m$ congruent triangles, and if a polygon can be decomposed into $n$ congruent triangles, it can be decomposed into $nk^{2}$ congruent triangles for any natural $k.$ The same article proves that no triangle can be decomposed into $7$ congruent triangles. It is a reasonable conjecture that this impossibility generalizes to $7$ congruent shapes.  --Lambiam 09:55, 23 October 2023 (UTC)

In addition to the cases already mentioned, there's a known decomposition of the triangle into 5 congruent (albeit disconnected) parts. GalacticShoe (talk) 12:45, 23 October 2023 (UTC)

Oh, also can't forget that it's trivial to dissect/slice a square into an arbitrary amount of congruent rectangles. GalacticShoe (talk) 05:53, 25 October 2023 (UTC)

The arbitrary decomposability of the square had already been mentioned.  --Lambiam 07:31, 25 October 2023 (UTC)

Oh whoops, I completely skipped over that part. GalacticShoe (talk) 13:53, 25 October 2023 (UTC)

This StackExchange post has some interesting results, including a decomposition of the triangle into 15 or 30 congruent parts. This StackExchange answer meanwhile mentions that Joseph Gallian Michael Reid apparently proved decomposition into pentiamonds for all triangles of length $5k$ when $k\geq 6$, implying $(3,5k^{2})$ (and also $(3,10k^{2})$ when you cut the pentiamond in half) for all $k\geq 6$. GalacticShoe (talk) 16:17, 25 October 2023 (UTC)

The person said there to have obtained this result is Michael Reid.  --Lambiam 20:43, 25 October 2023 (UTC)

Double whoops, I keep reading things too quickly for my own good. GalacticShoe (talk) 04:33, 26 October 2023 (UTC)

Can a(5), a(6), … in these sequences be written as expressions involving integers and the operations of addition, subtraction, multiplication, division, and the extraction of roots?

1. a(2) = Golden ratio x^2-x-1=0, a(3) = Plastic number x^3-x-1=0, a(n) = x^n-x-1=0
2. a(2) = Golden ratio x^2-x-1=0, a(3) = Supergolden ratio x^3-x^2-1=0, a(n) = x^n-x^(n-1)-1=0
3. a(2) = Golden ratio 1/x=x-1, a(3) = 1/x=(x-1)^2, a(n) = 1/x=(x-1)^(n-1)
4. a(2) = Golden ratio, a(3) = Tribonacci constant, a(4) = Tetranacci constant, a(5) = Pentanacci constant, a(6) = Hexanacci constant, …

The a(n) in these sequences requires to solve the algebraic equation with degree n, thus the a(n) in these sequences cannot be written as expressions involving integers and the operations of addition, subtraction, multiplication, division, and the extraction of roots if n>=5? 61.224.153.226 (talk) 04:43, 23 October 2023 (UTC)

The answer to what I think you are asking, is yes, they can be expressed that way. There is no closed-form solution for polynomials of degree greater than 4, though. Bubba73 ^{You talkin' to me?} 05:04, 23 October 2023 (UTC)

There is no general formula for the solution of algebraic equations of degree 5 or higher using radicals and the "Pentanacci constant" $1.965948236645485337...$ is indeed not expressible in radicals. But some specific instances of higher-degree equations may nevertheless be solvable with algebraic means. For example, we can solve the equation $x^{5}-2x^{3}-3x^{2}+6=0$ completely algebraically in radicals because its left-hand side can be factored into $(x^{2}-2)(x^{3}-3).$  --Lambiam 06:57, 23 October 2023 (UTC)

I remember that the solution of x^5-x-1 = 0 is not expressible in radicals, but the solution of x^5+x+1 = 0 is expressible in radicals, since x^5+x+1 can be factored. 61.224.153.226 (talk) 08:38, 23 October 2023 (UTC)

Indeed, $x^{5}+x+1=(x^{2}+x+1)(x^{3}-x^{2}+1).$  --Lambiam 08:55, 23 October 2023 (UTC)