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April 10

Largest regular hexagon inscribed in a square

What is the size of the largest regular hexagon that can be inscribed in a unit square? This is not a homework question, I'm trying to machine a hex bolt out of a square bolt[1]. Google got me nothing.Dncsky (talk) 02:02, 10 April 2013 (UTC)[reply]

I think its going to be one at 15° this touches an axis-aligned square at 4 points. A vertex of a hexagon with side length r will be at (r cos(15°),r sin(15°)). Now

\cos {\frac {\pi }{12}}=\cos 15^{\circ }={\tfrac {1}{4}}{\sqrt {2}}({\sqrt {3}}+1)\approx 0.9659258263\,

.Exact trigonometric constants So the max side length fitting inside a unit square will be 0.5/0.9659258263≈0.5176380902. The vertices will be (0.5,0.134),(0.134,0.5),(-0.366,0.366),(-0.5,-0.134),(-0.134,-0.5),(0.366,-0.366).--Salix (talk): 04:45, 10 April 2013 (UTC)[reply]

Compare this to a hexagon at 0° which has side length 0.5.--Salix (talk): 05:19, 10 April 2013 (UTC)[reply]

Thanks a lot!Dncsky (talk) 08:04, 10 April 2013 (UTC)[reply]

I agree with Salix - it's the one rotated by 15 degrees. 96.46.201.254 (talk) 04:28, 11 April 2013 (UTC)[reply]

Convergent or Divergent ?

$\lim _{n\to \infty }{\int _{0}^{\infty }{\underbrace {x^{{\color {Red}-}x^{\cdot ^{\cdot ^{x}}}}} _{{\color {Red}2}n}\ dx}\ =\ ?}$

For 2n = 2 we have 1.995⁺. For large values of 2n (up to about 50) we have 1.91⁺. But whether it actually converges (either to 0, or to a value close to 2) or diverges (towards infinity, or because of perhaps possessing an oscilant nature) is beyond me... — 79.113.210.163 (talk) 10:29, 10 April 2013 (UTC)[reply]

Unless I'm misunderstanding your notation, this doesn't converge for n > 1. I assume that for n = 2 the integrand would be x^(-x)^x^(-x). 70.162.4.242 (talk) 16:48, 10 April 2013 (UTC)[reply]

You are misunderstanding my notation. There's a single minus sign there. — 79.113.210.163 (talk) 17:38, 10 April 2013 (UTC)[reply]

In that case it still doesn't converge. 70.162.4.242 (talk) 17:52, 10 April 2013 (UTC)[reply]

The minus sign is not inside any paratheses. — 79.113.210.163 (talk) 18:15, 10 April 2013 (UTC)[reply]

This takes me back to old times. The function $x^{\cdot ^{\cdot ^{x}}}$ is known as tetration, and it has quite remarkable properties. As the number of x's goes to infinity:

If $x<e^{-e}$ , in the limit it oscillates between two asymptotes.
If $e^{-e}<x<e^{1/e}$ , it converges.
If $e^{1/e}<x$ , it diverges.

The result is that if you have an even number of x's, your integral actually does converge. Looie496 (talk) 18:25, 10 April 2013 (UTC)[reply]

I knew that too, I just can't see the connection. — 79.113.210.163 (talk) 19:03, 10 April 2013 (UTC)[reply]

It converges to 1. DTLHS (talk) 07:33, 12 April 2013 (UTC)[reply]

Except I shall see in his hands the print of the nails, and put my finger into the print of the nails, and thrust my hand into his side, I will not believe... — 79.113.224.31 (talk) 12:59, 12 April 2013 (UTC)[reply]

EXPONENTS AND NEGATIVE NUMBERS, Why is (-6)^2=36 but -6^2=-36 ??

I really don't understand. Aren't they both just -6x-6x-6x-6x-6x-6x ? And if so aren't both answers going to be positive for their being an even number of factors? How does the parenthesis change the answer? Please help me understand this, thank you. — Preceding unsigned comment added by 71.142.71.205 (talk) 17:51, 10 April 2013 (UTC)[reply]

See Order of operations in -6^2 the minus sign is applied last so -6^2 is interpreted as -(6^2)=-(36).--Salix (talk): 17:59, 10 April 2013 (UTC)[reply]

... also, you've misunderstood the power of 2 (squared). (-6)^2 is just "(-6) times (-6)". You are correct that any negative number raised to any even power will give a positive answer. Dbfirs 08:33, 11 April 2013 (UTC)[reply]