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April 16

Poncelet's closure theorem

Illustration of Poncelet's porism for n = 3, a triangle that is inscribed in one circle and circumscribes another.

By the n=3 case of Poncelet's closure theorem for circles, if the equation of Euler's theorem in geometry is satisfied, relating the inradius, circumradius, and distance between the incenter and circumcenter, then for given incircle and circumcircle, every point A on the circumcircle is a vertex of a triangle ABC having those circles. As we walk vertex A completely around the circumcircle from a given starting point X, dragging B and C along (with slippage) so as to maintain the given incircle, the original triangle occurs a total of three times: when A=X, when B=X, and when C=X.

For each of the three portions of this walkaround, what is the nature of the closed path traced out by (1) the triangle's centroid, and (2) its orthocenter? Does it trace a locus of a particular named type? Does the locus enclose a convex set? Loraof (talk) 01:22, 16 April 2017 (UTC)[reply]

The question can be visualized by this figure from the article, in which unfortunately the animation is too fast. I'm interested especially in the locus of centroids. Loraof (talk) 14:21, 17 April 2017 (UTC)[reply]

You can see a slowed-down version of the animation here - just paste in the URL of the animated gif. AndrewWTaylor (talk) 16:33, 17 April 2017 (UTC)[reply]

This came up in Richard Schwartz's Einstein Lecture at the AMS sectional meeting at Indiana U a couple of weeks ago. He has a java applet on his webpage. I am moderately confident that he asserted that the locus of the centroid is a circle. (As with many things in Euclidean geometry, this shouldn't be particularly difficult if you just throw down a coordinate system.) --JBL (talk) 19:42, 17 April 2017 (UTC)[reply]

Thanks to both of you! Loraof (talk) 14:42, 18 April 2017 (UTC)[reply]

Having gone through the calculations, out of curiosity: if we take R = 1 and the distance between the two centers to be d, then Euler's theorem gives r = (1 - d^2)/2. The locus of the centroid is a circle whose center is between the centers of the incircle and circumcircle, at distance 2d/3 from the circumcenter; it has radius d^2/3. I did not try to work out the details for the orthocenter. -JBL (talk) 20:31, 19 April 2017 (UTC)[reply]

Thanks, JBL—this is exactly what I wanted. I tried to work this out with Cartesian coordinates, but I became hopelessly lost in an algebraic morass. Could you give me a sketch of your derivation? Loraof (talk) 21:40, 19 April 2017 (UTC)[reply]

Nothing interesting or helpful, I'm afraid -- I just bashed through the calculations with the aid of a computer algebra system. Of course I've cleverly closed the file without saving anything, but if you like I can try to run through it again tomorrow and post the salient bits. (Or e-mail it.) --JBL (talk) 22:39, 19 April 2017 (UTC)[reply]

That's okay—the main thing is knowing the result. Loraof (talk) 00:30, 20 April 2017 (UTC)[reply]

Ternary and n-terms ratios equivalent representations

How many binary ratios are needed for an equivalent representation of a ternary ratio of numbers a:b:c? What is the general case for n-term ratios a:b:c:d:.:....:q? (Thanks.)--82.137.11.181 (talk) 13:47, 16 April 2017 (UTC)[reply]

n–1 are needed. E.g., for a:b:c, knowing a:b and b:c gives you a:b:c. In general, if you know all binary ratios of adjacent items, you can string together the whole n-item ratio. Loraof (talk) 16:39, 16 April 2017 (UTC)[reply]

And the necessary number of ratios cannot be less than n–1 because those n–1 sequential binary ratios are all independent pieces of information, and you need to have at least as many binary ratios as there are independent pieces of information to convey. Loraof (talk) 20:22, 16 April 2017 (UTC)[reply]

Hypercube/hypersphere volume question.

For number of dimensions n, take the n-hypercube of edge 2 units centered at the origin. Circumscribe the n-hypersphere around it. Divide the hypersphere's volume into n parts (Vn,V(n-1), ... V0) based on how many of coordinates of each point have absolute value<1. Is Vn (the n-hypercube) always the largest volume of the parts for each number of dimensions? For example if n=4, does the hypercube have more hypervolume than the points where only 3 of the 4 coordinates have absolute value<1? Naraht (talk) 18:03, 16 April 2017 (UTC)[reply]

An interesting question. I cannot answer it, but here is a formulation that could be helpful for an answer based on calculus. Define $V_{k,n}(r)$ the hypervolume of the part of the unit-radius hypersphere in n dimensions that verifies that at least k coordinates are smaller than r. Then the original question is whether the inequality $V_{k+1,n}(r)-V_{k,n}(r)\leq V_{n,n}(r)-V_{n-1,n}(r)$ holds for all k and n if $r={\sqrt {n}}/2$ (the side of the hypercube that fits into a sphere of radius 1). I imagine $V_{k,n}(r)$ could have an explicit formulation. Tigraan^{Click here to contact me} 18:32, 16 April 2017 (UTC)[reply]

The half-edge of the hypercube is 1, so the hypersphere's radius (which equals the distance from the hypercube's center to a vertex) is ${\sqrt {n}}.$ By n-sphere#Closed forms, when n is even the volume of a hypersphere with unit radius is ${\frac {\pi ^{n/2}}{(n/2)!}},$ so the volume of a hypersphere with radius ${\sqrt {n}}$ is the unit volume times the radius raised to the power n: ${\frac {(n\pi )^{n/2}}{(n/2)!}}.$ The volume of the hypercube with edge 2 is $2^{n}.$ So the fraction of the hypersphere's volume that is within the hypercube is ${\frac {(n/2)!}{(\pi n/4)^{n/2}}}.$ I think this declines sharply as n rises. For example, for n=12, this fraction is about .001. Since you have divided the hypersphere volume into only 13 parts, the hypercube Vn cannot be the largest. Loraof (talk) 23:54, 16 April 2017 (UTC)[reply]

That sharp decrease as n rises is sometimes called the curse of dimensionality. 50.0.136.56 (talk) 03:42, 18 April 2017 (UTC)[reply]

The example on that page is for the reverse, sphere inscribed in a cube, but I agree that a similar example could be set up for this.Naraht (talk) 15:15, 18 April 2017 (UTC)[reply]

Thanks for that link. And in fact, the section of that article Curse of dimensionality#Distance functions deals with the case of a hypersphere inscribed in a hypercube, and again the ratio of their volumes goes to 0 as n goes to infinity. Loraof (talk) 15:15, 18 April 2017 (UTC)[reply]