{"id":469,"date":"2013-12-01T01:00:16","date_gmt":"2013-12-01T01:00:16","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=469"},"modified":"2015-07-29T00:55:20","modified_gmt":"2015-07-29T00:55:20","slug":"deltoid-rolling-inside-astroid","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2013\/12\/01\/deltoid-rolling-inside-astroid\/","title":{"rendered":"Deltoid Rolling Inside Astroid"},"content":{"rendered":"
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\"Deltoid<\/a>

Deltoid Rolling Inside Astroid – Greg Egan<\/p><\/div>\n<\/div>\n

A deltoid<\/a><\/b> is a curve formed by rolling a circle inside a circle whose radius is 3 times larger. Similarly, an astroid<\/a><\/b> is a curve formed by rolling a circle inside a circle whose radius is 4 times larger. The picture here, drawn by Greg Egan<\/a>, shows a deltoid moving inside an astroid. Note that it fits in a perfectly snug way!<\/p>\n

It looks like it’s rolling. However, it doesn\u2019t truly \u2018roll\u2019 in the true sense of classical mechanics—-it slides a bit as it rolls.<\/p>\n

This pattern continues. The hypocycloid with $n$ cusps<\/a><\/b> is the curve formed by rolling a circle inside a circle whose radius is $n$ times larger. A hypocycloid with $n$ cusps moves snugly inside a hypocycloid with $n+1$ cusps.<\/p>\n

For another relation between the deltoid and astroid, see:<\/p>\n

• Astroid as catacaustic of deltoid<\/a>, Visual Insight<\/i>.<\/p>\n

This pattern does not<\/i> continue.<\/p>\n

To see why the hypocycloid with $n$ cusps moves snugly inside a hypocycloid with $n+1$ cusps, it helps to think about a related surprise. Recall that the special unitary group<\/b><\/a> $\\mathrm{SU}(n)$ consists of $n \\times n$ unitary matrices with determinant 1. In fact, the set of complex numbers that are the trace of some matrix in the group $\\mathrm{SU}(n)$ is filled-in hypocycloid with $n$ cusps! This is discussed here:<\/p>\n

• N. Kaiser, Mean eigenvalues for simple, simply connected, compact Lie groups<\/a>.<\/p>\n

But here is a fairly self-contained proof put together by Greg Egan, with some help from Shanthanu Bhardwaj and Aaron Wolbach.<\/p>\n

If you have a matrix in $\\mathrm{SU}(n+1)$, its $n+1$ eigenvalues can be any unit complex numbers that multiply to 1, and its trace is the sum of these numbers. We can take $n$ of them to be $e^{i \\theta}$, and then the remaining one has to be $e^{-i n \\theta}$. Then their sum is<\/p>\n

$$ n e^{i \\theta} + e^{-i n \\theta}. $$<\/p>\n

But this is also the curve traced out by a small circle of radius 1 rolling inside a big circle of radius $n+1$. Why? As it rolls, the small circle’s center moves around a circle of radius $n$, tracing out the curve $n e^{i \\theta}$. But as it rolls, the small circle turns in the opposite direction at an angular velocity that’s $n$ times higher. This gives the term $e^{-i n \\theta}$.<\/p>\n

In short, we have seen that<\/p>\n

$$ H_{n+1} = \\{ n e^{i \\theta} + e^{-i n \\theta} : 0 \\le \\theta \\le 2 \\pi \\} $$<\/p>\n

is a hypocycloid with $n+1$ cusps, and if we define<\/p>\n

$$\\mathrm{tr}(\\mathrm{SU}(n+1)) = \\{ \\mathrm{tr}(g) : g \\in \\mathrm{SU}(n+1) \\} $$<\/p>\n

then<\/p>\n

$$ H_{n+1} \\subseteq \\mathrm{tr}(\\mathrm{SU}(n+1)) .$$<\/p>\n

In fact, the hypocycloid $H_{n+1}$ is precisely the boundary of $\\mathrm{tr}(\\mathrm{SU}(n+1))$. To show this, note that the eigenvalues of any element of $\\mathrm{SU}(n+1)$ can be written as<\/p>\n

$$ e^{i\\phi_1} , \\; \\dots, \\; e^{i\\phi_{n}} , \\; e^{-i(\\phi_1+\\phi_2+ \\cdots +\\phi_{n})} $$<\/p>\n

so its trace is<\/p>\n

$$ e^{i\\phi_1} + \\cdots + e^{i\\phi_{n}} + e^{-i(\\phi_1+\\phi_2+ \\cdots +\\phi_{n})} $$<\/p>\n

where the angles $\\phi_i$ are arbitrary. When all the $\\phi_i$ equal the same angle $\\theta$, the trace gives a point in the hypocycloid $H_{n+1}$. But if we compute the derivative of the trace with respect to any angle $\\phi_i$ at a point where they\u2019re all equal, the derivative is always tangent to this hypocycloid: it\u2019s just $\\frac{1}{n-1}$ times the derivative of<\/p>\n

$$ n e^{i \\theta} + e^{-i n \\theta}$$<\/p>\n

with respect to $\\theta$. Except at the cusps, some neighborhood of the tangent line lies in the interior of the filled hypocycloid, so no change in the $\\phi_i$ can take you out of it. And at the cusps, moving along the tangent out of the hypocycloid would take you out of the disk of radius $n$, which is forbidden by the triangle inequality.<\/p>\n

Furthermore, any point inside the hypocycloid $H_{n+1}$ is an element of $\\mathrm{tr}(\\mathrm{SU}(n+1))$. To see this, note that $\\mathrm{SU}(n+1)$ is simply connected, and thus so is its image under the continuous map<\/p>\n

$$ \\mathrm{tr} \\colon \\mathrm{SU}(n+1) \\to \\mathbb{C} .$$<\/p>\n

Since its image includes the hypocycloid $H_{n+1}$, which bounds a set homeomorphic to a disk, its image must include this whole set. (Here we use a fact from topology, that a subset of a disk containing the boundary but missing some point in the interior cannot be simply connected.)<\/p>\n

In summary, $\\mathrm{tr}(\\mathrm{SU}(n+1))$ is precisely the closed set in the plane bounded by hypocycloid $H_{n+1}$. We can use this to see that a hypocycloid with $n$ cusps rolls snugly inside a hypocycloid with $n+1$ cusps. Recall that the eigenvalues of a matrix in $\\mathrm{SU}(n)$ are of the form<\/p>\n

$$ e^{i\\phi_1}, \\; \\dots , \\; e^{i\\phi_{n-1}}, \\; e^{-i(\\phi_1+\\phi_2+…+\\phi_{n-1})} $$<\/p>\n

where the angles $\\phi_i$ are arbitrary. On the other hand, the eigenvalues of any element of $\\mathrm{SU}(n+1)$ can be written as<\/p>\n

$$ e^{i\\theta}e^{i\\phi_1}, \\; \\dots , \\; e^{i\\theta}e^{i\\phi_{n-1}}, \\;e^{i\\theta}e^{-i(\\phi_1+\\phi_2+…+\\phi_{n-1})}, e^{-in\\theta} $$<\/p>\n

where the angles $\\phi_i$ and $\\theta$ are arbitrary. Thus we have<\/p>\n

$$ \\mathrm{tr}(\\mathrm{SU}(n+1)) = \\bigcup_{0 \\le \\theta \\le \\pi} e^{i\\theta} \\mathrm{tr}(\\mathrm{SU}(n)) + e^{-in\\theta} $$<\/p>\n

As $\\theta$ ranges from $0$ to $2\\pi$, this gives a filled-in hypocycloid with $n$ cusps moving snugly inside one with $n+1$ cusps!<\/p>\n

Egan’s picture above illustrates the case $n = 2$. The circling red dot shows what happens as $\\theta$ changes. Each of the colored lines shows what happens when we vary $\\phi_1$, while the progression from line to line sweeping out a filled-in deltoid is due to varying $\\phi_2$.<\/p>\n

For Egan’s movie of the case $n = 3$, and also some movies of nested<\/i> hypocycloids, each one moving in the next, see:<\/p>\n