Re: [math-fun] A pretty surface with surprising symmetry
I think both the genus g and the Euler characteristic bear on what symmetries a surface can have. The (orientable, compact) surface M_g of genus g always has a metric with an isometry of order g: Just arrange the holes in a circle. The surface of genus g = K + 1 always has a metric with an isometry of order K: Arrange one hole in the middle and the other K around it. The surface M_3 of genus 3 (with X(M_g) = -4) has a metric with an isometry of order 7. M_3 cannot be exhibited in 3-space with this 7-fold symmetry. It has other metrics, like one having 192 symmetries, including a subgroup of order 64. It has no metric with both a 7-fold symmetry and a subgroup of order 64, since a metric surface of genus g can have at most 168 * (g-1) symmetries and 7*64 = 448 > 336 would violate that theorem. —Dan Allan Wechsler wrote: ----- My thought is that if the genus is 73, the Euler characteristic is -144. 73 may seem like a number that is not very conducive to fancy symmetry, but -144 seems a lot more promising. I'm guessing that the Euler characteristic is more important to allowable symmetries than the genus is. Other examples can be harvested from the thickened skeleta of the platonic solids treated as surfaces. I expect the genus plus one to be a nicely divisible number. The tetrahedron's skeleton has genus 3; 4 has better divisibility. The cube has genus 5 but 6 is better. The octahedron: 7 versus 8. The dodecahedron: 11 versus 12. The icosahedron: 19 versus 20. In other words, I don't think the symmetries are juggling a set whose size is the genus; I think the relevant size is the genus plus 1. -----
Sorry, I think g-1 is what I meant. X = 2-2g = 2(1-g) = -2(g-1), so it would be g-1 that would correspond to the Euler characteristic. On the other hand, g+1 is significant for the polyhedral models: a thick spherical shell with n holes drilled in it has genus n-1, so g+1 = n. I don't know if there is any actual math here, or if I'm just being deluded by numerology. On Mon, Nov 2, 2020 at 12:42 PM Marc LeBrun <mlb@well.com> wrote:
=Allan Wechsler I expect the genus plus one to be a nicely divisible number.
? 73 *plus* one is 74, but 73 *minus* one is 72, which seems much more bigly nicely?
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What is the relation between the genus of X and the genus of Y when there is a d-to-1 map from X to Y? (Assume that around each y in Y we can find a disk whose preimage consists of d disks.) Do we have genus(X) = d genus(Y) ? Jim Propp On Mon, Nov 2, 2020 at 7:23 PM Allan Wechsler <acwacw@gmail.com> wrote:
Sorry, I think g-1 is what I meant. X = 2-2g = 2(1-g) = -2(g-1), so it would be g-1 that would correspond to the Euler characteristic.
On the other hand, g+1 is significant for the polyhedral models: a thick spherical shell with n holes drilled in it has genus n-1, so g+1 = n.
I don't know if there is any actual math here, or if I'm just being deluded by numerology.
On Mon, Nov 2, 2020 at 12:42 PM Marc LeBrun <mlb@well.com> wrote:
=Allan Wechsler I expect the genus plus one to be a nicely divisible number.
? 73 *plus* one is 74, but 73 *minus* one is 72, which seems much more bigly nicely?
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Jim's formula can't be the right one, because there are d-to-1 maps from the torus (R/Z)^2 to itself for any positive integer d. I think. On Mon, Nov 2, 2020 at 8:10 PM James Propp <jamespropp@gmail.com> wrote:
What is the relation between the genus of X and the genus of Y when there is a d-to-1 map from X to Y? (Assume that around each y in Y we can find a disk whose preimage consists of d disks.) Do we have genus(X) = d genus(Y) ?
Jim Propp
On Mon, Nov 2, 2020 at 7:23 PM Allan Wechsler <acwacw@gmail.com> wrote:
Sorry, I think g-1 is what I meant. X = 2-2g = 2(1-g) = -2(g-1), so it would be g-1 that would correspond to the Euler characteristic.
On the other hand, g+1 is significant for the polyhedral models: a thick spherical shell with n holes drilled in it has genus n-1, so g+1 = n.
I don't know if there is any actual math here, or if I'm just being deluded by numerology.
On Mon, Nov 2, 2020 at 12:42 PM Marc LeBrun <mlb@well.com> wrote:
=Allan Wechsler I expect the genus plus one to be a nicely divisible number.
? 73 *plus* one is 74, but 73 *minus* one is 72, which seems much more bigly nicely?
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participants (4)
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Allan Wechsler -
Dan Asimov -
James Propp -
Marc LeBrun