[math-fun] Draft of my April 2016 blog post
I started writing a new draft titled "The Paintball Party Problem, and the Habit of Symmetry" and would appreciate people's feedback. I plan on publishing it on April 17. Here are some issues I'm aware of: (1) What (if anything) should be cut, or at least relegated to the End Notes? Or maybe condensed somehow? (2) Can you think of ways to make the discussion of symmetry arguments clearer? (I think that's the part of the current draft that will cause readers the most trouble.) (3) Do you know of any good images and animations regarding finite geometries that might enhance the essay via embedded links? (4) Do you know of good animations of the symmetries of a cube? (If someone wants to make an animation that specifically shows my blue-and-red-colored cubes rotating, to show how the symmetries of the cube permute the set of colorings, that'd be fantastic!) I found one at https://www.youtube.com/watch?v=TggbcOrALMQ but it's not great. I'll probably make one tomorrow if I can find our family's Rubik's cube. (5) Links to animations of shear transformations would be appreciated. (6) On a more technical note, is there an easy way to see that the order of a Hadamard matrix must be 1, 2, or a multiple of 4? (7) I can't find a good way to make an "|R" (the symbol for the real numbers) in Wordpress; can someone help me find one? And of course I'm sure there are typos! Keep in mind that all feedback will be anonymous unless you include your name. As always, people who help me with the essay will be acknowledged by name unless they prefer not to be. Please leave your feedback here: https://mathenchant.wordpress.com?p=669&shareadraft=570be41c24d66 Title: The Paintball Party Problem, and the Habit of Symmetry Beginning: I was very surprised last fall, when I was running my nine-year-old son’s birthday party, to realize that the geometry of spaces built over finite fields (I’ll explain what that means later) was exactly what I needed to solve the very pressing problem of choosing teams for paintball in a fair way! Thanks, Jim Propp
(7) http://www.unicodeit.net/ is a fab site which gives you the corresponding Unicode character for a LaTeX command. I used it to obtain the character ℝ. For more complex maths, you can use Wordpress's built-in LaTeX rendering; there's some information on how to use it at https://en.support.wordpress.com/latex/. On Tue, 12 Apr 2016 at 05:51 James Propp <jamespropp@gmail.com> wrote:
I started writing a new draft titled "The Paintball Party Problem, and the Habit of Symmetry" and would appreciate people's feedback. I plan on publishing it on April 17. Here are some issues I'm aware of:
(1) What (if anything) should be cut, or at least relegated to the End Notes? Or maybe condensed somehow?
(2) Can you think of ways to make the discussion of symmetry arguments clearer? (I think that's the part of the current draft that will cause readers the most trouble.)
(3) Do you know of any good images and animations regarding finite geometries that might enhance the essay via embedded links?
(4) Do you know of good animations of the symmetries of a cube? (If someone wants to make an animation that specifically shows my blue-and-red-colored cubes rotating, to show how the symmetries of the cube permute the set of colorings, that'd be fantastic!) I found one at https://www.youtube.com/watch?v=TggbcOrALMQ but it's not great. I'll probably make one tomorrow if I can find our family's Rubik's cube.
(5) Links to animations of shear transformations would be appreciated.
(6) On a more technical note, is there an easy way to see that the order of a Hadamard matrix must be 1, 2, or a multiple of 4?
(7) I can't find a good way to make an "|R" (the symbol for the real numbers) in Wordpress; can someone help me find one?
And of course I'm sure there are typos!
Keep in mind that all feedback will be anonymous unless you include your name. As always, people who help me with the essay will be acknowledged by name unless they prefer not to be.
Please leave your feedback here: https://mathenchant.wordpress.com?p=669&shareadraft=570be41c24d66
Title: The Paintball Party Problem, and the Habit of Symmetry Beginning: I was very surprised last fall, when I was running my nine-year-old son’s birthday party, to realize that the geometry of spaces built over finite fields (I’ll explain what that means later) was exactly what I needed to solve the very pressing problem of choosing teams for paintball in a fair way!
Thanks,
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On Apr 12, 2016, at 12:50 AM, James Propp <jamespropp@gmail.com> wrote:
(6) On a more technical note, is there an easy way to see that the order of a Hadamard matrix must be 1, 2, or a multiple of 4?
Suppose the order is N>2. By multiplying rows by -1 you can always have the first three elements in each row be 1,1,1 or 1,1,-1 or 1,-1,1 or -1,1,1. Let the number of rows of these types be x,y,z,w. In addition to the constraint x+y+w+z=N, orthogonality of the first three columns gives three more constraints: x+y-w-z=0 x-y-w+z=0 x-y+w-z=0 These imply x=y=z=w, and therefore N=x+y+z+w is a multiple of 4. -Veit
Cool! That's a pretty subtle argument that doesn't look easy to discover. —Dan
On Apr 12, 2016, at 4:35 AM, Veit Elser <ve10@cornell.edu> wrote:
On Apr 12, 2016, at 12:50 AM, James Propp <jamespropp@gmail.com> wrote:
(6) On a more technical note, is there an easy way to see that the order of a Hadamard matrix must be 1, 2, or a multiple of 4?
Suppose the order is N>2. By multiplying rows by -1 you can always have the first three elements in each row be 1,1,1 or 1,1,-1 or 1,-1,1 or -1,1,1. Let the number of rows of these types be x,y,z,w. In addition to the constraint x+y+w+z=N, orthogonality of the first three columns gives three more constraints:
x+y-w-z=0 x-y-w+z=0 x-y+w-z=0
These imply x=y=z=w, and therefore N=x+y+z+w is a multiple of 4.
Veit's argument (which Seb pointed out to me as well) has the virtue of being easily translated into the paintball party context. Thanks! On Tuesday, April 12, 2016, Dan Asimov <asimov@msri.org> wrote:
Cool! That's a pretty subtle argument that doesn't look easy to discover.
—Dan
On Apr 12, 2016, at 4:35 AM, Veit Elser <ve10@cornell.edu <javascript:;>> wrote:
On Apr 12, 2016, at 12:50 AM, James Propp <jamespropp@gmail.com <javascript:;>> wrote:
(6) On a more technical note, is there an easy way to see that the order of a Hadamard matrix must be 1, 2, or a multiple of 4?
Suppose the order is N>2. By multiplying rows by -1 you can always have the first three elements in each row be 1,1,1 or 1,1,-1 or 1,-1,1 or -1,1,1. Let the number of rows of these types be x,y,z,w. In addition to the constraint x+y+w+z=N, orthogonality of the first three columns gives three more constraints:
x+y-w-z=0 x-y-w+z=0 x-y+w-z=0
These imply x=y=z=w, and therefore N=x+y+z+w is a multiple of 4.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (4)
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Christian Lawson-Perfect -
Dan Asimov -
James Propp -
Veit Elser