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Fw: Equal edge length Johnson Near Misses
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Eduard Baumann
Dec 31, 2020, 8:43:55 AM12/31/20
to 4D Groupe II
----- Original Message -----
From: Eduard
Baumann
Sent: Friday, December 25, 2020 12:26 PM
Subject: Equal edge length Johnson Near Misses
Hi,
Here comes a further collection of "repaired"
(equal edge lengthes) Johnson Near Misses.
Jim says:
"it's surprising how many of them will resolve
merely by allowing the square faces to become rhombi"
I add:
"It is also surprising that some Johnson Near
Misses cannot be modified (angles) to get equal edge lengthes. The planarity
condition for the m-gons with m>3 is not easy to fulfill."
The numbering of the Johnson Near Misses her is the
same as Jim's.
Solved: no 3, 4,
7,11
Certainly unsolvable: no
8
Probably unsolvable: no 9 and
10
No 3
No 4
No 7
No 11
Not solvable No 8
Kind regards
Ed
Marc Ringuette
Dec 31, 2020, 3:21:15 PM12/31/20
to hyper...@googlegroups.com
I don't recall learning about Johnson near misses before, so I did a
little reading. These two links from 2017 and 2019 are worth
checking out even if you know all about Johnson solids and near
misses already.
Evelyn Lamb brief intro article, 2017:
http://nautil.us/issue/49/the-absurd/the-impossible-mathematics-of-the-real-world
Blog post from Craig Kaplan of Waterloo, 2019, including discussion of the real-world TRAP cage molecule:
https://isohedral.ca/a-molecular-near-miss/
As background, this all builds on something near and dear to the hearts of 3D geometers and puzzle constructors, the 92 Johnson solids, enumerated and named by Norman Johnson in 1966.
https://en.wikipedia.org/wiki/Johnson_solid
The notion of "imperfect" mathematical structures seems like a really big deal to me: I imagine that the chemistry of the real world will inevitably be a zoo of geometric near misses that allow molecules to stay together.
I also liked the connection to music in Lamb's article: music seems to thrive on near-misses, frequency wise, and in particular, 2 to the power 7/12 is equal to 1.498, "close enough" to 3/2 for musical fifths to sound harmonious, and 2 to the power 5/12 is equal to 1.335, "close enough" to 4/3. So, a major chord (tonic, third, fifth, octave) is built on two such near-misses, and the variety of tempered musical scales play around with what is "close enough" in subtle ways that can alter the feel of the music.
Fun!
Marc
Evelyn Lamb brief intro article, 2017:
http://nautil.us/issue/49/the-absurd/the-impossible-mathematics-of-the-real-world
Blog post from Craig Kaplan of Waterloo, 2019, including discussion of the real-world TRAP cage molecule:
https://isohedral.ca/a-molecular-near-miss/
As background, this all builds on something near and dear to the hearts of 3D geometers and puzzle constructors, the 92 Johnson solids, enumerated and named by Norman Johnson in 1966.
https://en.wikipedia.org/wiki/Johnson_solid
The notion of "imperfect" mathematical structures seems like a really big deal to me: I imagine that the chemistry of the real world will inevitably be a zoo of geometric near misses that allow molecules to stay together.
I also liked the connection to music in Lamb's article: music seems to thrive on near-misses, frequency wise, and in particular, 2 to the power 7/12 is equal to 1.498, "close enough" to 3/2 for musical fifths to sound harmonious, and 2 to the power 5/12 is equal to 1.335, "close enough" to 4/3. So, a major chord (tonic, third, fifth, octave) is built on two such near-misses, and the variety of tempered musical scales play around with what is "close enough" in subtle ways that can alter the feel of the music.
Fun!
Marc
David Barr
Jan 1, 2021, 6:47:10 PM1/1/21
to hyper...@googlegroups.com
I enjoyed those articles. I made a magnet sculpture of the shape of the solid in the Kaplan article.
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Melinda Green
Jan 1, 2021, 10:25:18 PM1/1/21
to David Barr, hypercubing
Neat! It's particularly nice because magnetic balls are a nice
analog for molecules. My feeling is that it should be no surprise
that we find ourselves in a 3D universe because it's a bit of a mess
in relation to sphere packings which allows for richer chemistry
than in dimensions that admit perfect packings such as 2 and 4.
My favorite Johnson near miss is a false solution to the 554 problem you'll find a little way down on this page. What's the 554 problem? I'm glad you asked! I have a nice collection of Polydron tiles and once wondered whether one could construct a polyhedron that contained at least one vertex surrounded by exactly two pentagons and a square. The faces can be any regular polygons, and can even be nonconvex and self-intersecting. Sounds easy, right? Well I tried hard to find an example and failed. I was in a polyhedron mailing list for a time which included Magnus Wenninger, John Conway, and other geometry luminaries, and I eventually got them interested in the question. There was a flurry of activity for a time resulting in the above page among other results. It was funny to me because for a long time Conway insisted that he could settle the question if he put his mind to it, but once he did, he failed, as did everyone else. I encourage anyone here to take a crack at it. Find a true example or prove it impossible. Decades later it remains unsolved. I'll now offer a $100 prize for the first solution or a proof that it's impossible. I just now created this YouTube challenge which you can share with anyone else you think might enjoy taking a crack at it.
Good luck!
-Melinda
My favorite Johnson near miss is a false solution to the 554 problem you'll find a little way down on this page. What's the 554 problem? I'm glad you asked! I have a nice collection of Polydron tiles and once wondered whether one could construct a polyhedron that contained at least one vertex surrounded by exactly two pentagons and a square. The faces can be any regular polygons, and can even be nonconvex and self-intersecting. Sounds easy, right? Well I tried hard to find an example and failed. I was in a polyhedron mailing list for a time which included Magnus Wenninger, John Conway, and other geometry luminaries, and I eventually got them interested in the question. There was a flurry of activity for a time resulting in the above page among other results. It was funny to me because for a long time Conway insisted that he could settle the question if he put his mind to it, but once he did, he failed, as did everyone else. I encourage anyone here to take a crack at it. Find a true example or prove it impossible. Decades later it remains unsolved. I'll now offer a $100 prize for the first solution or a proof that it's impossible. I just now created this YouTube challenge which you can share with anyone else you think might enjoy taking a crack at it.
Good luck!
-Melinda
To view this discussion on the web visit https://groups.google.com/d/msgid/hypercubing/CACh%2BXar-H4uaNm0nWg0g8TgBqtiYkJ%2Bb9KamkW%2BhqTzuZ-y3LA%40mail.gmail.com.
Wagyx Xygaw
Jul 28, 2024, 3:09:55 AM7/28/24
to hypercubing
Hello there,
I am making a web page to gather near miss Johnson solids from across the internet and display them with an interactive 3D viewer.
https://asliceofcuriosity.fr/blog/posts/geometry18.html
I am making a web page to gather near miss Johnson solids from across the internet and display them with an interactive 3D viewer.
https://asliceofcuriosity.fr/blog/posts/geometry18.html
I hope you enjoy it and in the case you would like to help fill in the unknown names, authors and dates for some of the solids or add a solid or a resource I have yet to check, please contact me.
Have a good day !
Wagyx
Wagyx
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