Showing posts with label Edge with Multiple Beads. Show all posts
Showing posts with label Edge with Multiple Beads. Show all posts
Wednesday, December 26, 2012
Beaded trefoil knot using two beads per edge
Friday, November 9, 2012
Two artworks for the Mathematical Exhibition of Joint Mathematical Meeting
Chern and I submitted two artworks to the Joint Mathematical Beading, which were accepted today:
1.
Super Buckyball of Genus 31
2. Beaded Hilbert Curve, step two
In addition to the beadworks we submitted, we also noticed five Platonic bead models made by Ron Asherov. His bead models have multiple beads in an edge, which are similar to our works a few years ago. I labeled this type of bead models with Edge with Multiple Beads, where you can find all the posts. He doesn't seem to know our works along this direction, though.
He also mentioned that the Nylon string passes adjacent edges exactly once with carefully chosen path of string, which is simply the consequence of Hamiltonian path on the dual graph of the corresponding Platonic solids. We can view the whole beading process as a path through the face of polyhedron. Thus if there exists a Hamiltonian path through each face once (Hamiltonian path for the dual polyhedron), then the Nylon string will go through each beads exactly twice and only twice. Of course, you can also say the Nylon string will go through the adjacent edges exactly once. They are the same thing.
Here is a model made by Chern Chuang almost five years ago:
I was quite surprised by its rigidity when Chern showed me this model. At that time, people questioned me about the meaning of beads. I told some of my colleagues that spherical beads represent chemical bonds instead of atoms. Atoms are not shown in the bead model explicitly, instead, they are located at somewhere three beads meet. Most chemists feel uncomfortable with this connection. So Chern and I tried try to explore with the shape of beads and multiple beads and hope that they can better represent the shape of chemical bonds. So that is why we have these models in which multiple beads represent an edge.
But now, I have the valence sphere model of chemical bond as the theoretical foundation of bead models. Spherical beads are in fact the simplest possible approximation of electron pairs, in accord with the principle of Occam's razor. So to build a model of a molecule with only beads and strings is equivalent to performing a molecular analogue computation with beads. The result of computation is the approximate electron density of the corresponding molecule without referring to the Schrödinger equation or atomic orbitals (this is the comment I got from Prof. H. Bent). I have written an article on the connection between bead models and valence sphere model in Chinese for September issue of Science Monthly (科學月刊). I think I should write something about the molecular analogue computation with beads.
Of course, what I am saying above is to view bead models as molecular models. Results of mathematical beading do not need to have any connection to the molecular world. For instance, the beaded Hilbert curve accepted by the JMM mathematical art exhibition is a good example.
2. Beaded Hilbert Curve, step two
In addition to the beadworks we submitted, we also noticed five Platonic bead models made by Ron Asherov. His bead models have multiple beads in an edge, which are similar to our works a few years ago. I labeled this type of bead models with Edge with Multiple Beads, where you can find all the posts. He doesn't seem to know our works along this direction, though.
He also mentioned that the Nylon string passes adjacent edges exactly once with carefully chosen path of string, which is simply the consequence of Hamiltonian path on the dual graph of the corresponding Platonic solids. We can view the whole beading process as a path through the face of polyhedron. Thus if there exists a Hamiltonian path through each face once (Hamiltonian path for the dual polyhedron), then the Nylon string will go through each beads exactly twice and only twice. Of course, you can also say the Nylon string will go through the adjacent edges exactly once. They are the same thing.
Here is a model made by Chern Chuang almost five years ago:
I was quite surprised by its rigidity when Chern showed me this model. At that time, people questioned me about the meaning of beads. I told some of my colleagues that spherical beads represent chemical bonds instead of atoms. Atoms are not shown in the bead model explicitly, instead, they are located at somewhere three beads meet. Most chemists feel uncomfortable with this connection. So Chern and I tried try to explore with the shape of beads and multiple beads and hope that they can better represent the shape of chemical bonds. So that is why we have these models in which multiple beads represent an edge.
But now, I have the valence sphere model of chemical bond as the theoretical foundation of bead models. Spherical beads are in fact the simplest possible approximation of electron pairs, in accord with the principle of Occam's razor. So to build a model of a molecule with only beads and strings is equivalent to performing a molecular analogue computation with beads. The result of computation is the approximate electron density of the corresponding molecule without referring to the Schrödinger equation or atomic orbitals (this is the comment I got from Prof. H. Bent). I have written an article on the connection between bead models and valence sphere model in Chinese for September issue of Science Monthly (科學月刊). I think I should write something about the molecular analogue computation with beads.
Of course, what I am saying above is to view bead models as molecular models. Results of mathematical beading do not need to have any connection to the molecular world. For instance, the beaded Hilbert curve accepted by the JMM mathematical art exhibition is a good example.
Monday, June 11, 2012
Bead models of dodecahedron and icosahedron
Mr. Horibe started to play with polyhedral models very early. According to him, he has constructed Platonic models with a long cylindrical tube capped with two beads when he was still an undergraduate student in the 70s.
He demonstrated in front of us that dodecahedron and icosahedron are dual to each other using this kind of models. With long cylindrical tubes, the whole structure becomes very flexible, so you can compress it and put one inside the other. Students can realize, through this amazing way, why dodecahedron and icosahedron are dual to each other. I can only say that Mr. Horibe must be a very effective math teacher. I wish I could learn math from him when I was a high school student.
In some sense, this kind of bead models is very similar to the so-called tensegrity structures. BTW, Mr. Horibe usually uses a single elastic string for making his bead models, which makes his models even more flexible and bendable. I will comment more on this aspect later.
He demonstrated in front of us that dodecahedron and icosahedron are dual to each other using this kind of models. With long cylindrical tubes, the whole structure becomes very flexible, so you can compress it and put one inside the other. Students can realize, through this amazing way, why dodecahedron and icosahedron are dual to each other. I can only say that Mr. Horibe must be a very effective math teacher. I wish I could learn math from him when I was a high school student.
In some sense, this kind of bead models is very similar to the so-called tensegrity structures. BTW, Mr. Horibe usually uses a single elastic string for making his bead models, which makes his models even more flexible and bendable. I will comment more on this aspect later.
Wednesday, November 4, 2009
Saturday, December 27, 2008
Thursday, December 18, 2008
Tetrahedron
I try to experiment with the idea that using three beads to stand for an edge of polyhedron. In this case, I made a tetrahedron. Previously, I have posted some of this kind of models. The original idea came from Chuang again.
Friday, July 11, 2008
Monday, February 4, 2008
Sp3 carbons (one more example)
Here is one more beaded structure defined by the same net similar to that in the previous post except that we use compounded bonds (three beads for a single bond).
Wednesday, December 5, 2007
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