Showing posts with label Hamiltonian path. Show all posts
Showing posts with label Hamiltonian path. Show all posts
Thursday, January 3, 2013
Tetrahedral C28 and related structures
There are only three tetrahedral fullerenes with number of carbon atoms less than that of buckyball. They are C28, C40, and C44, respectively. The spiral code for the smallest tetrahedral fullerene, C28, is [1 2 3 5 7 9 10 11 12 13 14 15]. Following this code, we can easily make its bead model using the standard figure-eight stitch. We can see that, in this molecule, there are 12 pentagons, 3 in a group located at a vertex, and 4 hexagons located on the four faces of the tetrahedron. If we replace these pentagons by heptagons, we get a tetrapod-like structure, in which tri-pentagon vertices become tri-heptagon necks as shown in the following figure.
Using these tetrapods as building blocks, we can get the following diamond-like structure. In fact, this is exactly the structure Mr. Horibe put in the postcard. OK, if we start from other tetrahedral fullerenes such as C40 and C44, we can find out a lot more diamond-like structures.
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.
Tuesday, June 12, 2012
Two gifts from Mr. Horibe
I am usually the person who gave people beadworks as gifts. But I myself got a beaded C60 as a gift from Mr. Horibe on our visit to his home. And the next day, when he participated my workshop in Nagoya, I got another giant torus as a gift.
His model is quite different from mine because most often he used only a single elastic rubber band with exact length to make his beadworks. In this sense, the path of his elastic string exactly corresponds to the Hamiltonian path exactly and has, of course, the minimal length for a particular model. So, the elastic string passed every bead in the model twice and only twice. To avoid beads fall apart, Mr. Horibe then made a knot in the end. So you can pull or press his bead models to some extent.
Possibly due to his training as mathematics teacher, he always uses a single string even for structures that contain more several thousand beads. I was quite surprised when I just heard of it. Just imagine how one can use a single string to bead the giant green stellated dodecahedron as shown in the following photo. The problem is that it would become very difficult to bead with such a long string. The good thing is that the path of string in his case is really a Hamiltonian path no matter how many beads a model contains. I guess that he also needed to plan well before he started because it is not always trivial to find out the Hamiltonian path. Sometimes, we just got trapped as weaving process goes. I don't know how he managed to do it with only one string especially for certain structures that has more than several thousand beads.
Another practical aspect about using elastic string is that elastic string is much thicker than the Nylon string I typically used. I suspect that it may not easy to pass it through three times through a hole. But of course, it is not a good idea to pass string through some beads twice, but some other beads three times because that will make the tension generated by the rubber band uneven throughout the structure. There is no such problem for the Nylon string because the bead models made by Nylon string are usually quite strong and hard. So it is harder to deform them like the model made with elastic rubber band.
His model is quite different from mine because most often he used only a single elastic rubber band with exact length to make his beadworks. In this sense, the path of his elastic string exactly corresponds to the Hamiltonian path exactly and has, of course, the minimal length for a particular model. So, the elastic string passed every bead in the model twice and only twice. To avoid beads fall apart, Mr. Horibe then made a knot in the end. So you can pull or press his bead models to some extent.
Possibly due to his training as mathematics teacher, he always uses a single string even for structures that contain more several thousand beads. I was quite surprised when I just heard of it. Just imagine how one can use a single string to bead the giant green stellated dodecahedron as shown in the following photo. The problem is that it would become very difficult to bead with such a long string. The good thing is that the path of string in his case is really a Hamiltonian path no matter how many beads a model contains. I guess that he also needed to plan well before he started because it is not always trivial to find out the Hamiltonian path. Sometimes, we just got trapped as weaving process goes. I don't know how he managed to do it with only one string especially for certain structures that has more than several thousand beads.
Another practical aspect about using elastic string is that elastic string is much thicker than the Nylon string I typically used. I suspect that it may not easy to pass it through three times through a hole. But of course, it is not a good idea to pass string through some beads twice, but some other beads three times because that will make the tension generated by the rubber band uneven throughout the structure. There is no such problem for the Nylon string because the bead models made by Nylon string are usually quite strong and hard. So it is harder to deform them like the model made with elastic rubber band.
Friday, June 1, 2012
C76
There are two isomers of C76 that satisfy the isolated pentagon rule (IPR). Here is the bead model for C76 with Td symmetry just constructed by Yuan-Chia Fan.
The spiral codes for these two isomers of C76 are
C76:1 [1 7 9 11 13 18 26 31 33 35 37 39] D2
C76:2 [1 7 9 12 14 21 26 28 30 33 35 38] Td
C76:1 [1 7 9 11 13 18 26 31 33 35 37 39] D2
C76:2 [1 7 9 12 14 21 26 28 30 33 35 38] Td
Sunday, April 15, 2012
Three Kekule structures of C60
D. Vukicevic and M. Randic have figured out all possible distinct resonance (or Kekule)
structures a few years ago. According them,
buckminsterfullerene has 12500 Kekule structures grouped in 158 isomorphic classes. They also give
a complete list of all these 158 non-isomorphic Kekuke structures in a recent paper
entitled "Detailed Atlas of Kekulé Structures of the Buckminsterfullerene", in the book,
"The Mathematics and Topology of Fullerenes".
This is very convenient if we want to make any particular resonance form of C60. We can simply look at the Schlegel diagrams given in this paper, and pay attention to the single and double bond pattern as we bead. Here are three bead models for Kekule structures No. 134, 135 and 136 as shown in their paper.
This is very convenient if we want to make any particular resonance form of C60. We can simply look at the Schlegel diagrams given in this paper, and pay attention to the single and double bond pattern as we bead. Here are three bead models for Kekule structures No. 134, 135 and 136 as shown in their paper.
Tuesday, February 28, 2012
C84 - a tetrahedral fullerene
I posted a few bead models of C84 before, but I had never given the detailed beading procedure for this molecule. C84 is the smallest achiral fullerene with tetrahedral shape that satisfies the independent pentagon rule (i.e. no two pentagons are connected).
Here is another bead model of C84 consisting of 8mm beads I made yesterday.
The easiest way to make it is to follow the beading path as shown in the following Schlegel diagram. Note that this path does not correspond to the path give by a spiral code.
In principle, one can puncture holes on this molecule and use the resulting structures as building blocks to create more complicated structures or super fullerenes (fused C84 in this case). I will show how this can be done later.
Here is another bead model of C84 consisting of 8mm beads I made yesterday.
The easiest way to make it is to follow the beading path as shown in the following Schlegel diagram. Note that this path does not correspond to the path give by a spiral code.
In principle, one can puncture holes on this molecule and use the resulting structures as building blocks to create more complicated structures or super fullerenes (fused C84 in this case). I will show how this can be done later.
Saturday, December 3, 2011
Building blocks for the type-II high-genus fullerenes
The building block of type-II high-genus fullerenes can be chosen to be an arbitrary Goldberg polyhedron.
Puncturing three holes along three carefully chosen pentagons can create a basic unit with three coordination (or a trivalent unit).
I use C60 and its Schlegel diagram to illustrate how to puncture a hole on an arbitrary pentagon.
1. Schlegel diagram of C60
2. C60 with a hole punctured on a pentagon: one pentagon and five hexagons are replaced by five heptagons.
In principle, one can connect two this kind of unit with one hole to create a fused C120 with dumbbel-shape.
3. Of course, if we like, we can puncture two holes on a C60. There are three possible ways. Here I only show the situation with two pentagons separated by two hexagons. The resulting structure will contain two holes connected (or separated) by two heptagons.
There are two other different ways to puncture second hole. If the second pentagon separated from the first one by one CC bond are punctured, the resulting structure will have an octagon. The third situation is that the second pentagon is located at the antipodal position. I will talk about these situations later.
4. Punctured C60 with three holes:
It is easy to see that there are five heptagons and five more bonds are introduced around each hole. So one needs 105 beads for creating a single unit.
5. Here are two possible weaving path. I usually used the first path though. a. non-spiral path
b. spiral path
6. I am working on a project with teachers and students of the Taipei First Girls High School (北一女). We are going to make a giant buckyball consisting of sixty units of punctured C60s. Here are a few basic units I made:
105 12mm faceted beads are used for each unit.
I use C60 and its Schlegel diagram to illustrate how to puncture a hole on an arbitrary pentagon.
1. Schlegel diagram of C60
3. Of course, if we like, we can puncture two holes on a C60. There are three possible ways. Here I only show the situation with two pentagons separated by two hexagons. The resulting structure will contain two holes connected (or separated) by two heptagons.
4. Punctured C60 with three holes:
5. Here are two possible weaving path. I usually used the first path though. a. non-spiral path
6. I am working on a project with teachers and students of the Taipei First Girls High School (北一女). We are going to make a giant buckyball consisting of sixty units of punctured C60s. Here are a few basic units I made:
Wednesday, October 12, 2011
C70 beading procedure
C70. Point group D5h.
Shape: like a Rugby ball.
60 8mm red beads and 45 8mm white beads are used.
Ring spiral: [1 7 9 11 13 15 27 29 31 33 35 37]
The red and green parts of spiral are exactly the same as the spiral of C60. The orange part of the spiral is a ring of 10 hexagons inserted in between two hemispheres of C60! Of course, instead of inserting one ring of 10 hexagons, one can do it repeatedly to get different length of endcapped carbon nanotubes! So C70 is the shortest endcapped carbon nanotube.
Note also that the blue circles in this graph are not beads! Beads represent edges or chemical bonds of fullerenes.
I made a few bead models for Prof. Gillespie, who proposed the the famous VSEPR method, last week. This C70 model is one of them.
Shape: like a Rugby ball.
60 8mm red beads and 45 8mm white beads are used.
Ring spiral: [1 7 9 11 13 15 27 29 31 33 35 37]
The red and green parts of spiral are exactly the same as the spiral of C60. The orange part of the spiral is a ring of 10 hexagons inserted in between two hemispheres of C60! Of course, instead of inserting one ring of 10 hexagons, one can do it repeatedly to get different length of endcapped carbon nanotubes! So C70 is the shortest endcapped carbon nanotube.
Note also that the blue circles in this graph are not beads! Beads represent edges or chemical bonds of fullerenes.
I made a few bead models for Prof. Gillespie, who proposed the the famous VSEPR method, last week. This C70 model is one of them.
Friday, October 7, 2011
C60 beading procedure
I took a few photos for making bead model of C60. This will be helpful for learning the so-called figure-eight stitch (or right-angle weave) and the beading rule of C60 I mentioned before.
South hemisphere:
North hemisphere:
* General instruction:
1. There are 32 polygons consisting of 12 pentagons and 20 hexagons in a C60.
2. Every pentagon is separated from neighbored pentagons by eactly one CC bond.
3. If we choose one color of beads for pentagons and the other for the rest, one would find hexagons made of two colors alternatively.
4. It is better to view C60 as a sphere consisting of six layers of polygonal strips. For the south semisphere, they are basically a pentagon for the south pole, five hexagons next, 10 polygons consisting of 5 pentagons and 5 hexagons. Reverse the beading sequence, one gets the north semisphere. (This is what people called the spiral code.)
5. One has to check how many beads of group one wants to create in the next step. Some beads are already done, so one has to thread the fishing cord through these beads first, then add the remaining beads through the other end of fishing cord, and finally, form the n-bead group by threading the fishing cord through the last bead just added along the opposite direction.
Of course, one should always check the sequence of colors of beads one is going to bead and make sure that they satisfy the color coding mentioned in 3.
For beginners, this is usually the hardest part and mistakes occur easily. Most often, wrong number of beads are added or some beads are not threading through first. But, if one can pay attention to the number of beads in the next group one is going to make, it should be trivial to figure out how many beads are already there and how many more one should add.
If two colors of beads are used, one can simply pay attention to color. Beading process for a C60 becomes trivial.
The beading procedure can be summarized by the spiral code on the following Schlegel diagram of C60. For C60, it is [1 7 9 11 13 15 18 20 22 24 26 32], which is essentially the positions of pentagons along the spiral path starting from south pole to the north pole. It is not hard to find this code is the only information we need to create a bead model of C60. Similarly, one can create any other cage-like fullerene by its spiral code only if the fullerene possesses it.
A list of fullerenes up to 100 carbon atoms is given in the appendix of the book "An Atlas of Fullerenes" by P. W. Fowler and D. E. Manolopoulos. So one can create any fullerene easily by following its spiral code. The shape of the resulting bead model is basically consistent with the corresponding fullerene.
South hemisphere:
North hemisphere:
* General instruction:
1. There are 32 polygons consisting of 12 pentagons and 20 hexagons in a C60.
2. Every pentagon is separated from neighbored pentagons by eactly one CC bond.
3. If we choose one color of beads for pentagons and the other for the rest, one would find hexagons made of two colors alternatively.
4. It is better to view C60 as a sphere consisting of six layers of polygonal strips. For the south semisphere, they are basically a pentagon for the south pole, five hexagons next, 10 polygons consisting of 5 pentagons and 5 hexagons. Reverse the beading sequence, one gets the north semisphere. (This is what people called the spiral code.)
5. One has to check how many beads of group one wants to create in the next step. Some beads are already done, so one has to thread the fishing cord through these beads first, then add the remaining beads through the other end of fishing cord, and finally, form the n-bead group by threading the fishing cord through the last bead just added along the opposite direction.
Of course, one should always check the sequence of colors of beads one is going to bead and make sure that they satisfy the color coding mentioned in 3.
For beginners, this is usually the hardest part and mistakes occur easily. Most often, wrong number of beads are added or some beads are not threading through first. But, if one can pay attention to the number of beads in the next group one is going to make, it should be trivial to figure out how many beads are already there and how many more one should add.
If two colors of beads are used, one can simply pay attention to color. Beading process for a C60 becomes trivial.
The beading procedure can be summarized by the spiral code on the following Schlegel diagram of C60. For C60, it is [1 7 9 11 13 15 18 20 22 24 26 32], which is essentially the positions of pentagons along the spiral path starting from south pole to the north pole. It is not hard to find this code is the only information we need to create a bead model of C60. Similarly, one can create any other cage-like fullerene by its spiral code only if the fullerene possesses it.
A list of fullerenes up to 100 carbon atoms is given in the appendix of the book "An Atlas of Fullerenes" by P. W. Fowler and D. E. Manolopoulos. So one can create any fullerene easily by following its spiral code. The shape of the resulting bead model is basically consistent with the corresponding fullerene.
Saturday, February 5, 2011
Slides for making a beaded C60
I would like to thank Rochelle for pointing out that the procedure in the tabular form I posted before is incorrect. I have to admit that I have never used this kind of table for making C60. I think it is easy to make mistakes by just following this table literally and without thinking. If you are a little bit careful, you should be able to see the hidden rules for making the C60 just after about 10 steps.
Below is a few slides I used to teach people how to make a buckyball.
But, it is much easier to follow the simple mnemonic for making a buckyball.
If one wants to make a beaded C60 with two different colors, a single color for pentagons and two different colors alternatively for hexagons, one can use these two colors as a mnemonic for deciding whether one need to make a pentagon or hexagon in the next step. Remember that in a C60 every pentagon is surrounded by 5 hexagons and every hexagon is surrounded by 3 pentagons and 3 hexagons alternatively. Then one can start with a pentagon with a single color, then hexagons with two colors alternatively, eventually, one should get a beaded C60 correctly without using any other information.
Below is a few slides I used to teach people how to make a buckyball.
But, it is much easier to follow the simple mnemonic for making a buckyball.
If one wants to make a beaded C60 with two different colors, a single color for pentagons and two different colors alternatively for hexagons, one can use these two colors as a mnemonic for deciding whether one need to make a pentagon or hexagon in the next step. Remember that in a C60 every pentagon is surrounded by 5 hexagons and every hexagon is surrounded by 3 pentagons and 3 hexagons alternatively. Then one can start with a pentagon with a single color, then hexagons with two colors alternatively, eventually, one should get a beaded C60 correctly without using any other information.
Thursday, October 7, 2010
Weaving path for EMACs with 1,8-naphthyridine ligands
A simple weaving path for EMACs that contain 1,8-naphthyridine (萘啶) ligands:
I am not sure if Chern Chuang followed this path when he made the first beaded EMAC. Unlike EMACs with pyridyl-ligands, here one can use the same type of two-end weaving technique we used for making bead models of fullerenes.
I am not sure if Chern Chuang followed this path when he made the first beaded EMAC. Unlike EMACs with pyridyl-ligands, here one can use the same type of two-end weaving technique we used for making bead models of fullerenes.
Wednesday, October 6, 2010
Weaving path of EMACs
Qian-Rui told me the ingenious weaving path of EMACs he used. I made a schematic plot to show his weave path in the following figure. This path has a major difference from the ones we used for fullerenes before. I.e. one used only one end of fishing thread to weave pyridyl groups (hexagons, green) in the ligands. In doing so, he can weave the whole structure with only one long minimal fishing thread.
Tuesday, August 31, 2010
Spiral codes in "An atlas of fullerenes"
I finally got my own copy of Fowler and Manolopoulos's "An Atlas of Fullerenes" from the MIT coop at Kendall square last weekend. In the appendix of this book, there is a complete list of spiral codes for fullerenes with less or equal than 100 carbon atoms. So one can simply follow the spiral code to create the physical model of corresponding fullerene. More importantly, if we followed the simple weaving rule, no other information except the spiral code is required to make the correct physical model!
I made the only isomer of C72 and C74, which satisfies the IPR (isolated-pentagon rule) requirement, based on the spiral codes listed in the appendix immediately when I was back to Taiwan yesterday. Just as expected, the bead model of C72 has D6d symmetry, and the bead model of C74 has D3h symmetry as shown in the book.
Spiral codes:
C72: 1 7 9 11 13 18 22 24 27 34 36 38
C74: 1 7 9 11 14 23 26 28 30 32 35 38
I made the only isomer of C72 and C74, which satisfies the IPR (isolated-pentagon rule) requirement, based on the spiral codes listed in the appendix immediately when I was back to Taiwan yesterday. Just as expected, the bead model of C72 has D6d symmetry, and the bead model of C74 has D3h symmetry as shown in the book.
Spiral codes:
C72: 1 7 9 11 13 18 22 24 27 34 36 38
C74: 1 7 9 11 14 23 26 28 30 32 35 38
Saturday, February 13, 2010
Wednesday, November 11, 2009
Length of fishing thread
A liitle bit weaving technique is given here. This is important especially when one want to weave a long strip of beads (or equivalently zigzag carbon nanoribbon, ZGNR) with a width that consists of one or two hexagons, or more generally, odd or even numbers of hexagons. When we want to make a Mobius ZGNR, we will meet the situation like this. Basically, we need to weave a long strip first and then connect both ends in a particular way.
If we want to weave a long strip of ZGNR with a width of one hexagon, corresponding to a polyacene, we should observe that the length of fishing threads in both hands are running out at equal rate. It is easy to understand this by checking the following figure (a).
However, if we want to weave ZGNR with a width of two hexagons as shown in the bottom of the following figure (b), we should easily see that the one end of thread runs out much faster than the other end. It is also easy to understand why that happen by looking at the figure.
Of course, one can avoid this problem by following different weaving path. (see (c) of the following figure) I usually don't like to use weaving path (c). This is because weaving path (c) has the problem that it is much harder to make the structure constructed by this path tight.
If we want to weave a long strip of ZGNR with a width of one hexagon, corresponding to a polyacene, we should observe that the length of fishing threads in both hands are running out at equal rate. It is easy to understand this by checking the following figure (a).
However, if we want to weave ZGNR with a width of two hexagons as shown in the bottom of the following figure (b), we should easily see that the one end of thread runs out much faster than the other end. It is also easy to understand why that happen by looking at the figure.
Of course, one can avoid this problem by following different weaving path. (see (c) of the following figure) I usually don't like to use weaving path (c). This is because weaving path (c) has the problem that it is much harder to make the structure constructed by this path tight.
Wednesday, July 2, 2008
T120 weaving code
Here is the weaving procedure I used for creating T120 toroidal nanotube.
This code is resulting from my experience. The details may vary from time to time.
As I have addressed before, I separate the weaving procedure of a torus into two steps: in the first step, we weave the inner part of the torus; and in the second step, we weave the outer part. It is obvious why the inner part has to be built first, otherwise it will become difficult to weave the inner part if we weave from outer part inward.
You may wonder why I didn't follow the latitude completely for T120. The reason is that I want to avoid the the awkward position that may arise if I follow the latitude from inner part. If I weave all ten heptagons at the beginning, later on, I still need to return back the connected region between hexagons and heptagons. Usually, it is quite difficult to do that if I use 4mm beads. For larger beads, it may be ok to weave completely along latitude coordinate.
This code is resulting from my experience. The details may vary from time to time.
As I have addressed before, I separate the weaving procedure of a torus into two steps: in the first step, we weave the inner part of the torus; and in the second step, we weave the outer part. It is obvious why the inner part has to be built first, otherwise it will become difficult to weave the inner part if we weave from outer part inward.
You may wonder why I didn't follow the latitude completely for T120. The reason is that I want to avoid the the awkward position that may arise if I follow the latitude from inner part. If I weave all ten heptagons at the beginning, later on, I still need to return back the connected region between hexagons and heptagons. Usually, it is quite difficult to do that if I use 4mm beads. For larger beads, it may be ok to weave completely along latitude coordinate.
Tuesday, July 1, 2008
Saturday, November 24, 2007
Spiral Codes of Fullerenes
I re-checked my copy of the "Atlas" and found that there are actually no missing pages but with those pages arranged in reverse order. I don't know how'd that ever happen. For convenience, I made a digital copy of the spiral code table on our wiki site.
Spiral Codes of Fullerene Data Base
I only typed cases under C50 without IPR and will continue to finish the copy in a couple of days.
By the way, I found that Fowler actually left a fortran code for generating the spiral codes. I think that I'll do a Matlab version some other day.
Spiral Codes of Fullerene Data Base
I only typed cases under C50 without IPR and will continue to finish the copy in a couple of days.
By the way, I found that Fowler actually left a fortran code for generating the spiral codes. I think that I'll do a Matlab version some other day.
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