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  #1  
Old 09-26-2021, 12:57 PM
Laurence Finston Laurence Finston is offline
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Polyhedron Model: Pentagonal Hexecontahedron

I've just finished the plans for a model of a Pentagonal Hexecontahedron.
They can be found here: https://www.gnu.org/software/3dldf/g...s/pentahex.pdf

This is the web page with the GNU 3DLDF and TeX source code: The GNU 3DLDF Polyhedron Models Page

Information about the pentagonal hexecontahedron can be found at Wolfram Mathworld: Pentagonal Hexecontahedron -- from Wolfram MathWorld

This model is somewhat more complex than the one of the small rhombicosidodecahedron. It consists of 60 irregular pentagons.

A pentagonal hexecontahedron is an "Archimedean dual". It comes in two forms, a right-handed (dextro) and a left-handed (laevo) form, which are mirror images of each other (also known as "enantiomorphs"). See the article on "Chirality" at Wikipedia for more information (I couldn't get the link to work for some reason).

The plans include both the right-handed and the left-handed forms, each one with and without numbered pentagons. The numbered versions might be useful for coloring, when not printing the plans directly onto the paper or cardstock used for the model, or for other purposes.

These plans also illustrate the usefulness of using a 3D package for two-dimensional technical drawings. Since the plans are 2D, obviously a 2D graphics program could have been used to make them. In this case, the pentagons would have had to have been copied and then shifted ("translated") and rotated into the proper positions. In 3D, it's possible to just rotate them around an edge through the third dimension, which is much easier for the user of the program.

In addition, it's possible with a single command to create the left-handed version by rotating the entire plan around the x- or z-axis (in this case, the plans lie in the x-z plane). A mirror image could not be created by translations and rotations in a plane, at least not of the entire model. It would be possible by rotating and translating the individual pentagons, but it would be nearly as much work as doing the first version.

I hope that the tabs are all correct. I will test this and post any corrected versions to the website. I said in a previous posting that the tabs were strictly speaking just a convenience. This is true as far as it goes and it's not hard to cut off a tab that's not needed or replace a missing one with a strip of paper or a piece of tape. However, it would be a serious inconvenience if there were no tabs at all.

I've improved the macro used for making the tabs in these plans as compared with the ones in the plans for the small rhombicosidodecahedron. There are now four parameters that govern the size, shape and position of the tabs: The offset from each of the ends and the widths at each of the ends. There is one parameter governing the size of the pentagons, namely the length of the short sides.

Unlike some other polyhedra, this one is not made up of sections that can simply be copied and translated. They are rotated with respect to each other at lots of different angles. As a consequence of this, I've programmed the plans in such a way that there is a variable associated with each of the pentagons and each of the points. This makes for the maximum amount of flexibility when adjusting the plans for different sizes.

In this version, the plans are divided into two halves. The pentagons would have had to be very small for the complete plan to fit on one DIN A3 page. Even divided into two, they are fairly small. However, it would not be difficult to make larger versions, spread over more pages.

Last edited by Laurence Finston; 09-26-2021 at 01:10 PM. Reason: Addition
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Old 09-28-2021, 02:40 AM
Laurence Finston Laurence Finston is offline
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The plans after cutting out and creasing along with the tools used.

This is just ordinary printer paper for the purpose of checking whether the tabs are all correct. I discovered one minor error that I've corrected on the plans linked to above.

This time, I plan to use paste or glue to put the prototype together and try to make it look nicer than the one of the small rhombicosidodecahedron.
Attached Thumbnails
Polyhedron Model:  Pentagonal Hexecontahedron-escf0001.jpg   Polyhedron Model:  Pentagonal Hexecontahedron-escf0002.jpg  
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  #3  
Old 09-28-2021, 10:55 AM
AP40rocktruck AP40rocktruck is offline
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I saw these & have downloaded them. Very exciting, looking forward to taking a crack at one.

Cheers
AP40
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Old 09-28-2021, 11:07 AM
Laurence Finston Laurence Finston is offline
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Quote:
Originally Posted by AP40rocktruck View Post
I saw these & have downloaded them. Very exciting, looking forward to taking a crack at one.
AP40

Thank you, AP40. Please let me know how it works out or if you have any problems, suggestions, etc.
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Old 09-28-2021, 09:14 PM
Dave Pete Dave Pete is offline
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Looks like an interesting challenge. Look forward to seeing the final model. I downloaded the plans from the link for a future project. Thank you for providing the link. What are the two polished tools in your picture? Not seen anything like them before.
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Old 09-29-2021, 01:10 AM
Laurence Finston Laurence Finston is offline
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Quote:
Originally Posted by Dave Pete View Post
Looks like an interesting challenge. Look forward to seeing the final model.
Thanks. Actually, I like making the drawings for plans even more than I like building models, which is why I have a backlog of model plans that I've never executed. I would be thrilled if someone else did so, though.

I actually have another set of plans in the pipeline and I hope to post them this week or on the weekend.

The main thing needed for the polyhedron models is patience. They are straightforward to put together until the end, when it becomes a little like performing surgery. However, somehow I've always managed to get them together.

One thing that would probably help would be to sew the last faces together using long threads. This way, one can use them to pull them together from the outside, sort of like the way the masts of ships-in-bottles are pulled upright.

Quote:
Originally Posted by Dave Pete View Post
Thank you for providing the link.
My pleasure.

Quote:
Originally Posted by Dave Pete View Post
What are the two polished tools in your picture? Not seen anything like them before.
They are manicure tools from the company Zwilling (which means "twin" in German). The combination tool is still available: ZWILLING Kombi Manikure Instrument, Edelstahl mattiert kaufen

The other two have unfortunately been discontinued. I think I remember getting them just in time. I mostly use them for gluing. In this case, I used them for creasing. The sharp ends are very good for getting between the tabs and the tabletop.
Attached Thumbnails
Polyhedron Model:  Pentagonal Hexecontahedron-tools1.jpg   Polyhedron Model:  Pentagonal Hexecontahedron-tools2.jpg  
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  #7  
Old 09-29-2021, 02:41 PM
Laurence Finston Laurence Finston is offline
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There's been a delay in putting the prototype together: I was at a copyshop yesterday and printed out the plans on 170g/m2 paper. I don't know the weight of the paper I originally printed them out on, but I think it was probably 80g/m2.

The 170g paper is much stiffer but it was still possible to cut out the plans using scissors and I could fold rather than scoring. For a finished model I wouldn't recommend scissors, and I would recommend Bristol board rather than any kind of paper. However, for a prototype I think the results will be much better than with the 80g paper, which is pretty flimsy. I'm not sure whether much of a difference from the previous verson is visible on the photo.

I've started pasting the model together. Using paste works well; much better than tape. The faces don't need to be held together while the paste dries.

I've included a couple of photos of a simpler polyhedron model that I actually got as far as assembling. It's a truncated octahedron: Truncated Octahedron -- from Wolfram MathWorld
I used Bristol board and glued watercolor paper to the faces. I planned to give them a second coat but have never gotten around to it. Eventually, I put away the paints and I'll have to figure out which ones they were.

I don't have plans for this polyhedron on my website; I probably used AutoCAD for them a long, long time ago.

I was a bit worried about the watercolors, especially when I considered finishing it and giving it as a gift to a child. The pigments in watercolors can be poisonous (like cadmium) and it's probably not a good idea to handle the model a lot and then eat or put one's fingers in one's mouth or rub one's eyes.

I looked into non-poisonous colors and found some made by the company Stockmar. They make crayons, colored pencils and other art supplies for children: Welcome to STOCKMAR - STOCKMAR – for lots of creative fun I used to give them as presents when my friends had children of the right age. The company is a believer in J.W. von Goethe's color theory, which happens to be completely wrong and out-dated. However, their art supplies are still nice.

I bought all the watercolors from them that were available at the time, eight or so. They look alright but there's no comparison with really high-quality watercolors. I have a few from the company Old Holland which are really beautiful. I have a lot from the company Schmincke, which are very high quality and widely available in Germany, where I live. Otherwise, I have quite a few from the company Lukas, which are not as high quality as the others, but a good value. I have a few from Winsor and Newton, which are also very good, but overpriced in Germany.
Attached Thumbnails
Polyhedron Model:  Pentagonal Hexecontahedron-poly1.jpg   Polyhedron Model:  Pentagonal Hexecontahedron-poly3.jpg   Polyhedron Model:  Pentagonal Hexecontahedron-poly4.jpg   Polyhedron Model:  Pentagonal Hexecontahedron-poly5.jpg   Polyhedron Model:  Pentagonal Hexecontahedron-poly6.jpg  


Last edited by Laurence Finston; 09-29-2021 at 03:05 PM.
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  #8  
Old 09-30-2021, 08:26 AM
Laurence Finston Laurence Finston is offline
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I'm in the middle of gluing the model together. I expect to finish in an hour or so.

As you can see, you don't have to glue too many faces together before it begins to look three-dimensional.

I took one of the manicure tools, stuck it inside the thin paper prototype of the small rhombicosidodecahedron and pushed the faces out, which makes it look better, I think. It's kind of started to grow on me.
Attached Thumbnails
Polyhedron Model:  Pentagonal Hexecontahedron-poly7.jpg   Polyhedron Model:  Pentagonal Hexecontahedron-poly8.jpg   Polyhedron Model:  Pentagonal Hexecontahedron-poly9.jpg   Polyhedron Model:  Pentagonal Hexecontahedron-poly10.jpg   Polyhedron Model:  Pentagonal Hexecontahedron-poly11.jpg  

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  #9  
Old 09-30-2021, 09:49 AM
Laurence Finston Laurence Finston is offline
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Got the two halves pasted together. Waiting for the paste to dry so I can finally assemble the model.
Attached Thumbnails
Polyhedron Model:  Pentagonal Hexecontahedron-poly12.jpg   Polyhedron Model:  Pentagonal Hexecontahedron-poly13.jpg   Polyhedron Model:  Pentagonal Hexecontahedron-poly14.jpg   Polyhedron Model:  Pentagonal Hexecontahedron-poly15.jpg  
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  #10  
Old 09-30-2021, 11:07 AM
Laurence Finston Laurence Finston is offline
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Stuck for the moment

I've reached a point where I have to change my approach.

I will need some glue that sets faster than the paste I have been using so far. I think the best thing would be to use hide clue. I will have to get out my glue pot and the glue and start soaking the latter in water. It's been a long time since I've done this.

I think I will need to use the manicuring tools to hold the faces together in order to avoid burning my fingers and do the connections one-by-one, with pauses to let the glue cool.

I have a glue gun but I didn't have good results with it when I tried using it for cardboard models in the past.

It would have been better to make marks for sewing the model together and to punch holes. When I revise these plans I'll put in markings, as I did for some previous models. I'll also need to brush up my sewing skills.

There were quite a few tabs that needed to be cut off and some that were missing. I've updated the plans on the website. If you've viewed or downloaded the PDF or the source files before, you may need to refresh them in your browser. It may have cached them and not "notice" that they've changed. The timestamp on the revised version of the plans is 30 Sep 2021 6:47 p.m (on the upper right corner of each page).
Attached Thumbnails
Polyhedron Model:  Pentagonal Hexecontahedron-poly16.jpg   Polyhedron Model:  Pentagonal Hexecontahedron-poly17.jpg  

Last edited by Laurence Finston; 09-30-2021 at 11:08 AM. Reason: Typo
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