polytest

Polyhedra is the plural of polyhedron. A polyhedron is a many sided three-dimensional object. A cube is a common example. As most of you know, I love math and am intrigued by shapes. I have made a whole variety of different kinds. Most of what is on here is made from paper products that have been glued or slided together. Every now and then I like to challenge myself with something more difficult like pencils or straws. I have also made may polyhedra from modular origami. I have some at my parents house, but most of it was given to my cooperating teacher at South and to Maighie's mom.

Things I'm Planning

• Another Weaved Icosadodecahedron
• Possibly Something Made from Toothpicks

• I saw a smaller and more colorful version of something exactly like this at a place I subbed at. I decided to go for it. I made some calculations to get the angles right. All of it is made of 3 angles. It's basically a dodecahedron in sphere form. I constructed the design and printed off 60 strips. Those strips folded into the spherical triangles. I glued those strips, the bits, the pieces and ended with a nice end product. I used some paper clips to help the glue set better. I also had a little fun with it in the middle of the project. It wasn't as easy to take off. In the end it is about 11 in. in diameter and took about 13 hours to make. That includes calculation time. I also want to point out that I did those 13 hours of work within 24 hours. If I would ever do this again or were to give suggestion, I would use thicker paper so the end product as very straight arches.
  
  
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    Calc
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    Strips
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    Fold
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    5 Strips Glued
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    Part
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    Helmet
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    Done
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    Done
  
  




• I decided to do another. On my list of three the catenary arch sounded like the best fit for what I was willing to do. Plus I've been thinking about it for a while. This project that I have put the most mathematical work on of all of them. First I planned my dimensions. I wanted it to be in the shape of an inverted catenary (e.g. St. Louis Arch) The general formula being y=-a*cosh(x/a)+10+a for me (cosh is a hyperbolic cosine). It was to be 12" tall and 14" wide on the outside and 10" tall and wide on the inside as well as 2" thick. I made 12 divisions thus making 13 pieces. They were cut in very specific places. I graphed a middle catenary (dotted) and made line division perpendicular to that curve every half-step on the horizontal (i.e. x=.5, 1.5, 2.5, 3.5...). I did this to only one side because it reflects vertically. I found the intercestions between the lines and curves and plotted them to scale on graph paper. Using Illustrator I plotted polygons and made my cutouts. Building wasn't too bad. I used the support method here.

  The reason I wanted a catenary was because I didn't want it being supported by anything else. In the last arch you can see 2 coasters cholding it together. This shape has the benifit of letting gravity work with it. It is not a parabola, though they look similar and ahve been mistakien may time thoughout history. You can make your own catenary by getting a uniform piece of string and letting it hang while holding each end of the string.

  
  
  
  
  
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    Calc
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    Plan
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    Layout
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    Pieces
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    Parts
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    Together
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    Done
  
  



• This is one step in an experimental project. I was weaving together what would best be described an icosadodecahedron. I made each color strip from three strips of paper and glued the ends together. My first attempt went horrible because I went along blindly with no planning and ended up with a large mess. The strips were too thin and I had weaved improperly. It wasn't supporting itself in a ball and the glue wasn't holding from being ripped apart so much. This is my second attempt. I calculated the width to be perfect so there was no wiggle room in the strips. Unfortunately, as you can see, this was an undercalculation. I'm still not sure where I went that wrong. At least this one can hold some shape and is weaved correctly. I haven't had the will nor the mathmatical insipration to make a third go at it. I'm sure I'll try again eventually.
  
  
  
  
  
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    Weave
  
  
  




• I wanted to use some material other than paper. I noticed that a deck of cards is pretty cheap and, for the most part, they are already cut. I then decided to make a polyhedra with a few of the faces missing (pentagons and triangles) seeing as rectangles make horrible pentagons and triangle, but make pretty good squares. I wanted to use as many as I could. My only real choice was the rhombicosadoedcahedron. Since some of the faces are missing you could see it as a icosahedron where the cards represent edges. (You could also call it a dodecahedron if the cards are edges that go the other way) I wanted the cards to slide together nicely, that way it would hold itself together better. I calculated the angle of the cut to be arctan(tan54*sin54) which is approximately 48.07 degrees. I cut the cards and started to assemble. I realized then my calculations were wrong. I assembled the shape with tape due to my gross miscalculation. Thank heavens I wasn't sending a man into space. My recalculation was arctan(2*tan54*cos54) which is approximately 58.28 degrees. I should be shot. I made another one for a fellow teacher with the correct angle and I also made the cards closer together so it would REALLY hold without tape. I don't have a picture of that one.
  
  
  
  
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    5 tetra
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    5 tetra
  
  
  




• I had made something similar to this before in WI, but I decide to go for style and a challenge. What I had made before consisted of 11 pieces and the blocks had square cross-sections. I also simplified my math (Note: I was planning the constructing while donating plasma) by assuming the side view was a semi-circle instead of a semi-22-gon. To build that one I made a truss for the inside and removed it on completion.

  I took it a level higher on this one. First, I made it with 13 pieces. Second, the cross-sections were triangles. Third, i didn't assume the side view was a circle and did ALL the math. (Note: it really wouldn't have made that much of a difference if I hadn't since I had so many pieces.) Lastly, construction would not be as easy. A truss wouldn't work as well in this case because the pieces would not be able to lie flat. Now that I think of it. I could have made two slightly inflated trusses and had the pieces wedge in-between the two. Of course, that would require me to construct two trusses, and I didn't want to spend the time nor use the cardboard and tape to make them. I wanted something easy and quick to make. I decided to support it from the outside since that was nice and flush. I made a long strip of paper and lined the pieces up as shown. I was able to lift it up and rotate it around with ease.

  Specs: The inside arch has a diameter of 9 inches and the outside 13 inches. These are the only non-uniform polyhedra that I have posted so far (i.e. All the sided aren't regular polygons).
  
  
  
  
  
  
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    all
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    pieces
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    line up
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    U see
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    done
  
  
  
  




• I have made this twice before and I'm not sure why I made another one. Maybe because of the loss of my rhomicosadodecahedron. This is more modular origami. They are five compound tetrahedrons in the shape of a dodechedron. I got the instructions off the internet. It's pretty annoying to make. The first time I tried it in Oregon, I smashed what I had cuz I couldn't get the fourth terahedron in properly. (Update: the one I smashed I rebuild later and is the colorful one.) There are 30 pieces to put together. In all, it takes an hour to build if you have a cutting board. I also have computer model which IS made by me. I made it so you can somewhat see the dodecahedron.
  
  
  
  
  
  
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    5 tetra
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    5 tetra
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    Model
  
  
  
  



• I was sick of doing things with paper. And what better to appose paper than a pencil. Well maybe scissors, but pencils are definitely second. I had 36 golf pencils left from the wedding. I didn't want to make something too small and I wanted it to keep shape. The wire holds shape well, but I wanted the faces to look real nice. Equilateral triangles are hard to screw up compared to regular pentagons. All this in mind I went with the icosahedron. It only required 30 pencils and all faces are triangles. While building there were two things I wanted to be mindful of : the directions of the pencils and the amount of wire on each pencil. I didn't want any vertex to have 4 or 5 erasers at a point nor 4 or 5 tips. The first picture shows my plan. In my head I made a path to follow that would allow each pencil to be wrapped exactly twice. The second picture shows what a pencil wrapped once looks like. My plan required me to cut the wire so the polyhedron has two separate wires. I'm sure there is, if my discrete math skill are still with me, a way of doing it without cutting, but that would have required more thinking and would have been slightly more annoying to build. Well, I referenced my discrete Math book and, yes, I could have done it without cutting.
  
  
  
  
  
  
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    prep
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    wrap
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    starting
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    halfway
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    almost
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    still wrapping
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    done
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    good
  
  
  
  




• I wanted to make something bigger than I had ever made before. I think I got in over my head. I decided to make a snub dodecahedron. It's got 12 pentagons and 80 triangles. I put about 13 hours into it cumulatively. My longest project ever. Second is an origami rhombicosahedron. I'm making this one with the flaps inside, so I need to tape it for extra support. It's about 14" in diameter. I had trouble putting the last piece on. The flaps made some trouble with that one. An interesting thing about snub polyhedron is that it has rotational, but not reflection symmetry. Because of that there are two physically different snub dodecahedrons (as well with the snub cube). If you want to know what the other would look like, then look at the picture in the mirror.
  
  
  
  
  
  
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    Half snub
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    Half snub
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    Whole snub
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    Whole snub
  
  
  




• This is something I made out of boredom and curiosity. It's basically two tedrahedra (4 sides, a triangular pyramid) within each other and the faces are a of a sacred geometry that kind of looks like the duck choking plastic remnants of a non-existent 3-pack a beer (pop in my case) with the circles clipped, so the ducks don't get strangled. It's glued together. It kinda looks cool, but is way to jumbled to look cool. It's a start.
  
  
  
  
  
  
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    tetra thing
  
  
  
  



• I tested an idea that was in my head for a few months. Modular circle polyhedrons (don't worry there are two pluralizations, like fish). No glue and quite stable. A plus in my book. I make the circles n stuff myself. If you're interested I use Adobe Illustrator. The first below is the making of a rhombicosadodecahedron (62 sided). The yellow shapes are "pentagons", the blue are "squares", and the purple are "triangles". The quotations are there because they are really circles cut to represent those shapes. It took me about 2 1/2 hours to cut and make. Cutting takes a majority of the time. It's 9" in diameter. The last is of a dodecahedron and an icosahedron. It is interesting to know that the circles trace out a sphere. Not so much with the icosahedron because the circle should intersect a little more. In the model they overlap.
  
  
  
  
  
  
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    Paper
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    Cut Out
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    build
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    Build
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    inside
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    rhomicos...
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    More
  
  
  




• This lamp shades is in the shape of an icosa-(20)hedron. I also made a lamp shade in a dodeca-(12)hedron, but I trashed it cause I broke it on the trip here. The icosahedron was a present for Alison. It is about, if memory serves me correctly, 15" in diameter. It is adorned with a fairy scene, also made by me. She still has it to my knowledge.
  
  
  
  
  
  
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    Lamp
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    Lamp