Shapes of toruses

   After spending a lot of time working with my tube beads, I spent some time this week going back to my molecular shapes.  My idea was to make several long narrow toruses (I suppose it's actually "tori", but I can't bring myself to say that) that I would join together, probably with metal tube structures. I found I couldn't make what I wanted, so I took some time to just try and figure out how shaping these structures works.
  The basic idea is that if you build a tube out of hexagons ( 6-bead circles serve as hexagons), it will be just that--a straight tube.  If you want it to curve, you add  heptagons on the inside and  pentagons on the outside of the curve.  It takes 12 heptagons on the inside and 12 pentagons on the outside to make a full  rotation and create a torus.  Actually, you can do it with 10 and 10, and that's what many of the toruses in the beaded molecules blog do, but to get it all around without having to force it you need 12.  Often you get a firmer structure by making it do something it doesn't quite "want" to do, but here I was trying to not do that.
  What I've come to realize, after making a jillion of the things, is that what you really need is just to add 12 extra beads on the inside and a corresponding number left out on the outside, and you can do that in any configuration.  The easiest, and one I had done before, is to use 6 octagons on the inside and 6 squares on the outside. The picture here is taken from one of my very first posts, and shows  a hexagonal torus done that way on the right of the piece.  It's easy to see because the octagons are green, the squares blue and the rest red-brown. The big square and triangle of the necklace were made by trying to stretch those same octagons into ever tighter angles, and show how much I didn't know what I was doing back then.  On the other hand, it's nice to read that post and see that I've actually learned something since then.
  The 6 octagon/6 square torus makes for a very blocky shape that I don't much like, so I haven't used it often.  But now I can see that there are lots of ways to add 12 extra beads on the inside of the circle, the number 12 being divisible in so many ways.

The first torus in this picture was made by making the center out of 4 circles of 9 beads.  That way each circle has 3 extra beads for a total of 12 extra. It's a bit odd because 9 is an odd number, so sometimes you're adding more to the back and sometimes to the front, so it doesn't lie quite flat.  But you get a sort of 4-cornered torus. The center is actally a short fat diamond on 1 side and a tall skinny one on the other, so it's a bit odd. The second torus uses 3 circles of 10 beads each, and so makes a triangle.  And the last one uses 2 circles of 12 beads each, and makes a sort of long skinny shape.  I can now see that if I had added some hexagons (6-bead circles) between the 12-bead ones, I'd have gotten a longer torus, which was what I was originally looking for.  But by then I was into exploring torus shapes.Actually in any of these configurations, you could make the donut hole larger by just putting 6-bead circles between the larger circles.

  Having gotten  handle on the inside of the torus, I started playing with the outside.  I liked the triangular center best, so I used a center of 3 10-bead circles in each of these.  On the outside you need to take 12 beads away from your structure of hexagons.  You can do that with 12 pentagons, or with 6 squares, or with a combination of the 2.  Using all squares ( the 1st one) makes the hardest angles.  Using some or all pentagons makes the curves gentler, which I prefer.  Also it makes a difference whether you put the smaller circles near the points of the triangle or along the flat sides.  If you could space your small circles evenly around the torus, you'd get a circular outside with a triangular hole in the center.  I couldn't quite figure out how to do that. but I came close in #2.  It has 6 5-bead circles and 3 4-bead ones.  There are 2 pentagons on each flat side and a square at each point of the triangle.  The last 2 are both done with 12 pentagons, so they have nice, gentle curves.  #3 has 4 pentagons at each point of the triangle, so it maintains it's triangular shape. #4 has 2 pentagons at each point and 2 on each flat side.
  I have no idea if anyone is still reading, or if this makes any sense at all.  But I do like being able to talk out my ideas, even if no one is listening.  And if you actually are reading this and have questions, be sure and ask.