Rings, Tori, Pigtails
A round ring is known to most all people. A torus to some of us.
In my chapter on Bars, Rods, Columns, and Prisms; if I formed any of these solid figures into a proportionate circular shape I would create rings. From a side view, anything outside the centerline is expanded, and anything inside the centerline is contracted.
The volume for the ring remains the same as the original rod or bar. The lateral surface area also remains the same. These values can be calculated as with the bars and rods... using the centerline as the height. (Centerline once around, one circumference.) Use a full perpendicular cross section slice for a base area. See Fig- 28 & 29.
Did you know that all the above applies, if the bars and rods are sliced and formed proportionately into any regular polygon ring? As shown in Fig - 30, as an equilateral triangle ring. Again using the centerline, the math as with rods and bars applies.
And, if you twist the bar or rod proportionately, as well as forming it into some polygon ring, the math is still the same.
During the forming of the rings, the centerline is held constant. What area and volume gained on the outside is lost on the inside of the centerline, and maintains a balance. The common "Slinky" toy (or a spring) is a way to demonstrate and visualize these forming actions.
Now look at Fig. - 31 & 32. I call this a Pig Tail for lack of another name. It is a regular square base pyramid that has been proportionately formed into a ring, with the centerline height, as one circumference back to itself. This could just as well have been a cone. It could also have been a heptagon or octagon base pyramid.
In this case the math we used on cones and pyramids applies. The centerline circumference is as with the cone or pyramid height. The lateral line we had as side height, S, would be a line drawn on the side surface, as looking at the ring from the open side, and overlaying the centerline. (S, from the side view would have the same circumference as the centerline, but is actually longer as it is drawn on the outside surface.)
These figures could also be twisted as long as they are kept proportional, and the centerline is maintained.
I have not gone any further with these figures, but I suspect there is more to know, and I would think it interesting to evolve these figures into spirals and such, while keeping the math more simple than I suspect it presently may be? And, maybe finding or keeping similarities between circles and regular polygons?
