After I published the last post, which I did not originally intend to have two parts, this comment was left by one of my blog’s followers. My answer is also shown.

A torus can be viewed as a flexible rectangle rolled into a donut shape, and I had used 24 small rectangles by 24 small rectangles as the settings for *Stella 4* for the torus, and its dual, in the last post — which, due to the nature of that program, are actually rendered as toroidal polyhedra. To investigate my new question, I increased 24×24 to 90×90, and these three images show the results. The first shows a 90×90 torus, the second shows its dual, and the third shows the compound of the two.

When I compare these images to those in the previous post, it is clear that these figures are approaching a limit as n, in the expression “nxn rectangle,” increases. What’s more, I recognize the dual now, of the true torus, at the limit, as n approaches infinity — it’s a cone. It’s not a finite-volume cone, but the infinite-volume cone one obtains by rotating a line around an axis which intersects that line. This figure, not a finite-volume cone, is the cone used to define the conic sections: the circle, ellipse, parabola, and hyperbola.

What’s more, I smell calculus afoot here. I do not yet know enough calculus.

“Learn a lot more about calculus” is definitely on my agenda for the coming Summer, for several reasons, not the least of which is that I plainly need it to make more headway in my understanding of geometry.

[Note: *Stella 4d*, the program used to make these images, may be found at http://www.software3d.com/Stella.php.]

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Thanks Robert. You’ll need a lot of calculus for this one, it looks like differential geometry to me (a topic I know almost nothing about!)

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There exist precise definition (as mathematical operation) of several kinds of duality. Some of them lead to different shapes in the same space, others lead to different spaces. Even the same duality generally used in polyhedron analysis gives different results depending on whether you consider a figure as its bounds or also its interior. Consider a pentagon. Its dual is also a pentagon (vertex edge, 5 of each). However the same pentagon viewed as a facet of dodecahedron gives icosahedron, wich is completely other story.

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Thank you for the info!

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I smell topology!

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Very interesting! I too smell topology!

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How many holes does the dual of the torus have?

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I think the answer is zero, but I could be wrong.

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Ive been mathematician for a long time but only recently discovered this concept of duals. This concept goes far deeper than geometry but intimately ties to the nature of reality. The dual of a torus is a hyperboloid as you have found via software assisted construction. Well this is according to Ken Wheeler the nature of magnetic fields, space and counter space coexisting and interweaving. And according to Victor Shauberger the universe is 2 way motion not 1 way as modern physics likes to believe. For example yes a river flows downhill but there are countercurrents flowing back upstream which Schauberger noticed fish used to maintain perfectly still in a seemingly raging current. Schauberger watched in shock as he saw a fish “levitate” up a steep waterfall! He pointed out that centripetal/implosion impetus is the way forward while mankind uses highly inefficient explosive centrifugal methods which Schauberger was extremely critical of. These two motions are undoubtedly duals of one another but man has a propensity to only see the Euclidean options while nature chooses its dual when trying to accomplish things. As such nature experiences no friction, perfect efficiency, and growth while man gets inefficiency , pollution, and decay. Walter Russell similarly pointed out that yes apples fall from tree but the apple then decomposes and radiates back outward into space. Gravity and radiation , dielectric and magnetic, upstream and downstream , torus and hyperboloid….Schauberger used hyperbolic funnels (I own one) to energize water .

Very exciting to start to see my love of mathematics and my love of “alternative” science start to merge 🙂

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