Re: [math-fun] 2300 yr old news
I'm not sure how polar reciprocation determines the exact size of a dual polyhedron. —Dan Mike Stay wrote: ----- On Sun, May 26, 2019 at 6:10 PM Dan Asimov <dasimov@earthlink.net> wrote:
I've long been confused about the exact metric definition of "dual polyhedron". Sure, define one "new vertex" for each old face, and one "new edge" for each pair of old faces that share an edge — that gives the dual combinatorially. But as for the exact shape & size, I'm deeply confused. (We're in R^3.)
Polar reciprocation: https://math.wikia.org/wiki/Dual_polyhedron -----
The definition I remember is this: Imagine that your R^3 is the w=1 hyperplane through an R^4 parameterized by (w, x, y, z). Then to find the polar reciprocal of the point P = (1, x, y, z), take the line L through P and the origin (0, 0, 0, 0). Construct the hyperplane H through the origin, perpendicular to L. H intersects the w=1 hyperplane in a "hyperline" F, which is actually a plane in R^3, and F is the defined polar reciprocal of the original point P. On Sun, May 26, 2019 at 10:10 PM Dan Asimov <dasimov@earthlink.net> wrote:
I'm not sure how polar reciprocation determines the exact size of a dual polyhedron.
—Dan
Mike Stay wrote: ----- On Sun, May 26, 2019 at 6:10 PM Dan Asimov <dasimov@earthlink.net> wrote:
I've long been confused about the exact metric definition of "dual polyhedron". Sure, define one "new vertex" for each old face, and one "new edge" for each pair of old faces that share an edge — that gives the dual combinatorially. But as for the exact shape & size, I'm deeply confused. (We're in R^3.)
Polar reciprocation: https://math.wikia.org/wiki/Dual_polyhedron -----
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I always thought of the dual as defined so that the edges of the original polyhedron and its dual cross (for Platonic polyhedra, at their midpoints). - Cris
On May 26, 2019, at 8:25 PM, Allan Wechsler <acwacw@gmail.com> wrote:
The definition I remember is this:
Imagine that your R^3 is the w=1 hyperplane through an R^4 parameterized by (w, x, y, z).
Then to find the polar reciprocal of the point P = (1, x, y, z), take the line L through P and the origin (0, 0, 0, 0). Construct the hyperplane H through the origin, perpendicular to L. H intersects the w=1 hyperplane in a "hyperline" F, which is actually a plane in R^3, and F is the defined polar reciprocal of the original point P.
On Sun, May 26, 2019 at 10:10 PM Dan Asimov <dasimov@earthlink.net> wrote:
I'm not sure how polar reciprocation determines the exact size of a dual polyhedron.
—Dan
Mike Stay wrote: ----- On Sun, May 26, 2019 at 6:10 PM Dan Asimov <dasimov@earthlink.net> wrote:
I've long been confused about the exact metric definition of "dual polyhedron". Sure, define one "new vertex" for each old face, and one "new edge" for each pair of old faces that share an edge — that gives the dual combinatorially. But as for the exact shape & size, I'm deeply confused. (We're in R^3.)
Polar reciprocation: https://math.wikia.org/wiki/Dual_polyhedron -----
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I always thought of the dual as defined so that the edges of the original polyhedron and its dual cross (for Platonic polyhedra, at their midpoints).
For instance, if we take the convex hull of a cube and its dual octahedron, we get the rhombic dodecahedron. Similarly the dodecahedron and icosahedron give the rhombic triacontahedron. - Cris
On May 26, 2019, at 8:25 PM, Allan Wechsler <acwacw@gmail.com> wrote:
The definition I remember is this:
Imagine that your R^3 is the w=1 hyperplane through an R^4 parameterized by (w, x, y, z).
Then to find the polar reciprocal of the point P = (1, x, y, z), take the line L through P and the origin (0, 0, 0, 0). Construct the hyperplane H through the origin, perpendicular to L. H intersects the w=1 hyperplane in a "hyperline" F, which is actually a plane in R^3, and F is the defined polar reciprocal of the original point P.
On Sun, May 26, 2019 at 10:10 PM Dan Asimov <dasimov@earthlink.net> wrote:
I'm not sure how polar reciprocation determines the exact size of a dual polyhedron.
—Dan
Mike Stay wrote: ----- On Sun, May 26, 2019 at 6:10 PM Dan Asimov <dasimov@earthlink.net> wrote:
I've long been confused about the exact metric definition of "dual polyhedron". Sure, define one "new vertex" for each old face, and one "new edge" for each pair of old faces that share an edge — that gives the dual combinatorially. But as for the exact shape & size, I'm deeply confused. (We're in R^3.)
Polar reciprocation: https://math.wikia.org/wiki/Dual_polyhedron -----
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participants (3)
-
Allan Wechsler -
Cris Moore -
Dan Asimov