[math-fun] Why isn’t PentakisDodecahedron Johnson 93??
Johnson 93? <https://www.wolframcloud.com/objects/810fbe92-6ab3-480b-9f86-e4869990cf46> —rwg And why can't we tumble that figure?
I see 6 triangles meeting at a vertex. If the triangles are regular, then either they are coplanar or the polyhedron is concave. Tom Bill Gosper writes:
Johnson 93? <https://www.wolframcloud.com/objects/810fbe92-6ab3-480b-9f86-e4869990cf46> —rwg And why can't we tumble that figure? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
It looks like the tumble problem has been fixed. Rotating it makes it clear that the triangles are not regular. Wikipedia says they are isosceles, and that the Pentakis Dodecahedron is a Catalan Solid. Tom Tom Karzes writes:
I see 6 triangles meeting at a vertex. If the triangles are regular, then either they are coplanar or the polyhedron is concave.
Tom
Bill Gosper writes:
Johnson 93? <https://www.wolframcloud.com/objects/810fbe92-6ab3-480b-9f86-e4869990cf46> —rwg And why can't we tumble that figure? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Here's a colored, auto-rotating rendering. You can also click- and-drag to manually rotate it (requires a mouse, not a touch screen): http://www.karzes.com/polyhedra/polyhedron.html?ph=v5.6.6 Tom Tom Karzes writes:
It looks like the tumble problem has been fixed. Rotating it makes it clear that the triangles are not regular. Wikipedia says they are isosceles, and that the Pentakis Dodecahedron is a Catalan Solid.
Tom
Tom Karzes writes:
I see 6 triangles meeting at a vertex. If the triangles are regular, then either they are coplanar or the polyhedron is concave.
Tom
Bill Gosper writes:
Johnson 93? <https://www.wolframcloud.com/objects/810fbe92-6ab3-480b-9f86-e4869990cf46> —rwg And why can't we tumble that figure?
The solid is a regular dodecahedron augmented with a pentagonal pyramid on each face. The height of the pyramids can be adjusted so that the triangles are equilateral, but that makes the edges between pentagonal pyramids concave. On Mon, Mar 11, 2019 at 3:47 PM Tom Karzes <karzes@sonic.net> wrote:
Here's a colored, auto-rotating rendering. You can also click- and-drag to manually rotate it (requires a mouse, not a touch screen):
http://www.karzes.com/polyhedra/polyhedron.html?ph=v5.6.6
Tom
Tom Karzes writes:
It looks like the tumble problem has been fixed. Rotating it makes it clear that the triangles are not regular. Wikipedia says they are isosceles, and that the Pentakis Dodecahedron is a Catalan Solid.
Tom
Tom Karzes writes:
I see 6 triangles meeting at a vertex. If the triangles are regular, then either they are coplanar or the polyhedron is concave.
Tom
Bill Gosper writes:
Johnson 93? < https://www.wolframcloud.com/objects/810fbe92-6ab3-480b-9f86-e4869990cf46> —rwg And why can't we tumble that figure?
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"Peraugmented Dodecahedron"? And what about (Mono)Augmented? Ortho-, Meta, & Para Biaugmented?, <Help-me-Polya> Triaugmented? . . . Undecaaugmented?? What am I missing? —rwg On Mon, Mar 11, 2019 at 11:44 AM Bill Gosper <billgosper@gmail.com> wrote:
Johnson 93? <https://www.wolframcloud.com/objects/810fbe92-6ab3-480b-9f86-e4869990cf46> —rwg And why can't we tumble that figure?
The missing fact, I think, is that if you augment two adjacent pentagons of a dodecahedron, the edge between the two pyramids will be a valley and not a ridge, so the resulting polyhedron won't be convex. The Wikipedia article on "Johnson solid" lists the four possible ways to augment a dodecahedron: on one face, two different ways to do it to two faces, and on three faces. You can't pick four faces of a dodecahedron without two being adjacent, so that's all there is. This is one of those enumeration problems where the Frobenius-Burnside-Polya-Uncle-Tom-Cobbley-And-All counting formula is just overkill. On Mon, Mar 11, 2019 at 4:45 PM Bill Gosper <billgosper@gmail.com> wrote:
"Peraugmented Dodecahedron"? And what about (Mono)Augmented? Ortho-, Meta, & Para Biaugmented?, <Help-me-Polya> Triaugmented? . . . Undecaaugmented?? What am I missing? —rwg
On Mon, Mar 11, 2019 at 11:44 AM Bill Gosper <billgosper@gmail.com> wrote:
Johnson 93? < https://www.wolframcloud.com/objects/810fbe92-6ab3-480b-9f86-e4869990cf46> —rwg And why can't we tumble that figure?
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participants (3)
-
Allan Wechsler -
Bill Gosper -
Tom Karzes