Talk:Near-miss Johnson solid

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[edit] possible vertices

I'm missing something somewhere. It's "obvious" that two regular triangles cannot form a vertex with a third polygon whose angle is 2π/3 or more; and yet the vertex figure \cos{\frac{\pi}{3}} : \cos{\frac{\pi}{3}} : \cos{\frac{\pi}{N}} obeys the triangle inequality for all N. Can you show me the vertex figures for 3.3.6, 3.3.7? —Tamfang 19:18, 2 April 2006 (UTC)

Sorry, I see I was wrong in listing existence merely by positive angle defects, but I missed requirement that internal angle of only face can't exceed sum of angles of the other two faces! No time now for me to check again now, so if you're happy with what you changed (that I reverted), I trust your corrections. Tom Ruen 02:19, 4 April 2006 (UTC)
Incidentally, a while ago I made a test article listing vertex figures used in the uniform and johnson solids. A better article would link them all to articles, but a bit of a pain to complete links to all the long formal names. User:Tomruen/Polyhedra_by_vertex_figures. Tom Ruen 02:29, 4 April 2006 (UTC)

[edit] vocabulary

I used the word compound for the eleven Js that contain a rotunda. Is there a better word? —Tamfang 20:39, 20 May 2006 (UTC)

Hmmmm... Looking at Johnson solid terminology, seems like augmented or augumentations may be the most general term. Tom Ruen 05:13, 21 May 2006 (UTC) ... WELL, looking again, Augmented means that a pyramid or cupola has been joined to a face of the solid in question. is limited, but seems as good as anything. Tom Ruen 05:18, 21 May 2006 (UTC)

[edit] Number of examples

As I write this, there are six examples of near-misses shown, with wikilinks to near-misses that have separate pages. Should we add more, perhaps even going for a comprehensive listing? After all, the Johnson solid page lists (and shows images of) all 92. Just a question for discussion.... RobertAustin 16:50, 30 December 2006 (UTC)

If you could explain a measure for near-misses, then it makes sense to me to create a longer table of examples ordered by that measure, or alternately subtables by symmetry type C/D/T/O/I perhaps? Tom Ruen 22:03, 30 December 2006 (UTC)
Okay, converted to data table like Johnson solids. Looks like Jim McNeil's measure is his own, unpublished besides at his website [1]. Perhaps a simpler measure would be to limit the list under some VEF count, and models that are close enough to be built with rigid models. I admit the 3 truncated Catalan solids will fail a construction test for nonplanar faces or unfoldable vertex figures (like 6.6.6)! Tom Ruen 22:30, 30 December 2006 (UTC)

[edit] Poor examples

Okay, I removed these as near-misses, since they are "too far" to be built with regular polygons. Tom Ruen 22:33, 30 December 2006 (UTC)

Okay, added back Truncated triakis tetrahedron since its on McNeil's list, but as a physical model, the regular hexagons must be nonplanar to fit. Tom Ruen 23:14, 30 December 2006 (UTC)

If you allow pentagonal distortion, though, I would think that you could use regular flat hexagons. RobertAustin 13:30, 5 January 2007 (UTC)
Agreed since that's what the truncated forms are - irregular pentagons and perfect hexagon (perfect 'red polygons in general below). I was just being conservative in the removal to here. Best to show the most exact models on the page. Tom Ruen 23:20, 5 January 2007 (UTC)
Name Image verfs. V E F F3 F4 F5 F6 F8 F10 Symmetry
Truncated triakis tetrahedron 4 (5.5.5)
24 (5.5.6)		 28 42 16     12 4     Td
Truncated rhombic dodecahedron 24 (4.6.6)
8 (6.6.6)		 32 48 18   6 12       Oh
Truncated rhombic triacontahedron 60 (5.6.6)
20 (6.6.6)		 80 120 42   12 30       Ih