Talk:Near-miss Johnson solid

message: 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)
replies: text: 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)

text: 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)
thread_title: possible vertices

message: 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)
replies: text: 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)
thread_title:  vocabulary 

message: 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)
replies: text: 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)

text: Okay, converted to data table like Johnson solids. Looks like Jim McNeil's measure is his own, unpublished besides at his website . 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)
thread_title:  Number of examples 

message: 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)
replies: text: If you allow pentagonal distortion, though, I would think that you could use regular flat hexagons.  RobertAustin 13:30, 5 January 2007 (UTC)

text: 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)
thread_title:  Poor examples 

message: Where does this 7 sided equal area but different faced solid with 4 equalateral tringles and 3 diamonds
fit in?  http://www.frankchester.com/2010/chestahedron-geometry/ Is a new section needed?  — Preceding unsigned comment added by 78.238.220.41 (talk) 22:24, 4 November 2013 (UTC)
replies: text: Faces must be regular polygons in this list. The Chestahedron is given as a special case at Diminished trapezohedron. Tom Ruen (talk) 23:09, 4 November 2013 (UTC)
thread_title:  Diamond/rhombic faces?

message: http://www.cgl.uwaterloo.ca/csk/projects/nearmisses/ Holy (talk) 18:04, 16 March 2017 (UTC)
thread_title:  A source for more candidates 

message: The article should include explanations for why each of these is a near-miss, e.g. "The pentagons are not quite equiangular" or "The square faces are actually slightly rectangular", perhaps with some measure for just how far off of regular they are.  This measure would also help for defining what does and does not belong on the list, say you include everything that's within 5% of being a true Johnson solid (comparing longest and shortest edges, or largest and smallest angles on a supposedly "regular" face), then you can have a criteria for what doesn't qualify. 75.112.52.7 (talk) 14:36, 9 October 2019 (UTC)
replies: text: Many of these figures can be built in different ways by deforming different shapes. Again, this concept is really not well-defined, and it's study is pretty much exclusively recreational. – OfficialURL (talk) 21:05, 13 April 2020 (UTC)
thread_title:  Explanations for near misses