Geometry : Near-miss Johnson solids

Created on June 22, 2024. Last update on August 02, 2024.

TL;DR : The goal for this web page is to gather near-miss Johnson solids found by people across the world in an interactive gallery.

In the realm of convex solids with regular faces, I have already made galleries displaying the Platonic solids, the Archimedean solids, the prism and antiprism infinite families and the Johnson solids. Beyond that there exists a set of solids that seems like they would be part of the Johnson solids but not quite, there faces are not quite regular but can be very close. For this reason, they are called near-miss Johnson solids. There is no strict definition for them though the usual rules is that the faces are close to planar, that the edges length do not deviate too much from 1 and that the angles of the faces do not deviate from the regular angle for the face. It is "commonly" admitted that the edge ratio (maximum over minimum edge length) should be less than 1.1 with planar faces but that still leaves a lot of room for deviation so the boundaries of this set are very fuzzy. It is up to personal or expert judgement to decide whether a certain solid qualifies or not. All the solids that are discarded belong to the set of near near-miss Johnson solids. If you wish to dig into this topic yourself, I have set up very good references at the bottom of this page, particularly the Polytope Wiki.

Alex Doskey defines multiple metrics to quantity the deviation of a solid to a true convex solid with regular faces (CRF). Based on his work, I have computed several metrics. The model is scaled so that the mean edge length is equal to one.

  • Edge Ratio (ER) : the ratio of the maximum edge length over the minimum edge length.
  • Aggregate Edge Deformation (E) : the sum over each edge of the absolute value of the difference between its length and the mean length (=1).
  • Aggregate Angle Deformation (A) : the sum over each angle of each face of the absolute value of the difference between the angle and the regular angle, in degrees.
  • Polygonal Deformation (P) : the sum over each diagonal of each face of the absolute value of the difference between the diagonal length and the corresponding length in a regular convex face.
  • Planarity (PL) : the sum over each point over each face of the distance to the best plane (according to least square) for this face.
For any CRF, the edge ratio is one and all other metrics are zero. I could also have gone for the standard error instead of the accumulated absolute error. Alex Doskey adds that "The form shown must be at a minimum distortion configuration with distortion confined to the smallest faces.". Of course, satisfying this constraint depends very heavily of the metrics considered to measure distortion and the weights used to combine them into an optimization function for the solver. As you may see through the gallery, the planarity if often zero, so this metric has a very high (maybe infinite) weight. The solids presented here are one solution to this problem, most of the time obtained using Hedron or Stella (which has two solvers).

One question is why would anyone care about such an irregular and badly defined category of solids that some would call them blasphemous and look away ? One answer is that sometimes the deviations are small enough that building them out of paper, magnets, carboard, wood, steel, whatever with the same making process used for Johnson solids works (thank to you tolerances, sometimes you are cursed, here you are blessed). They really are fake Johnson solids that would ravish the common eye, bringing variety, and confuse experts. Another answer is why not. Realization may also change the shape as the constraints of the real world and making process are different from the mathematical ones.

The models are organized by source so some may appear multiple times (the latest count is 161). Date is expressed as day/month/year. If you happen to know the author or date of discovery for any solid tagged as unknown please let me know and give a source if possible. Same if you find a mistake. If you wish to add your own to the list, either provide a file with the geometry (.off, .obj, .stel, ...) or send me a bunch of picture so that I can try to make it (success is not guaranteed).

If you would like to search through the solids to check if the one you made already exists or not, please use this alternative viewer which has a search bar. Details for the search bar are available through the bottom left icon.

Each model view is interactive, the controls are the following:

  • Click and drag with the mouse right click to rotate the model.
  • Click and drag with the mouse left click to pan the model.
  • The mouse middle click zooms in or out.
  • Press R to reset the model position or triple click.
  • Press F to trigger full screen or double click.
  • Press P to pause/resume the animation.
It should work on touch screens too, if not, I am very sorry for you. The web techs may have changed so please try on a computer or with a different browser (checked with Chrome and Firefox).

Galleries

Index

From Stella Software

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5-5-4 Acrohedron
Author: Melinda Green
Date: Unknown
Source OFF
Decagonal-faced Near Miss
Author: Unknown
Date: Unknown
Source OFF
Dodecagonal-faced Polyhedron
Author: Craig S. Kaplan, Georges W. Hart, Alex Doskey
Date: 2001
Source OFF
Enneagonal-faced Polyhedron
Author: Craig S. Kaplan, Georges W. Hart
Date: 2001
Source OFF
Equal-edged Near Miss 1
Author: Robert Webb
Date: Unknown
Source OFF
Equal-edged Near Miss 2
Author: Robert Webb
Date: Unknown
Source OFF
Half-truncated Truncated Icosahedron
Author: Daniele Barbaro
Date: 1569
Source OFF
Hendecagonal-faced Polyhedron
Author: Craig S. Kaplan
Date: ~2001
Source OFF
Hexagonal-Heptagonal Mixed Prism
Author: Jim McNeill
Date: Unknown
Source OFF
Pentagonal-Hexagonal Mixed Prism
Author: Jim McNeill
Date: Unknown
Source OFF
Square-Pentagonal Mixed Prism
Author: Jim McNeill
Date: Unknown
Source OFF
Tetrated Dodecahedron
Author: Alex Doskey, Robert Austin
Date: 2002, 2003
Source OFF
The Wedge
Author: Unknown
Date: Unknown
Source OFF
Truncated Truncated Icosahedron
Author: Craig S. Kaplan
Date: Unknown
Source OFF
Zonish Truncated Icosahedron
Author: Robert Austin
Date: Unknown
Source OFF

From Orchid Palms

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#1 : Augmented Pentagonal Prism
Author: Alex Doskey
Date: 01/06/2006
Source OFF
#2 : Myers 3
Author: Bill Myers
Date: 08/03/2010
Source OFF
#3 : Bi-augmented J49
Author: Roger Kaufman
Date: 07/08/2006
Source OFF
#4 : Trisquare Hexadecatrihedron
Author: Jim McNeill
Date: 12/05/2003
Source OFF
#5 : J91-like
Author: Roger Kaufman
Date: 06/06/2006
Source OFF
#6 : 5-5-4 Acrohedron
Author: Melinda Green
Date: 07/08/2006
Source OFF
#7 : Wedge 4-5
Author: Jim McNeill
Date: 20/04/2004
Source OFF
#7 : Wedge 4-5 Augmented
Author: Jim McNeill
Date: 08/05/2006
Source OFF
#7 : Wedge 4-5 Augmented 2
Author: Jim McNeill
Date: 08/05/2006
Source OFF
#8 : J85 Pentagonal Prism
Author: Roger Kaufman
Date: 24/01/2010
Source OFF
#9 : Hexawedge
Author: Mick Ayrton
Date: 24/04/2006
Source OFF
#10: J85 Altered
Author: Roger Kaufman
Date: 21/11/2009
Source OFF
#11 : Myers 2s
Author: Bill Myers
Date: 06/03/2010
Source OFF
#12 : 4-5-6 Acrohedron
Author: Alex Doskey
Date: 05/06/2006
Source OFF
#13 : 6-5-5 b05
Author: Robert Webb
Date: 17/04/2006
Source OFF
#14 : Pentagonal Square Barrel
Author: Jim McNeill
Date: 1999
Source OFF
#14 : Hexagonal Square Barrel
Author: Roger Kaufman
Date: 28/04/2006
Source OFF
#15 : 6-5-5 b05ex
Author: Roger Kaufman
Date: 14/09/2006
Source OFF
#16 : J85 Altered 2
Author: Roger Kaufman
Date: 21/11/2009
Source OFF
#17 : 5-4-3 nm
Author: Robert Webb and Alex Doskey
Date: 2002
Source OFF
#18 : J26 Augmented
Author: Roger Kaufman
Date: 07/08/2006
Source OFF
#19 : Myers 6
Author: Bill Myers
Date: 12/03/2010
Source OFF
#20 : X2
Author: Roger Kaufman
Date: 21/04/2006
Source OFF
#21 : UM1
Author: Ulrich Mikloweit
Date: 01/12/2009
Source OFF
#22 : Fullerene c28
Author: Robert Austin and Roger Kaufman
Date: 27/04/2006
Source OFF
#23 : 5-5-4 Acrohedron Tetra-augmented
Author: Roger Kaufman
Date: 09/05/2007
Source OFF
#24 : Myers 4
Author: Bill Myers
Date: 09/03/2010
Source OFF
#25 : Quasi Gyro J90
Author: Jim McNeill
Date: 07/08/2006
Source OFF
#26 : 5-5-4 Acrohedron Bi-augmented
Author: Roger Kaufman
Date: 09/05/2007
Source OFF
#27 : UM2
Author: Ulrich Mikloweit
Date: 01/12/2009
Source OFF
#28 : Myers 7
Author: Bill Myers
Date: 29/04/2010
Source OFF
#29 : A56
Author: Ulrich Mikloweit
Date: 04/01/2007
Source OFF
#30 : Saucer
Author: Mick Ayrton
Date: 07/08/2006
Source OFF
#31 : J90 Truncated
Author: Roger Kaufman
Date: 08/05/2007
Source OFF
6-5-3 Acrohedron nm
Author: Roger Kaufman
Date: 21/10/2005
Source OFF
Enneagonal Polyhedron
Author: Unknown
Date: 12/05/2003
Source OFF
9-3-5-3 Acrohedron
Author: Robert Webb
Date: 27/03/2006
Source OFF
Hendecagonal Polyhedron
Author: Craig Kaplan
Date: 12/05/2003
Source OFF
12-5-4 Near miss
Author: Alex Doskey
Date: 01/05/2003
Source OFF
Tetrated Dodecahedron
Author: Alex Doskey and Robert Austin
Date: 26/12/2003
Source OFF
Tetrahedrally Expanded Tetrated Dodecahedron
Author: Unknown
Date: 04/08/2009
Source OFF
Snub Expanded Tetrated Dodecahedron
Author: Mick Ayrton
Date: 03/10/2005
Source OFF
Curvy Octahedroid
Author: Mick Ayrton
Date: 24/04/2006
Source OFF
Tripentagonal Snub Dodecahedron
Author: Mick Ayrton
Date: 05/05/2004
Source OFF
Bipentagonal Snub Dodecahedron
Author: Mick Ayrton
Date: 05/05/2004
Source OFF
Hexagonally Expanded Snubbed Dodecahedron
Author: Mason Green
Date: 27/03/2006
Source OFF
Wedge 5-6
Author: Unknown
Date: 20/04/2004
Source OFF
Wedge 6-7
Author: Unknown
Date: 20/04/2004
Source OFF
Wedge 7-8
Author: Unknown
Date: 21/04/2004
Source OFF

From Polytope Wiki

^ Back to index ^

Sesquitruncated square pyramid
a.k.a. Heptagonal-faced polyhedron
Author: Tom Shapiro
Date: 03/01/2022
Source OFF
Sesquitruncated triangular bipyramid
a.k.a. Octagonal-faced polyhedron
Author: Tom Shapiro
Date: 03/01/2022
Source OFF
Sesquitruncated octahedron
a.k.a. Enneagonal-faced polyhedron
Author: Craig S. Kaplan, George W. Hart
Date: 2001
Source OFF
Sesquitruncated rhombic dodecahedron
a.k.a. Decagonal-faced polyhedron
Author: Robert Webb
Date: Unknown
Source OFF
Sesquitruncated trapezo-rhombic dodecahedron
Author: Tom Shapiro
Date: 03/01/2022
Source OFF
Sesquitruncated pentagonal bifrustum
Author: Tom Shapiro
Date: 03/01/2022
Source OFF
Sesquitruncated hexagonal bifrustum
Author: Tom Shapiro
Date: 03/01/2022
Source OFF
Sesquitruncated pentagonal icositetrahedron
a.k.a. Hendecagonal-faced polyhedron
Author: Craig S. Kaplan
Date: ~2001
Source OFF
Sesquitruncated deltoidal icositetrahedron
Author: Tom Shapiro
Date: 03/01/2022
Source OFF
Sesquitruncated gyrodeltoidal icositetrahedron
a.k.a. Sesquitruncated dual of J37
Author: Tom Shapiro
Date: 03/01/2022
Source OFF
Sesquitruncated tetradeltoctapentagonal icositetrahedron
a.k.a. Sesquitruncated dual of J45
Author: Tom Shapiro
Date: 03/01/2022
Source OFF
Sesquitruncated order-6 truncated tetrakis hexahedron
Author: Tom Shapiro
Date: 03/01/2022
Source OFF
Sesquitruncated cuboctahedron
Author: Tom Shapiro
Date: 03/01/2022
Source OFF
Sesquitruncated icosidodecahedron
Author: Tom Shapiro
Date: 03/01/2022
Source OFF
Truncated order-6 truncated triakis tetrahedron
Author: Craig S. Kaplan
Date: 2016
Source 1 Source 2 OFF
Rectified order-6 truncated triakis tetrahedron
Author: Unknown
Date: Unknown
Source OFF
Expanded order-6 truncated triakis tetrahedron
Author: Craig S. Kaplan, Georges W. Hart, Alex Doskey
Date: 2001
Source OFF
Tetrated dodecahedron
Author: Alex Doskey, Robert Austin
Date: 2002, 2003
Source OFF
Chiral tetrated dodecahedron
Author: Unknown
Date: Unknown
Source OFF
Tetrahedrally expanded tetrated dodecahedron
Author: Tadeusz E. Dorozinski
Date: 2008 ?
Source 1 Source 2 OFF
Snub expanded tetrated dodecahedron
Author: Mick Ayrton
Date: Unknown
Source OFF
Hexagonally expanded snub dodecahedron
a.k.a. Pentahexagonal pyritoheptacontatetrahedron
Author: Mason Green
Date: 2006
Source OFF
Bipentagonal snub dodecahedron
Author: Unknown
Date: Unknown
Source OFF
Order-4 truncated triangular bipyramid
a.k.a. Three-dimensional associahedron
Author: Dov Tamari, Jim Stasheff
Date: 1951, 1963
Source OFF
Disdyakis order-4 truncated triangular bipyramid
Author: Robert Austin
Date: March 2016
Source OFF
Partially expanded disdyakis order-4 truncated triangular bipyramid
Author: Unknown
Date: Unknown
Source OFF
Elongated disdyakis order-4 truncated triangular bipyramid
Author: Robert Austin
Date: March 2016
Source OFF
Expanded disdyakis order-4 truncated triangular bipyramid
Author: Unknown
Date: Unknown
Source OFF
Triated dodecahedron
Author: Unknown
Date: Unknown
Source OFF
Partially truncated trapezo-rhombic dodecahedron
Author: Unknown
Date: Unknown
Source OFF
Bisquare icosahedron
Author: Tom Shapiro
Date: 01/06/2022
Source OFF
Semi-rectified pentagonal bipyramid
Author: Unknown
Date: Unknown
Source OFF
Diminished triaugmented truncated octahedron
Author: Unknown
Date: Unknown
Source OFF
Diminished triaugmented truncated icosahedron
Author: Unknown
Date: Unknown
Source OFF
Diminished rectified order-6 truncated triakis tetrahedron
Author: Unknown
Date: Unknown
Source OFF
Sphenomegalunacorona
Author: Alex Doskey
Date: 2006
Source OFF
Tetrasquare icosioctahedron
Author: Bill Myers
Date: March 2010
Source OFF
Bisquare dodecahedron
Author: Roger Kaufman
Date: 2006
Source OFF
Trisquare enneadecahedron
Author: Jim McNeill
Date: Unknown
Source OFF
Belted bilunarotunda
Author: Roger Kaufman
Date: 2006
Source OFF
5-5-4 acrohedron
Author: Melinda Green
Date: Unknown
Source OFF
Diminished sphenocorona
Author: Jim McNeill
Date: Unknown
Source OFF
Name: OrchidPalms #8
Author: Roger Kaufman
Date: Unknown
Source OFF
Name: OrchidPalms #9
Author: Mick Ayrton
Date: 2006
Source OFF
Trisquare icosiheptahedron
Author: Roger Kaufman
Date: 2009
Source OFF
Name: OrchidPalms #11
Author: Bill Myers
Date: 2010
Source OFF
Name: OrchidPalms #12
Author: Unknown
Date: Unknown
Source OFF
Semi-rectified hexagonal bipyramid
Author: Robert Webb
Date: Unknown
Source OFF
Semi-rectified order-6 truncated hexagonal antitegum
Author: Jim McNeill
Date: 1999
Source OFF
Bidiminished expanded semi-rectified hexagonal bipyramid
Author: Roger Kaufman
Date: 2006
Source OFF
Tetrasquare icosioctahedron
Author: Roger Kaufman
Date: 2009
Source OFF
Name: OrchidPalms #17
Author: Alex Doskey
Date: 2002
Source OFF
Trisquare tridecahedron
Author: Roger Kaufman
Date: 2006
Source OFF
Name: OrchidPalms #19
Author: Bill Myers
Date: 2010
Source OFF
Name: OrchidPalms #20
Author: Roger Kaufman
Date: 2006
Source OFF
Tetrasquare pentadecahedron
Author: Ulrich Mikloweit
Date: 2009
Source OFF
Order-6 truncated triakis tetrahedron
Author: Robert Austin, Roger Kaufman
Date: Unknown
Source OFF
Pentakis 5-5-4 acrohedron
Author: Roger Kaufman
Date: 2007
Source OFF
Name: OrchidPalms #24
Author: Bill Myers
Date: 2010
Source OFF
Quasigyrodisphenocingulum
Author: Jim McNeill
Date: Unknown
Source OFF
Parabiaugmented 5-5-4 acrohedron
Author: Roger Kaufman
Date: 2007
Source OFF
Diminished tetrasquare pentadecahedron
Author: Ulrich Mikloweit
Date: 2009
Source OFF
Second trisquare enneadecahedron
Author: Bill Myers
Date: 2010
Source OFF
Name: OrchidPalms #29
Author: Ulrich Mikloweit
Date: 2007
Source OFF
Gyrate partially expanded disdyakis order-4 truncated triangular bipyramid
Author: Tom Shapiro
Date: 15/11/2021
Source OFF
Diminished disphenocingulum
Author: Roger Kaufman
Date: 2007
Source OFF
The Edshammar polyhedron
11E
Author: Edshammar
Date: 1969
Source OFF
14E
Author: Lidin et al.
Date: 07/1992
Source OFF
Chamfered cube
Author: Unknown
Date: Unknown
Source OFF
Chamfered dodecahedron
Author: Unknown
Date: Unknown
Source OFF
Rectified truncated icosahedron
Author: Daniele Barbaro
Date: 1569
Source OFF
Truncated truncated icosahedron
Author: Craig S. Kaplan
Date: Unknown
Source OFF
Rectified chamfered dodecahedron
a.k.a. Rectified truncated rhombic triacontahedron
Author: Robert Austin
Date: March 2015
Source OFF
Icosahedrally expanded chamfered dodecahedron
a.k.a. Dodecadiminished expanded truncated icosahedron
Author: Unknown
Date: Unknown
Source OFF
Expanded truncated icosahedron
Author: Robert Austin
Date: November 2013
Source OFF
Snub rectified truncated icosahedron
Author: Unknown
Date: Unknown
Source OFF

By Tom Shapiro

^ Back to index ^

sesq1
Author: Tom Shapiro
Date: 2022
OFF
sesq2
Author: Tom Shapiro
Date: 2022
OFF
sesq3
Author: Tom Shapiro
Date: 2022
OFF
sesq4
Author: Tom Shapiro
Date: 2022
OFF
Sesquitruncated Square Antitegum
Author: Tom Shapiro
Date: 2022
OFF
Sesquitruncated Dual of Semi-rectified Order-6 Truncated Hexagonal Antitegum
Author: Tom Shapiro
Date: 2022
OFF
Snub Cube but with 1 Pentagon
Author: Tom Shapiro
Date: 08/11/2021
Source OFF
Snub Cube but with 2 Pentagons
Author: Tom Shapiro
Date: 08/11/2021
Source OFF
Snub Cube but with 4 Pentagons
Author: Tom Shapiro
Date: 08/11/2021
Source OFF
17-hedron
Author: Tom Shapiro
Date: 08/11/2021
Source OFF
Totally Asymmetric 26-hedron
Author: Tom Shapiro
Date: 08/11/2021
Source OFF
Non-Rhrombic Triacontahedron
Author: Tom Shapiro
Date: 09/11/2021
Source OFF
Totally Asymmetric 22-hedron
Author: Tom Shapiro
Date: 09/11/2021
Source OFF
Totally Asymmetric 18-hedron
Author: Tom Shapiro
Date: 09/11/2021
Source OFF
Diminished Thrice Hexakis Truncated Octahedron
a.k.a.Diminished Triaugmented Truncated Octahedron
Author: Tom Shapiro
Date: 10/11/2021
Source OFF
Snub
Author: Tom Shapiro
Date: 11/08/2023
Source OFF
Sesquitruncated Order-5 Truncated Deltoidal Hexecontahedron
Author: Tom Shapiro
Date: 10/08/2024
Source OFF

From Polytope Discord

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guy
Author: the p o kekodjek
Date: 23/07/2023
Source OFF
Snub Triangular Prism
Author: H A M
Date: 23/04/2024
Source OFF
Snub Pentagonal Prism
Author: H A M
Date: 23/04/2024
Source OFF
Pentadecasquare Icositrihedron
Author: Puffer
Date: 20/11/2021
Source OFF
Hexadecahedron
Author: Puffer
Date: 20/11/2021
Source OFF
Icosiheptahedron
Author: Puffer
Date: 26/11/2021
Source OFF
Truncated Pentagonal Prism
Author: the p o kekodjek
Date: 27/12/2021
Source OFF
Bi-augmented Truncated Pentagonal Prism 1
Author: the p o kekodjek
Date: 27/12/2021
Source OFF
Bi-augmented Truncated Pentagonal Prism 2
Author: the p o kekodjek
Date: 27/12/2021
Source OFF
Bi-augmented Truncated Pentagonal Prism 3
Author: the p o kekodjek
Date: 27/12/2021
Source OFF
Icosagonal Tetraconheptahedron
Author: Tom Shapiro
Date: 05/01/2022
Source OFF
Cursed Esquigybcu
Author: the p o kekodjek
Date: 07/01/2022
Source OFF
Octatruncated Triakis Octahedron
Author: Livio Zucca
Date: 08/04/2022
Source OFF

From Hedra Web

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Unnamed
Author: David Smith
Date: 27/04/2016
Source OFF
Unnamed Augmented
Author: David Smith
Date: 27/04/2016
Source OFF
Unnamed Augmented Bis
Author: David Smith
Date: 27/04/2016
Source OFF
Unnamed BiAugmented
Author: David Smith
Date: 27/04/2016
Source OFF
Unnamed BiAugmented Bis
Author: David Smith
Date: 27/04/2016
Source OFF
Unnamed BiAugmented Ter
Author: David Smith
Date: 27/04/2016
Source OFF
Unnamed 2
Author: David Smith
Date: 27/04/2016
Source OFF
Unnamed 3
Author: David Smith
Date: 27/04/2016
Source OFF
Unnamed 4
Author: David Smith
Date: 27/04/2016
Source OFF
TriPenta 2
Author: David Smith
Date: 06/05/2016
Source OFF
TriPenta 3
Author: David Smith
Date: 06/05/2016
Source OFF
Unnamed 5
Author: David Smith
Date: 04/08/2024
Source OFF
Unnamed 6
Author: David Smith
Date: 04/08/2024
Source OFF
Unnamed 7
Author: David Smith
Date: 04/08/2024
Source OFF

From RobertLovesPi.net

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Zonish Cuboctahedron
Author: Robert Austin
Date: June 2014
Source OFF
Augmented Truncated Dodecahedron
Author: Robert Austin
Date: June 2014
Source OFF
Trick Johnson No1
Author: Hiroshi Nakagawa, Nobuaki Muto
Date: June 2010
Source OFF

From Stella Forum

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Nadia's Icosahedron (minimum edge ratio)
Author: Nadia
Date: Unknown
Source OFF
Nadia's Icosahedron (flat faces)
Author: Nadia
Date: Unknown
Source OFF

Sources, Credits and Thanks

^ Back to index ^

Thanks to Tom Shapiro for filling in most on the Polytope Wiki pages on Near miss solids and finding many himself.