Geometry : Near-miss Johnson solids

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.

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. Near Miss Johnson Solids Viewer

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.

Galleries

Index

• From Stella Software
• From Orchid Palms
• From Polytope Wiki
• By Tom Shapiro
• From Polytope Discord
• From HedraWeb
• From RobertLovesPi.net

From Stella Software

^ Back to index ^

From Orchid Palms

^ Back to index ^

From Polytope Wiki

^ Back to index ^

By Tom Shapiro

^ Back to index ^

From Polytope Discord

^ Back to index ^

From Hedra Web

^ Back to index ^

From RobertLovesPi.net

^ Back to index ^

From Stella Forum

^ Back to index ^

Sources, Credits and Thanks

^ Back to index ^

• Near-miss Johnson solid on the Polytope Wiki
• Orchid Palms
• Modified Johnson Near Misses By Ed Baumann
• Wikipedia
• Stella software by Robert Webb
• The Polytope Discord
• Craig S. Kaplan
• The 554 Problem by Melinda Green
• RobertLovesPi.net
• Hedraweb - David Smith
• A Facebook Group : Polyèdres et particularités mathématiques.
• Two polyoxovanadate-based metal-organic polyhedra with undiscovered “near-miss Johnson solid” geometry
• Hendecagonal Near-miss Polyhedral Cages
• Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons

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