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).