__Near Misses__

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**Johnson Solid Near Misses**

When Norman Johnson was cataloging the convex polyhedra in the 1960's, he apparently came across "a number of tantalisingly close misses". A selection of such polyhedra are shown below:

A 'near-miss' polyhedron is one in which the faces are not quite regular. In many cases physical models can be made without any apparent distortion or 'forcing' of the faces. The 'nearness' of a miss is the subject of some debate amongst polyhedronists. No good definition exists. My own quantification (which I admit has its faults) is to define a triplet of figures:

'E'is the sum of the distortions of all of the edge lengths (assuming an edge length of 1)

'P'is the polygonal distortion, it is expressed as the sum of the distortion of all of the diagonals across each polygon. For polygons other than triangles, forms exist with all edge lengths 1 which are not regular (eg a rhombus).

'A'is the aggregate angular error. The absolute difference is determined between the angle of a regular polygon and the angle in the modelled polygon. These are summed for each angle in each polygon.

There are two ways in which these
numbers can be
read. '**E**' and '**P**' are both measures of side
distortion ('**E**'
is edges; '**P**' is diagonals), there is an element of trade off
between the
two. An approximation to the extent of the distortion is given by
'**E**+**P**'. The
accepted practice is that the distortion is confined to the smallest
faces.

'**A**' is a standalone measure of
distortion,
and is an attempt to quantify the 'nearness' of a near-miss in a single
value. This has its shortcomings though as (for example) a
rectangular face
would give an '**A'** value of zero.

Presented are below are the
closest Johnson Solid near misses ranked according to the parameter '**A'**.
Unless
numbers get out of hand, I will extend this list to include any
polyhedron meeting the criteria below with a value of **A **< 60°

To qualify as a Johnson Solid near miss (for the purposes of this list), a polyhedron must meet the following criteria:

- It must be convex.
- If all faces are assumed to be regular, there must be an angle deficit at each vertex. (i.e. the sum of the internal angles must be less than 360°)
- The form shown must be at a minimum distortion configuration with distortion confined to the smallest faces.

Further submissions are welcome.

**Number 1:
A=5.1****°**

This
augmented pentagonal prism was discovered by
Alex Doskey
in 2006. It can be formed by taking a pentagonal
prism and augmenting two adjacent square faces with pyramids.
Then augment the two adjacent triangular faces that result with tetrahedra.
The result can be seen here.
Note the two
'peaks' of the tetrahedra almost
co-incide
and there are a number of coplanar triangles. If a model is
formed with
these vertices together a convex figure results, the coplanar
triangles in
the original figure now have a dihedral angle 179.665º. (Stress map).
Distortion (**E**=0.027,
**P**=0
, **A**=5.07°). A convex
variation
can be formed where the opposite square face of the pentagonal
prism is also augmented with a pyramid,
the
distortion statistics are the same.

**Number 2:
A=7.6****°** **New
Entry 08 Mar 10**

This near miss was discovered by Bill
Myers in
March 2010. Only the
triangular faces are distorted. (Stress
map). Distortion:
(**E**=0.045, **P**=0, **A**=7.6º). The pair of
triangles which
look almost coplanar have a dihedral angle of 177.3º.

**Number 3:
A=10.9° **

This near miss was discovered
by Roger Kaufman in
2006. Only the triangular faces are distorted. (Stress
map). Distortion:
(**E**=0.059, **P**=0, **A**=10.9º). It can be
formed by
augmenting an augmented triangular
prism (J49)
with two pyramids (one on a square face and one on an adjacent face on
the
original augmentation) and then 'filling' the gap with a tetrahedron.

**Number 4: A=18.3****°**

This trisquare hexadecatrihedron has 16 triangular
and 3 square faces, and looks somewhat like a cube embedded in
an icosahedron (hence my informal
name of 'cubicos'),
. The squares are
regular and the aggregate distortion in the lengths of the triangular
edges
is only about 0.1 in total (stress map).
Distortion
(**E**=0.10, **P**=0
, **A**=18.3°).

**Number 5:
A=24.0****°**

This near miss, discovered by
Roger Kaufman in 2006, resembles
the Johnson Solid 'bilunabirotunda'
with one of the 'rotundas' replaced by a chain of three square faces.
The
pentagons are regular and the distortion is confined to the squares and
the
triangles. (stress map). Distortion (**E**=0.134, **P**=.300
,
**A**=24.0°).

This near miss containing a 5-5-4 acron was discovered by Melinda
Green and modelled by Alex
Doskey.
The triangular and square faces are distorted, but the pentagonal
faces
are regular. (Stress map). Distortion:
(**E**=0.028, **P**=0.718,**A**=24.4º). Note: the
pentagons can
be augmented resulting in further figures, some of which can be formed
with
regular squares and triangles. Such examples are listed
separately
below.

**Number 7:
A=34.0°**

A
family of 'wedges' can be obtained by
attaching an n-gon to an
(n+1)-gon by one square and a ring of triangles. Above the {3}-(4} wedge the wedges cease to be regular
and exhibit gradually higher levels
of distortion. This distortion can always be confined to
the
triangular faces and is most apparent for the faces close to the square
face. For the {4}-(5} wedge, the
triangle-triangle edges are distorted by 0.19 (stress
map), Distortion (**E**=0.194, **P**=0
,
**A**=34.0°).

Roger Kaufman has pointed out to me that either of the square faces in this polyhedron can be augmented with a square pyramid to form distinct Johnson Solid Near Misses. As the square face that is being augmented is regular, the augmentation does not affect the distortion statistics. Top square augmented (stress map). Side square augmented (stress map)

**Number 8:
A=34.7° New
Entry 24 Jan 10**

This near miss based on the Snub
Antiprism was discovered by Roger Kaufman. It is
generated
in similar fashion to his other J85 based discoveries
but
In this case, it involves part of a pentagonal prism fitting into a
space made
on one side. Only the
triangular faces are distorted. (Stress
map) Distortion:
(**E**=0.185, **P**=0, **A**=34.7º).

**Number 9:
A=35.4° **

This near miss was discovered by Mick
Ayrton in
2006. The distortion is confined to the triangles. (stress
map) Distortion: (**E**=0.134,
**P**=0, **A**=35º).
If the condition that the distortion is confines to the smallest faces
is
relaxed such that all faces are allowed to distort (here,
stress map) then the distortion
becomes
(**E**=0.0212, **P**=0.017, **A**=6.5º), a value of **A**
that
would claim a much higher position.

This near miss based on the Snub
Antiprism was discovered by Roger Kaufman in November 2009.
In Roger's words: "Take
the Snub Antiprism J85 and remove one square and two opposite adjoining
triangles to the square. What is left is now flexible. Flex it until
the
triangular gaps are 90 degrees and insert 2 squares. Now there will be
two
triangular holes left and fill those in." Only the
triangular faces are distorted. (Stress
map) Distortion:
(**E**=0.191, **P**=0, **A**=40.2º).
See also the
associated model with both squares of the Snub
Antiprism replaced.

__Number 11: A=42.1° New
Entry 06 Mar 10__

This near miss was discovered by Bill
Myers in
March 2010. Only the
triangular faces are distorted. (Stress
map) Distortion:
(**E**=0.192, **P**=0, **A**=42.1º). The pair of
triangles which
look almost coplanar have a dihedral angle of 177.5º.

**Number 12: A=43.1° **

This near miss containing a 6-5-4
acron was
discovered by Alex
Doskey in
2006. Only the triangular and
square faces are
distorted. (Stress map) Distortion:
(**E**=0.0387, **P**=0.535, **A**=43º).

This near miss containing a 6-5-5 acron was
discovered by Robert
Webb. Only the triangular
faces are
distorted. (Stress map) Distortion:
(**E**=0.170, **P**=0, **A**=44.7º).

**Number 14: A=45.3°
**

This near miss was discovered by myself
in 1999
(unnamed) and re-discovered independently by Roger Kaufman in 2006,
Roger calls
it an hexagonal 'square barrel'. The distortion
is confined to the triangular and square faces (stress map). Distortion:
(**E**=0.029, **P**=1.36, **A**=45º).

This polyhedron is a member of a family of polyhedra where for one particular value of n/d the polyhedron would be regular. Roger and Adrian Rossiter have calculated that this value is:

Values of n/d between 5 and 6 which have been tested are as follows:

n/d = 5 (stress map). Distortion: (

E=0.064,P=3.88,A=87º).

n/d = 11/2 (5.5) (stress map). Distortion: (E=0.042,P=2.42,A=67º).

n/d = 40/7 (5.714...) Distortion: (E=0.013,P=0.66,A=20º).

n/d = 23/4 (5.75) Distortion: (E=0.007,P=0,36,A=11º).

n/d = 6 (stress map). Distortion: (E=0.029,P=1.36,A=45º)

n/d = 23/4 shows the minimal value of A to date for this family, although it should be noted that non integer values of n/d are not near miss Johnson Solids as they are not convex polyhedra. Further (un-modelled) values of n/d which should give low values for A are: 63/11, 86/15, 103/18, 109/19, 149/26 ... but by this point the star polygon is too complex to model. For a really close value (to Roger's prediction) try 9109/1589. Alex Doskey has also pointed out that 3365/587 and 2872/501 should give the best values for d<1000.

It is interesting to note the almost
linear
relationship between **E **(and to a lesser extent **A**) and
the product
of the error from Roger's value and d (i.e. the number of loops around
the
symmetry axis.

Roger Kaufman has also (in 2013)
discovered a 'great square barrel' by changing the sign in the above
equation

The above image shows the n/d=11/5 example - this is an isomorph to
the n/d =
11/2 square barrel.

**Number 15: A=45.7°
**

This near miss containing dodecagons was
discovered by Roger Kaufman in 2006. It is an expanded version of
Robert
Webb's 6-5-5
acron shown
above. Only the triangular and square faces are
distorted. It is the closest near miss discovered to date to contain an
'apolydronic'
polygon. (Stress map)
Distortion:
(**E**=0.369, **P**=1.07, **A**=45.7º).

This near miss based on the Snub
Antiprism was discovered by Roger Kaufman in November 2009.
Proceed as per the associated
model with one square of the Snub Antiprism replaced, then repeat
the
process for the other square. Only the triangular faces are
distorted. (Stress map)
Distortion:
(**E**=0.217, **P**=0, **A**=46.7º).

This near miss was discovered
independently by
both Robert
Webb and Alex
Doskey in 2002. Only the triangular faces are distorted. (Stress
map) Distortion:
(**E**=0.263, **P**=0, **A**=47.7º). See also an
augmented form.

**Number 18: A=48.5°**:

This near miss was discovered by Roger
Kaufman in
2006. Only the triangular faces are distorted. (Stress
map) Distortion:
(**E**=0.245, **P**=0, **A**=48.5º). Roger
discovered this
figure by 'knocking parts off a sphenomegacorona'
but
it can as Roger also points out be made by augmenting a gyrobifastigium.

__Number 19: A=50.8° __**New
Entry 12 Mar 10**

This near miss was discovered by Bill
Myers in
March 2010. Only the
triangular faces are distorted. (Stress
map). Distortion:
(**E**=0.282, **P**=0, **A**=50.8º). It can be
regarded as an
augmented form of this polyhedron.

**Number 20: A=50.9° **

This near miss was discovered by Roger
Kaufman in
2006. Only the triangular faces are distorted. (Stress
map) Distortion:
(**E**=0.234, **P**=0, **A**=51º).

In Roger's words: "I found this while messing around with Stewart X. But it can also be derived by cutting J91 in half. Do a self augmentation of that half. Then there is a non-convex part. By augmenting pyramids on that it came close to closing. It can then be faceted and sprung into shape such that only those triangles are distorted."

__Number 21: __**A=52.1°**** New
Entry 3 Jan 2010**

The
above polyhedron was discovered by Ulrich
Mikloweit in 2009, there is also a variation
with a
pentagonal pyramid removed. Only the triangular faces are
distorted. (Stress
map) Distortion:
(**E**=0.273, **P**=0 , **A**=52.1°)

**Number 22: ****A=52.47°****
**

The
above polyhedron is an example of a 'fullerene',
and was brought to my attention by Robert Austin and Roger
Kaufman. It has
tetrahedral symmetry with 12 pentagons and 4 hexagons and as such is
the closest
near miss with higher than prismatic symmetry. The above model
has
the distortion confined to the pentagons, stress map.
Distortion (**E**=0.316,
**P**=3.07
, **A**=52.47°). If the hexagons are allowed to distort (here,
stress map) then the
distortion becomes (**E**=0.059,
**P**=2.19 , **A**=108°)

**Number 23: ****A=52.50°**** **

This near miss was discovered by Roger
Kaufman in
2007. It is an augmented form of Melinda Green's
5-5-4 acron.
All four pentagonal faces are replaced with pentagonal pyramids.
This
allows the distortion to be confined to the triangular faces, with the
square
faces remaining regular. (Stress
map).
Distortion:
(**E**=0.264, **P**=0, **A**=52.50º).

**Number 24:
A=53.6****°** **New
Entry 09 Mar 10**

This near miss was discovered by Bill
Myers in
March 2010. Only the
triangular faces are distorted. (Stress
map). Distortion:
(**E**=0.274, **P**=0, **A**=53.6º).

**Number 25: ****A=54.0°**

A
short personal diversion here: This
'potato' is the first near miss I came across, I was actually
attempting to
model a disphenocingulum from Polydron
and I inadvertently rotated one of the sphenoid caps by one edge length
with respect to the
other. Having then discovered I had not it in fact generated a disphenocingulum,
I contacted George
Hart to ask what it
was. George kindly did some research of his own (including
correspondence
with Norman Johnson) and
replied telling
me it was a 'near-miss', he also offered to host a model of this on his
website
should I manage to generate one. Three months later I had
succeeded, and
the first (quick-basic form of) **HEDRON**
was
born. I term it a 'quasi-gyro-disphenocingulum'. All of the
distortion in the linked model is contained in the triangular
faces (stress map).
Distortion (**E**=0.261,
**P**=0, **A**=54.0°). If the triangular faces are
forced to be
regular, the square faces become rhombi with acute angles of just over
79
degrees (here).

**Number 26:
A=54.1° **

This near miss was discovered by Roger
Kaufman in
2007. It is an augmented form of Melinda Green's
5-5-4 acron.
Two pentagonal faces are replaced with pentagonal pyramids. This
allows
the distortion to be confined to the triangular faces, with the square
faces
remaining regular. (Stress map).
Distortion:
(**E**=0.243, **P**=0, **A**=54.1º). Note:
augmenting only one
of the pentagons, or two vertex adjacent pentagons results in a figure
which
requires the triangles and the squares to be distorted, such figures
can be
regarded as Melinda Green's 5-5-4 acron with a
regular
pentagon augmented and so are not seperately listed. A further
version has
three pentagons augmented but has the same distortion as this figure
and so can
be regarded as an augmentation.

__Number 27 __**A=54.7°**** New
Entry 3 Jan 2010**

The
above polyhedron was discovered by Ulrich
Mikloweit in 2009, there is also a variation
with the
pentagon replaced by a pentagonal pyramid. Only the
triangular faces are distorted. (Stress
map) Distortion:
(**E**=0.265, **P**=0 , **A**=52.1°)

**Number 28:
A=54.9° ****New
Entry 29 April 2010 **

This near miss was discovered by Bill
Myers in
April 2010. Only the
triangular faces are distorted. (Stress
map). Distortion:
(**E**=0.304, **P**=0, **A**=54.9º).

**Number 29:
A=56.0° **

This near miss was discovered
by Ulrich
Mikloweit in 2007. Only the triangular faces are
distorted. (Stress
map) Distortion:
(**E**=0.208,
**P**=0, **A**=56.0°)

**Number 30:
A=56.4°**

This
discovery of Mick Ayrton's (which he terms a 'saucer')
contains
two caps from the Johnson Solid
'trigyrate rhombicosidodecahedron'. Stress map.
Distortion (**E**=0.214,
**P**=0, **A**=56.4°). Interestingly, if the pentagons
are replaced by pentagonal pyramids (here),
the
model
becomes regular. Mick has pointed out that this occurs to a
surprising
number of near misses.

**Number 31:
A=57.6°**

This near miss was discovered by Roger
Kaufman in
2007. It can be considered a truncated form of the disphenocingulum.
The distortion is confined to the triangular faces. (Stress map).
Distortion:
(**E**=0.318, **P**=0, **A**=57.6º).