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The Johnson Solids

The Johnson Solids

2023-12-23 06:48:56


The Johnson solids are 3D polyhedra which are convex and have
common polygons for his or her faces, however aren’t among the many
Platonic solids or uniform 3D
polyhedra
. Norman Johnson enumerated the total listing of 92 such polyhedra,
which have been named after him.

Family portrait of the 92 Johnson
solids
[Full-size image]

Within the above picture, 97 solids are proven, as a result of 5 of the Johnson solids are
chiral: they’re distinct from their mirror photos. These are
J44–J48. They’re paired with their mirror photos above, thus making a
whole of 97 solids.

The Johnson solids happen as cells of the so-called CRF
polychora
, a generalization of the Johnson solids to 4D.

Pyramids, Cupolae, and Rotundae

The primary six Johnson solids are the pyramids, cupolae, and rotundae. A
pyramid is fashioned by including a degree (the apex) over a polygon and
erecting triangles connecting the purpose to the sides of the polygon. A cupola
is fashioned by inserting a polygon and its Stott enlargement on parallel planes and
connecting them with triangles and squares. A rotunda is a hemispherical
dome-shaped polyhedron; there is just one in Johnson’s listing: the pentagonal
rotunda, J6.

Modifications of J1–J6

The subsequent solids in Johnson’s listing are fashioned by pasting collectively numerous
mixtures of the primary six plus some prisms and antiprisms. The listing is
fairly lengthy as a result of combinatorial explosion.

The prefix ortho- means two elements are in matching orientation;
whereas gyro- means one half rotated with respect to the opposite. Thus,
an elongation or gyroelongation is the lengthening of a polyhedron by attaching
or inserting a prism or antiprism, respectively. A fastigium is a
roof-like form, referring to a triangular prism in J26.

†The final 5 of those Johnson solids, J44–J48, are
chiral: they’re distinct from their mirror photos. These are the one
chiral Johnson solids.

Prism Augmentations

Subsequent are augmentations of the polygonal prisms. An augmentation is the
attaching of a number of smaller polyhedra (often pyramids or cupolae) to a
bigger one. There’s some overlap between this class and the earlier one,
however we listing them individually right here as a result of their 4D analogues, the duoprism
augmentations, kind their very own class of CRFs.

Modifications of Uniform Polyhedra

Following this, we come to the augmentations and diminishings of the common
and uniform polyhedra. A diminishing is an advert hoc chopping off of 1
or extra smaller elements from a polyhedron no matter symmetry (versus
truncation, the place the chopping is mostly achieved with respect to the
polyhedron’s full symmetry.)

Crown Jewels

The previous few Johnson solids are particular circumstances that can’t be obtained by
easy cut-and-paste operations on the uniform polyhedra. They’re nicknamed
crown jewels as a result of they’re uncommon, troublesome to search out from first
ideas, and have distinctive properties.

See Also

The final two of those, J91 and J92, are
not directly associated to the icosahedron through a particular,
modified type of Stott enlargement. Making use of this operation to numerous 4D
uniform polychora produces a large variety of 4D
CRF crown jewels, a lot of which comprise J91 and J92 as
cells.

The names of those previous couple of solids as defined by Johnson himself:

If we outline a lune as a fancy of two triangles hooked up to reverse sides
of a sq., the prefix spheno- refers to a wedgelike advanced fashioned
by two adjoining lunes. The prefix dispheno- denotes two such
complexes, whereas hebespheno- signifies a blunter advanced of two lunes
separated by a 3rd lune. The suffix -corona refers to a crownlike
advanced of eight triangles, and -megacorona, to a bigger such advanced
of 12 triangles. The suffix -cingulum signifies a belt of 12
triangles.

Final up to date 23 Might 2019.

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