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European Journal of Applied Sciences – Vol. 11, No. 1
Publication Date: January 25, 2023
DOI:10.14738/aivp.111.13776.
Meurant, R. C. (2023). The Morphology of the Regular & Semi-Regular Polyhedra and Tessellations According to the Separation of
Facial Polytopes. European Journal of Applied Sciences, 11(1). 147-168.
Services for Science and Education – United Kingdom
The Morphology of the Regular & Semi-Regular Polyhedra and
Tessellations According to the Separation of Facial Polytopes
Robert C. Meurant
Institute of Traditional Studies; Adjunct Professor,
Seoul National University PG College of Eng.;
Exec. Director, Research & Education, Harrisco Enco
4/1108 Shin-Seung Apt, ShinGok-Dong 685 Bungi,
Uijeongbu-Si, Gyeonggi-Do 11741, Republic of Korea
ABSTRACT
In previous work, inspired by Critchlow, and by Grünbaum & Shephard, I
proposed an integral 2.5D cubic schema of the regular and semi-regular
polyhedra and polygonal tessellations of the plane for each class of symmetry,
which could be differentiated into an upper and lower layer of 4 polytopes each,
and characterized by corresponding pairs, so that upper polytope always
corresponds to lower. I explored the motif of paired two-step sequences of first
alternating facial separation and morphological transformation, and second
facial morphological transformation and separation, which in the 2D
consideration of the 2.5D schema are disposed about the vertical axis, as
characterized by the correspondence between the PPs of the lower and upper
squares (diamonds or rhombi). Following intensive research, I here focus on a
deeper pattern of morphological transformation of the primary prototypes that
is characterized by the separation of one gendered set of the negative (−ve),
neutral (ntrl), or positive (+ve) facial polytopes along the Y, Z, & X axes of the
cubic schema. As one set of faces separates, the other two sets morph/ project
if polar/neutral, through null→regular or quasi-regular→double facial levels
(0→α|β→2) of the rhombic schema or its reflection. Each facial set separates just
once: d=0→1. The cubic schema reveals significant three-fold symmetry by
gender. The separation of faces provides an adequate schema for the
morphology of the three classes of the regular and semi-regular polyhedra of
{2,3,3}, {2,3,4}, and {2,3,5} symmetry, and two classes of polygonal tessellations
(tilings) of {2,3,6} and {2,4,4} symmetry.
Keywords: polyhedra, tessellations, morphology, separation of faces
PAIRING OF POLYHEDRA BY THE SEPARATION OF FACES: CLASS II AND GENERIC
Further to previous work, I have discovered that the pairings of polyhedra within any one
class can be characterized by the separation of one set of the negative, neutral, or positive
surface polytopes on the Y, Z, or X axes, respectively, of the 2.5D schema, as in Fig. 1 below.
Therefore, three significant kinds of pairings of PP s can be made in the 2.5D schema,
dependent on the schema orthogonal axis. For descriptive convenience, this is described
for Class II of {2,3,4} symmetry, where the −ve, neutral, and +ve axes of the class, and thus
of each of its individual polytopes, are conveniently the √1, √2, and √3 (100, 110, 111) axes
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European Journal of Applied Sciences (EJAS) Vol. 10, Issue 6, December-2022
Services for Science and Education – United Kingdom
of the cube, respectively. This class is of 3D polyhedra, and the +ve and −ve polar polytopes
take different forms (unlike Class I of 3D polyhedra, where the polar polytopes take the
same form of the tetrahedron, though in alternative orientation, or Class V of 2D polygons,
where both polar polytopes also take the same form of the square, but in different location).
In the other classes, the symmetry axes are not in general orthogonal; in addition, Class II
(leaving aside the transitional Snub form) precisely constitutes the PP s of the Class III
honeycombs, so are the primary components of the Class III honeycomb periodic all-space- filling arrays. Subsequently, I compare this Class II with Class IV, to illustrate the differences
between 3D polyhedral and 2D polygonal form, while considering classes with different
polar polytopes, as opposed to having the same, but reoriented (3D) or relocated (2D) form.
The beautiful integrity of interrelationship can be apprehended through contemplation of
Fig. 1:
Fig. 1: Pairings of the Class II polyhedra according to −ve (left), neutral (upper), or +ve
(right) faces, which separate from adjoining (sharing a V or E) to adjacent by distance unit
1 = edge length. CB and OH are considered the −ve and +ve polar polytopes, respectively,
with facial PTs shown as −ve (cyan) & +ve (magenta),
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Meurant, R. C. (2023). The Morphology of the Regular & Semi-Regular Polyhedra and Tessellations According to the Separation of Facial Polytopes.
European Journal of Applied Sciences, 11(1). 147-168.
URL: http://dx.doi.org/10.14738/aivp.106.13776
respectively, while neutral polytopes are shown in yellow, or thick black edge.
Table I. Separating PP pairs for Class II and their source and goal polytopes.
Negative Neutral Positive
Separating
facial PTs
Source
polytope
Goal
polytope
Separating
facial PTs
Source
polytope
Goal
polytope
Separating
facial PTs
Source
polytope
Goal
polytope
V
− VP2 OH V
0 VP2 CO V
+ VP2 CB
SQ− CB SRCO Eα
0 OH TO TR+ OH SRCO
RS− CO TO Eβ
0 CB TC RT+ CO TC
OG− TC GRCO SQ0 SRCO GRCO HX+ TO GRCO
NB. In this paper, I modify my previous conventions, so Vertex VT → V; neutral vertex NV → V
0
; edge
EG → E, neutral edge NE → E
0
; neutral square NS → SQ0
; Facial polytope → F; on-axis 0D V
0
(the 1-
gon, of 1 E & 1 V
0
) & 1D E
0
(the 2-gon, of 2 E & 2 V
0
), & 2D polygons (TR, HX, SQ, ...), are considered
F. See Nomenclature, p.17.
On the Y-axis of the schema (going leftwards), negative faces separate (cyan; lower left).
Adjoining (coincident) V
− s of the VP separate to adjacent V
− s of the OH (its nodes);
adjoining SQ−s of the CB separate to adjacent SQ−s of the SRCO; adjoining RS−s of the CO
separate to adjacent RS− s of the TO ; and adjoining OG− s of the TC separate to adjacent
OG−s of the GRCO. In each case, adjoining pairs of negative polytopes of a PP that share a
V
0 or E
0
separate by edge length unit distance 1 to adjacent negative polytopes of its PP
pair.
On the Z-axis of the schema (going upwards), neutral faces separate (yellow; upper).
Adjoining (coincident) V
0
s of the VP separate to adjacent V
0
s of the CO (its nodes);
adjoining E
0
s of the OH separate to adjacent E
0
s of the TO ; adjoining E
0
s of the CB
separate to adjacent E
0
s of the TC; and adjoining SQ0
s of the SRCO separate to adjacent
SQ0
s of the GRCO. In each case, adjoining pairs of neutral surface polytopes of a PP, sharing
a V or E that need not be +/0/−ve, e.g., of SRCO, separate by d=1 to adjacent neutral Fs of
its PP pair.
On the X-axis of the schema (going rightwards), positive faces separate (magenta; lower
right). Adjoining (coincident) V
+s of the VP separate to adjacent V
+s of the CB (its nodes);
adjoining TR+s of the OH separate to adjacent TR+s of the SRCO; adjoining RT+s of the
CO separate to adjacent RT+s of the TC; and adjoining HX+s of the TO separate to adjacent
HX+s of the GRCO. In each case, adjoining pairs of positive polytopes of a PP, sharing a V
0
or E
0
, separate by distance 1 to adjacent positive polytopes of its PP pair.
These various correspondences by separation of facial polytopes by unit distance can be
combined into the one illustration (Fig. 2), shown at left & middle for Class II, and at right
for all classes. Again, in each case of facial separation, adjoining pairs of polytopes of a PP
(d=0) separate by unit distance d=1 (= length of polytope side) to adjacent polytopes of its
PP pair: