U.S. patent number 3,645,535 [Application Number 05/031,175] was granted by the patent office on 1972-02-29 for block construction.
Invention is credited to Alexander Randolph.
United States Patent |
3,645,535 |
Randolph |
February 29, 1972 |
BLOCK CONSTRUCTION
Abstract
A block system useful as a puzzle or game consisting of a group
of solids adapted to be assembled into a larger regular polyhedron.
The solids each have shapes that are determined by an assembly of
components that include a tetrahedron and one or more fractional
sections of an octahedron. The fractional sections comprises
one-eighth and one-fourth sections of a regular octahedron.
Inventors: |
Randolph; Alexander
(Shinjuku-ku, Naka Ochiai, Tokyo, JA) |
Family
ID: |
21858018 |
Appl.
No.: |
05/031,175 |
Filed: |
April 23, 1970 |
Current U.S.
Class: |
273/157R;
52/DIG.9; 273/160; 434/403 |
Current CPC
Class: |
A63F
9/12 (20130101); Y10S 52/09 (20130101) |
Current International
Class: |
A63F
9/06 (20060101); A63F 9/12 (20060101); A63f
009/12 () |
Field of
Search: |
;273/156,157R,160 ;35/72
;34/34 ;46/17 |
References Cited
[Referenced By]
U.S. Patent Documents
Primary Examiner: Oechsle; Anton O.
Claims
I claim:
1. A block system consisting of solids adapted to be assembled in
facing relation to form a regular polyhedron, a plurality of said
solids, each having shapes determined by an assemblage of
components, consisting of eight regular tetrahedrons and fractional
sections of four regular octahedrons, at least one said solid
comprising an assemblage solely of a tetrahedron, a quarter section
of an octahedron and an eighth section of an octahedron.
2. A block system as set forth in claim 1 including at least one
solid comprising an assemblage solely of a tetrahedron, two quarter
sections of an octahedron and one eighth section of an
octahedron.
3. A block system consisting of solids adapted to be assembled in
selected relations to form a hexahedron having a line through its
geometrical center, a plurality of said solids each having shapes
determined by an assemblage of components including a tetrahedron
and at least one fractional section of an octahedron, each said
tetrahedron having an edge extending from a point on said center
line of said hexahedron to an outer edge defining part of said
hexahedron.
4. A block system as set forth in claim 3 wherein said hexahedron
is a cube and the point from which said edge of each said
tetrahedron extends is the center of said cube said edge extending
to the midpoint of an outer edge defining part of said cube.
5. A block system as set forth in claim 4 wherein said plurality of
solids consists of eight solids each including a tetrahedron, with
pairs of tetrahedrons of said eight tetrahedrons having abutting
edges extending from the center of said cube to the midpoint of an
edge of said cube, four pairs comprising said eight tetrahedrons,
each pair having abutting edges extending from the center of said
cube to the midpoint of different edges of said cube.
6. A block system as set forth in claim 3 wherein said hexahedron
is a cube and the point from which said edge of each said
tetrahedron extends is the center point of an outer face of said
cube, said edge extending to a corner defined by three edges of
said cube.
7. A block system consisting of solids adapted to be assembled in
facing relation to form a polyhedron, a plurality of said solids,
each having shapes determined by an assemblage of components, at
least one assemblage of components including a tetrahedron, a
quarter section of an octahedron, and an eighth section of an
octahedron, and at least one other assemblage of components
including a tetrahedron, two quarter sections of an octahedron, and
an eighth section of an octahedron.
8. A block system consisting of solids adapted to be assembled in
facing relation to form a polyhedron, a plurality of said solids,
each having shapes determined by an assemblage of components, at
least one assemblage of components including a tetrahedron, and two
quarter sections of an octahedron, and at least one other
assemblage of components including a tetrahedron and four quarter
sections of an octahedron.
Description
SUBJECT MATTER OF INVENTION
The present invention relates to a block system useful as a game,
puzzle, geometric or sculpture.
BACKGROUND OF INVENTION
Games and toys have been built using a plurality of polyhedric
shapes including cubes, rectangular blocks, pyramids, and the like.
Some attempts have been made to develop games or toys utilizing the
mathematical or geometric interrelationship of various solids.
Insofar as known, however, no one has yet devised a block system
that makes use of certain geometrical relationship of regular
polyhedrons and fractional sections thereof to form a larger
regular polyhedron, and in particular, the relationship of
polyhedrons herein described. Regular polyhedrons or platonic
solids are geometrically interrelated. Thus, the tetrahedron, the
cube or hexahedron, and the octahedron are related in a number of
interesting ways. These interrelationships, for example, include
the fact that diagonals and opposite faces of the hexahedron or
cube at right angles to each other form the edges of a tetrahedron.
Further, the midpoints of the faces of a cube are vertices of an
octahedron; and conversely, the midpoints on the faces of an
octahedron are the vertices of the cube. Such interrelationships
have heretofor never been utilized in the design of a block system
of toys, games or puzzles.
SUMMARY OF INVENTION
It is an object of the present invention to provide a block system
which may be used as a game, puzzle, sculpture, or toy. It is also
an object of the present invention to provide such a system
utilizing a plurality of the platonic solids or regular polyhedrons
or fractional sections thereof in a geometrically interrelated
fashion that illustrates such relations. It is also the object of
the present invention to provide an improved block system
comprising a group of solids that are adapted to be assembled into
a regular polyhedron with the solids each defined by an assembly of
components that include a tetrahedron and a fractional section of
an octahedron. One further object of the present invention is to
provide an improved game or puzzle formed of a plurality of solids
with each solid having a shape defined by an assembly of regular
polyhedrons or fractional portions thereof. One further object of
the present invention is to provide an improved game or puzzle
defined by regular tetrahedrons and quarter and/or eighth sections
of a few regular octahedrons. Another object of the present
invention is to provide a means wherein eight regular tetrahedrons
and quarter and one-eighth sections of four corresponding regular
octahedrons may be preassembled or shaped into a plurality of
geometric solids that are adapted to be put together into a regular
polyhedron in a variety of different configurations. One further
object of the present invention is to provide a block system formed
of a plurality of solids, each having shapes determined by an
assembly of components that include a tetrahedron and fractional
sections of an octahedron with the system adapted to be used either
as a puzzle or as a sculptured system.
BRIEF DESCRIPTION OF DRAWINGS
These and other objects and advantages of the present invention
will be more clearly understood when considered in conjunction with
the accompanying drawings in which
FIGS. 1 to 5 inclusive each comprise a perspective exploded view of
a different solid which, together with other solids of the shapes
illustrated in FIGS. 1 to 5, may be assembled into a cube;
FIGS. 6 to 13 inclusive illustrate the types of solids illustrated
in FIGS. 1 to 5 in relative orientations or positions necessary for
assembly into a cube;
FIGS. 14 to 17 inclusive each comprise a perspective exploded view
of a different solid which, together with other solids of the
shapes illustrated in FIGS. 14 to 17 may be assembled into a
cube;
FIGS. 18 to 25 inclusive illustrate the types of solids illustrated
in FIGS. 14 to 17 and their relative orientations or positions
necessary for assembly into a cube;
FIGS. 26 to 31 inclusive each comprise a perspective exploded view
of a different solid which, together with other solids of the
shapes illustrated in FIGS. 26 to 31 may be assembled into a
cube;
FIGS. 32 to 39 inclusive illustrate the types of solids illustrated
in FIGS. 26 to 31 and their relative orientations or positions
necessary for assembly into a cube.
DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS
Reference is first made to FIGS. 1 through 13 inclusive. Of these
figures, FIGS. 1 to 5 illustrate the structure of the different
style solids that are used in varying numbers to form a cube, and
FIGS. 6 through 13 inclusive illustrate the specific orientation of
these solids to form a cube of the size illustrated in the dotted
outline.
In this embodiment of the invention, the set consists of eight
separate solids having five different configurations. Solid 1A,
illustrated in FIG. 1, consists of three components including a
tetrahedron 1, a quarter section of a regular octahedron 2, and an
eighth section of a regular octahedron 3. Each tetrahedron, quarter
and eighth section of an octahedron herein described are the same
in size in each solid of each embodiment. The tetrahedrons are, of
course, formed with four identical equilateral faces 11. The
quarter section of an octahedron 2 has a pair of faces 10B that are
identical isosceles triangles having a common edge 10D. The other
two faces 10 are identical equilateral triangles. The eighth
section of the regular octahedron 3 is formed with a base face 12
that is an equilateral triangle which is dimensioned to the same
size in this invention to the faces of the tetrahedron and faces
10, 10A of the quarter section of the octahedron. The other three
faces 13 are equally dimensioned right isosceles triangles with the
hypotenuses of each defined by an edge of the base face 12.
Equilateral face 10 of the quarter section of the octahedron 2 is
aligned with an identically shaped face 11 of the tetrahedron 1
while equilateral face 12 of the eighth section of octahedron 3 is
aligned with an identical face 11 of the tetrahedron 1. These three
components are secured together by suitable means to form a solid
1A. Obviously, the solid may be integrally formed as a single piece
having a polyhedric shape consistent with the assemblage of three
components illustrated in FIG. 1. This solid, as well as other
herein described, may be made of any suitable material and may, for
example, be formed by conventional injection molding techniques of
a suitable styrene or other plastic material.
Shape 1AM, illustrated in FIG. 2, is the mirror image of shape 1A,
illustrated in FIG. 1. In this arrangement, a tetrahedron component
1 is secured or integrally formed with a quarter section of an
octahedron 2 and an eighth section of an octahedron 3. A face 11,
the hidden face of the tetrahedron in FIG. 2, is arranged in facing
relation with the identically shaped face 12 of the eighth section
of octahedron 3. A face 10 of the quarter section of the octahedron
2 is integrally secured to a face 11 of identical shape to the
tetrahedron 1.
Shape 1B is illustrated in FIG. 3. This shape consists of a
tetrahedron 1, two quarter sections of octahedrons 2 and 2' and an
eighth section of an octahedron 3. One face 11 of the tetrahedron 1
is secured or formed integrally with an equilateral face 10 of one
quarter section of an octahedron 2', while the other quarter
section of an octahedron has an equilateral face 10 secured or
integrally formed with another face 11 (not shown) of the
tetrahedron 1. Equilateral face 12 of the eighth section of
octahedron is secured to a third face 11 (not shown) of the
tetrahedron 1. As in the case in each solid, the facing or
integrally formed faces are each identical in shape and size with
their facing or integrally formed face.
Solid 1C is illustrated in FIG. 4. This solid is formed of four
components including one tetrahedron 1, two quarter sections of an
octahedron 2, 2' and a one-eighth section of an octahedron 3.
Equilateral face 10 (not shown) of one quarter section of an
octahedron 2' is secured or formed integrally with a face 11 of the
tetrahedron 1. An equilateral face 10 (not shown) of the other
quarter section of the octahedron is secured or integrally formed
with another face 11 of the tetrahedron, while equilateral face 12
(not shown) of the eighth section of the octahedron 3 is secured or
integrally formed with a third face 11 of the tetrahedron 1.
Shape 1CM is illustrated in FIG. 5 and comprises four components
including a tetrahedron 1, two quarter sections of an octahedron
2,2' and an eighth section of an octahedron 3. A face 11 of the
tetrahedron 1 is secured or integrally formed with an equilateral
face 10 (not shown) of one quarter section of an octahedron 2',
while the other octahedron 2 is formed with its equilateral face 10
facing or integral with another face 11 (not shown) of the
tetrahedron 1. Equilateral face 12 (not shown) of the one-eighth
section of the octahedron 3 is secured in facing relation or
integral with a third face 11 of the tetrahedron 1.
These five solids illustrated in their relationship in FIGS. 1 to 5
inclusive may be assembled into a cube as illustrated in FIGS. 6
through 13 inclusive. In this set, eight pieces are provided
including two solids 1A, two solids 1AM, two solids 1B, one solid
1C, and one solid 1CM. For convenience in understanding the
orientation of the solids with relation to one another when
oriented to form a cube, the overall outline of a cube divided in
eighths is illustrated in FIGS. 6 through 13. For further
convenience in identifying the relative positions of each solid
when so oriented in this embodiment of the invention as well as in
others, intersections of each of these dividing lines in each
figure is correspondingly identified by a letter or letters in the
alphabetical sequence of A to Z inclusive and AA. In the set
illustrated in FIGS. 1 through 5, two solids 1A are used. One of
these solids is shown in proper orientation in FIG. 6, and the
other is shown in proper orientation in FIG. 13. As illustrated,
solid 1A is arranged in FIG. 6 with its one-eighth section of an
octahedron occupying points D, G, H and P. The quarter section of
an octahedron component occupies points Z, H, P and N. The
tetrahedron component occupies points D, H, N and P. In FIG. 13,
the other solid 1A is arranged with the one-eighth section of an
octahedron component occupying points A, B, D and J. The
one-quarter section of an octahedron component occupies points B,
D, H and N. The tetrahedron component occupies points B, D, J and
N.
Two solids 1AM are provided and are oriented as illustrated in
FIGS. 7 and 11. The solid shown in FIG. 7 is arranged with the
quarter section of an octahedron component occupying points B, L, N
and T. The tetrahedron occupies points L, N, T and X. The
one-eighth section of an octahedron component occupies points L, T,
X and U. In FIG. 11, the solid 1AM is arranged with the eighth
section of an octahedron component occupying points B, C, F and L.
The quarter section of an octahedron component occupies points B,
H, F and N. The tetrahedron occupies points F, B, N and L.
The two solids 1B are oriented as illustrated in FIGS. 9 and 12. In
FIG. 9 the one-eighth section of an octahedron component occupies
points Z, AA, R and X. The tetrahedron occupies points Z, R, N and
X. One of the two quarter sections of an octahedron component
occupies points Z, T, N and X. The other quarter section of an
octahedron component occupies points N, R, L and X. The solid 1B
shown in FIG. 12 is oriented with the one-eighth section of an
octahedron component occupying points J, S, V and T. One of the two
one-quarter sections of an octahedron component occupies points J,
V, N and D. The other quarter section of an octahedron component
occupies points B, J, D and N. The tetrahedron occupies points V,
J, T and N.
Component 1C is oriented as illustrated in FIG. 10. The one-eighth
section of an octahedron component occupies points V, Y, Z and P.
One one-quarter section of an octahedron occupies points V, P, N
and D. The other one-quarter section of the octahedron component
occupies points Z, T, V and N. The tetrahedron occupies points Z,
P, N and V.
Component 1CM is oriented as illustrated in FIG. 8. The one-eighth
section octahedron component occupies points F, I, H and R. One
quarter section of the octahedron occupies points H, R, Z and N.
The other quarter section of the octahedron occupies points L, R, F
and N. The tetrahedron occupies points F, H, N and R.
As indicated above and in FIGS. 6 through 13, at least one edge of
each tetrahedron extends from the center of the cube and each such
edge abuts the edges of an adjacent tetrahedron.
Although the solids described in this embodiment are illustrated in
the form of a puzzle intended to be assembled into a perfect cube,
it should be understood that these solids may also be used to form
a wide variety of interesting geometric shapes. For example, these
solids may be reassembled in the shapes resembling a variety of
animals, such as dogs and cats, birds such as owls, and other
imaginative shapes. Such shapes as well as a cube may be used as
game objectives in a variety of games designed for use with the
components.
The second embodiment of the present invention may be formed from
solids having the configuration illustrated in FIGS. 14 to 17
inclusive. In this arrangement, one solid 2A is formed of five
geometric components as illustrated in FIG. 14 in exploded form.
These components include a tetrahedron 1 having identical
equilateral faces 11 and four quarter sections of an octahedron 2,
2', 2" and 2'". Each of these quarter sections of the octahedron
have a pair of equilateral faces 10 and a pair of right isosceles
faces 10B having a common edge 10D. Each of the quarter sections of
the octahedron 2, 2', 2'" has an equilateral face 10 integrally
formed with one of the faces 11 of the tetrahedron 1 as
illustrated. In the present embodiment, two of such solids 2A are
provided.
In FIG. 15 there is illustrated a solid 2B. This solid is formed
with a tetrahedron shape 1 and two quarter sections of an
octahedron 2 and 2'. These shapes are integrally formed with an
equilateral face of each quarter of a section of an octahedron 2
and 2' integral with similarly shaped faces of the tetrahedron 1.
It will be noted in this arrangement the longest edges 10D of each
of the quarter sections of the octahedron emanate from different
corners of the tetrahedron 1. In this case, the end 20 of one edge
10D of one quarter section of an octahedron is coincident with the
corner 20' of the tetrahedron, while the end 21 of the other long
edge 10D of the other quarter section of an octahedron is
coincident with the corner 21' of the tetrahedron 1.
FIG. 16 illustrates solid shape 2BM, which is made up of the same
number of components as FIG. 15 except that they are arranged to
form the mirror image of the shape 2B. In this arrangement the
configuration of a tetrahedron 1 and one quarter section of an
octahedron 2' may be identical as that in FIG. 15. The positioning
of the component 2, however, is such that its long edge 10D has an
end 22 that is coincident with the corner 22' of tetrahedron 1
rather than corner 23 which would be the corner to which it would
be secured or coincident if it were to be arranged identical to
shape 2B of FIG. 15.
FIG. 17 illustrates shape 2C. This shape 2C consists essentially of
a tetrahedron 1 and a quarter section of an octahedron 2 with a
long side 10D of the octahedron extending from one corner of the
tetrahedron.
Two shapes 2A, one shape 2B, one shape 2BM and four shapes 2C may
be assembled into a cube as illustrated in FIGS. 18 to 25. For
convenience in understanding the orientation of these solids or
shapes the same cube orientation identification used in FIGS. 6 to
13 are used in FIGS. 18 to 25 inclusive. A shape 2B is illustrated
in FIG. 18. Here the quarter section of the octahedron occupies
points I, AA, Q and O. The other quarter section of an octahedron
occupies points Q, M, E and O, while the tetrahedron occupies
points Q, E, I and O.
A shape 2C is illustrated in FIG. 19. The tetrahedron of this shape
occupies points W, K, O and U of the cube. The quarter section of
the octahedron of this shape occupies points O, AA, U and W.
FIG. 20 illustrates a shape 2C. The quarter section of an
octahedron of this shape occupies points A, E, G and M of the cube.
The tetrahedron of this shape occupies points A, E, K and M.
FIG. 21 illustrates a shape 2A. One quarter section of an
octahedron of this shape occupies points A, C, E and K. A second
quarter section of an octahedron of this shape occupies points C,
E, I and O. A third quarter section of an octahedron of this shape
occupies points C, U, O and K. A fourth quarter section of an
octahedron of this shape occupies points C, O, K and E. In FIG. 22
there is illustrated shape 2A. In this shape, one quarter section
of an octahedron occupies points A, S, M and K. A second quarter
section of an octahedron occupies points K, M, Q and W. A third
quarter section of an octahedron occupies points S, U, K and W. A
fourth quarter section of an octahedron occupies points S, Y, M and
W. The tetrahedron occupies points S, M, K and W.
In FIG. 23 there is illustrated a shape 2C. The quarter section of
an octahedron of this shape occupies points G, I, Q and E. The
tetrahedron occupies points S, M, K and W.
In FIG. 23 there is illustrated a shape 2C. The quarter section of
an octahedron of this shape occupies points G, I, Q and E. The
tetrahedron occupies points M, G, E and Q.
In FIG. 24 there is shown another shape 2C. The quarter section of
an octahedron of this shape occupies points G, M, Q and Y. The
tetrahedron occupies points M, Q, Y and W.
In FIG. 25 there is shown the location of shape 2BM. Here, the
tetrahedron occupies points Q, O, AA and W. One quarter section of
an octahedron occupies points Q, K, O and W. The other quarter
section of an octahedron occupies points Q, Y, AA and W. When
assembled in the locations illustrated in FIGS. 18 to 25, the
various shapes referred to will form a cube. As previously
indicated, the same shapes in the same or fewer numbers may be
arranged to form other configurations, including for example
animals and birds, such as owls, eagles, bears, dogs, cats, in
relatively abstract forms.
Referring now to the embodiment illustrated in the remaining
figures, there is illustrated an arrangement wherein eight pieces
or six different shapes may also be formed into a perfect cube. One
solid illustrated in FIG. 26 comprises a tetrahedron, a single
eighth section of an octahedron and two quarter sections of an
octahedron integrally formed to form a solid 3D. In this
arrangement, the tetrahedron 1 has the eighth section of an
octahedron 3 secured or integrally formed to it on commonly sized
faces. One quarter section of an octahedron 2' is integrally formed
with a second face 11 of the tetrahedron 1 with the long edge 10D'
of the quarter section of an octahedron ending in corner 30 common
to the tetrahedron 1 and eighth section of the octahedron 3. The
other quarter section of an octahedron 2 is integrally formed with
face 11A of the tetrahedron 3, but its long edge 10D is formed with
its corner 31 common to corner 31 of the tetrahedron 1. This corner
31 is not common with any corner of the eighth section of the
octahedron 3. FIG. 27 illustrates the mirror image of shape 3D and
is identified as shape 3DM. In this arrangement, the specific
arrangement of the tetrahedron 1 and the eighth section of an
octahedron 3 and one quarter section of an octahedron 2 may be
identical to that of shape 3D. However, the other quarter section
of an octahedron 2' is arranged in common facing relation with one
of the other of the two remaining faces of the tetrahedron. In this
case, this remaining face is identified as 11B. The corners 30'
common to the long edge 10D' of the quarter section of the
octahedron 2' and a corner of the tetrahedron correspond with
corner 30 of shape 3D.
In FIG. 28 there is illustrated a shape 3B. Two of these shapes are
used in this embodiment.
As illustrated in FIG. 28, shape 3B comprises one tetrahedron, a
one-quarter section of an octahedron and a one-eighth section of an
octahedron. In this arrangement, the tetrahedron 1 has a common
face with the eighth section of the octahedron 3. The quarter
section of an octahedron 2 is arranged with its long edge 10D
common at one end 34 with corner 34 of tetrahedron 1. Corner 34 of
tetrahedron 1 is the corner opposite to the corner 35 of the eighth
section of an octahedron. Corner 35 is the one that is surrounded
and defined by the right triangular faces 36 of the eighth section
of the octahedron.
FIG. 29 illustrates solid 3C. This solid consists of a tetrahedron,
two quarter sections of an octahedron, and an eighth section of an
octahedron. Two of these solids are used in this embodiment. In
this arrangement, the tetrahedron 1 is arranged with faces common
with the two quarter sections of an octahedron 2 and 2'. The long
edges 10D and 10D', respectively, of the quarter sections of the
octahedrons 2 and 2', respectively, are coplanar and have ends
coincident with the corner 37 of the tetrahedron 1. The eighth
section of the tetrahedron 3 is arranged with the face that has a
size common with the faces of tetrahedron 1, integral with the face
11A of the tetrahedron 1 that is opposite to corner 37.
In FIG. 30 there is illustrated shape 3A. One such solid is
required. This solid consists of a tetrahedron, a quarter section
of an octahedron and an eighth section of an octahedron. In this
configuration, the tetrahedron 1 has one face 11A common with the
similarly size shape face of the eighth section of the octahedron
3. The quarter section of the octahedron 2 is arranged with its
long edge 10D having an end 39 common with corners 39 of the
tetrahedron and the eighth section of the octahedron.
The shape 3AM is illustrated in FIG. 31. This shape is similar in
configuration to FIG. 30, except that it forms a mirror image which
is attained by forming the quarter section of a tetrahedron 2 with
its long edge having an end 40 coincident with a different corner
of the tetrahedron and eighth section of the octahedron.
These shapes or solids are arranged relative to one another to form
a cube in one form of the game or puzzle, as illustrated in FIGS.
32 to 39 inclusive. FIG. 32 shows the orientation of one solid 3C.
In this arrangement, one quarter section of the octahedron occupies
points A, C, E and K. The eighth section of the octahedron occupies
points E, K, N and O. The second quarter section of an octahedron
occupies points C, U, O and K. The tetrahedron occupies points C,
E, O and K.
FIG. 33 shows the orientation of solid 3AM. In this arrangement,
the quarter section of an octahedron occupies points O, U, W and
AA. The tetrahedron occupies points O, Q, AA and W. The eighth
section of the octahedron occupies points Q, AA, Z and W.
FIG. 34 illustrates the orientation of shape 3A. In this
arrangement, the eighth section of an octahedron occupies points E,
D, G and M. The quarter section of a tetrahedron occupies points G,
E, I and O. The tetrahedron occupies points G, M, Q and E.
FIG. 35 illustrates the orientation of solid 3C. One quarter
section of the tetrahedron occupies points C, I, E and O. The
tetrahedron occupies points E, I, Q and O. The eighth section of
octahedron occupies points E, Q, O and N. The other quarter section
of an octahedron occupies points I, Q, O and AA.
FIG. 36 illustrates the orientation of shape 3D. Here one quarter
section of an octahedron occupies points A, S, M and K. The eighth
section of the octahedron occupies points A, D, E and M. The other
quarter section of an octahedron occupies points E, K, Q and M. The
tetrahedron occupies points A, E, K and M.
FIG. 37 illustrates the orientation of shape 3DM. In this
arrangement, one quarter section of an octahedron occupies points
G, M, Q and Y. The eighth section of an octahedron occupies points
Q, Y, W and Z. The other quarter section of an octahedron occupies
points M, Q, O and W.
FIG. 38 illustrates the orientation of shape 3B. In this
arrangement, the quarter section of an octahedron occupies points
S, Y, M and W. The eighth section of the octahedron occupies points
M, K, N and W. The tetrahedron occupies points S, M, K and W.
FIG. 39 illustrates the orientation of another shape 3B. In this
arrangement, the quarter section of a tetrahedron occupies points
S, U, K and W. The eighth section of an octahedron occupies points
K, N, O and W; and the tetrahedron occupies points K, O, U and
W.
* * * * *