Aid to understanding Shashibo.

Aid to understanding Shashibo. Aid to understanding Shashibo.
Version of Tuesday 4 March 2025.
Dave Barber's other pages.

Links to elsewhere within this site (but read this page first):

The term "chunk" appearing in previous versions of this report has been replaced with "atom", to accord it with a usage of reddit user Aware_Secretary5979.

§1. Introduction.

§1A. The Shashibo is a "patented magnetic puzzle cube", as described in the factory website; the present report covers many of its geometric properties. By contrast, little attention is given to the character and placement of its magnets, which are essential to its operation. United States patent 10,569,185, which is a public document, explains the Shashibo in extensive detail.

A Shashibo consists of twelve tetrahedra, each hinged to two others to form a loop. All the tetrahedra are shaped the same and hinged similarly, but the factory makes over two dozen versions of the puzzle differing merely in the artwork appearing on the faces of the tetrahedra.

§1B. Although the Shashibo can be manipulated into dozens of shapes, two of them are given special attention here. One is the cube, and the other is the rhombic dodecahedron ("rhodo" for short). This report calls them the primary shapes.

The Shashibo is shipped from the factory in the cube shape, which contains no hollow regions. Meanwhile, the rhodo contains one hollow region that is exactly the size and shape of the cube. Further, the faces that appear on the outside of a cube are precisely those that appear on the inside of the rhodo. Thus it is reasonable to regard the rhodo as an "inside-out" cube.

Further explanation about the rhodo is necessary. It has 12 rhombic faces. On the Shashibo, each rhombus is composed of two triangular faces from two different tetrahedra. Those faces are isosceles triangles that are adjacent base to base, and are coplanar. Hence the pair is congruent to a rhombus.

§1C. All the images that appear below can be downloaded.

§2. Any tetrahedron has:

Four vertices. In this report they are denoted by the letters A, B, C, D.
Six edges, each connecting two vertices. They are named according to the vertices they connect; hence AB, AC, AD, BC, BD, CD.
Four triangular faces, denoted by their vertices; hence ABC, ABD, ACD, BCD.

The name of edge AB can equivalently be written BA; face ABD as DAB; et cetera. But alphabetical order is used unless there is a particular reason not to.

For brevity, a face is ofen denoted by one letter rather than three, indicating the sole vertex that does NOT bound it. Hence face A = face BCD; B = ACD; C = ABD; D = ABC.

The Shashibo tetrahedron can be described in a three-dimensional Cartesian coordinate system using the familiar format (x, y, z). The coordinates of the four vertices of a unit-size Shashibo tetrahedron at a convenient location in space are:

A = ( ½, ½, ½ )

B = ( 0, 1, 0 )

C = ( 1, 0, 0 )

D = ( 0, 0, 0 )

With the definition of point E = (½, ½, 0), this tetrahedron is symmetric about the plane ADE. Thus it is not chiral.

To find the size in millimeters of a Shashibo tetrahedron as commercially manufactured, multiply by 61:

A = ( 30½, 30½, 30½ )

B = ( 0, 61, 0 )

C = ( 61, 0, 0 )

D = ( 0, 0, 0 )

For brevity, this report will use the term "atom" for any tetrahedron geometrically similar to these.

§3. Here are some measurements of an atom, assuming unit size.

§3A. Lengths and dihedral angles of edges:

edges AB, AC AD BC BD, CD

length exactly ½√3 ½√3 √2 1

numerically 0.8660 0.8660 1.4142 1.0

dihedral angle π/3 rad = 60 deg 2π/3 rad = 120 deg π/2 rad = 90 deg π/4 rad = 45 deg

edges	`AB`, `AC`	`AD`	`BC`	`BD`, `CD`
length	exactly	½√3	½√3	√2	1
numerically	0.8660	0.8660	1.4142	1.0
dihedral angle	π/3 rad = 60 deg	2π/3 rad = 120 deg	π/2 rad = 90 deg	π/4 rad = 45 deg

All four faces are isosceles triangles.

All the dihedral angles are divisors of 360 degrees, which is the number of degrees in a circle. This is one reason that multiple Shashibos are candidates for three-dimensional tessellation.

§3B. Areas of faces:

faces ABC, ABD, ACD BCD

exactly ¼√2 ½

numerically 0.3536 0.5

faces	`ABC`, `ABD`, `ACD`	`BCD`
exactly	¼√2	½
numerically	0.3536	0.5

§3C. Volume is ¹⁄₁₂ ≈ 0.0833. This is consistent with the fact that the twelve atoms of the Shashibo can be arranged to fill a cube of edge 1 unit. In fact, a Shashibo is shipped from the factory in this cube form.

§3D. Plane angles between edges:

faces ABC ABD, ACD BCD

edges AB & CB
BC & AC CA & BA AB & DB
AD & BD
AC & DC
AD & CD BA & DA
CA & DA CB & DB
BC & DC BD & CD

as inverse cosine acos (2/√6) acos (−1/3) acos (1/√3) acos (1/3) acos (1/√2) acos (0)

numerically 35.26 deg 109.47 deg 54.74 deg 70.53 deg 45 deg 90 deg

reference letter e f g h i j

faces	`ABC`	`ABD`, `ACD`	`BCD`
edges	`AB` & `CB` `BC` & `AC`	`CA` & `BA`	`AB` & `DB` `AD` & `BD` `AC` & `DC` `AD` & `CD`	`BA` & `DA` `CA` & `DA`	`CB` & `DB` `BC` & `DC`	`BD` & `CD`
as inverse cosine	acos (2/√6)	acos (−1/3)	acos (1/√3)	acos (1/3)	acos (1/√2)	acos (0)
numerically	35.26 deg	109.47 deg	54.74 deg	70.53 deg	45 deg	90 deg
reference letter	e	f	g	h	i	j

Various relations among the angles:

2e + f = 2g + h = 2i + j = 180 deg

f + h = 180 deg 2g = f

e + g = 90 deg 2e = h

Faces B = ACD and C = ABD are equal, and for some purposes interchangeable. The symbol B?C can be used when they need not be distinguished.

Two instances of face D = ABC, adjacent base to base, are congruent to two instances of face(s) B?C, adjacent base to base.

§3E. Solid angles at vertices, given in steradians:

vertices A B, C D

angle π/3 sr π/12 sr π/6 sr

vertices	`A`	`B`, `C`	`D`
angle	π/3 sr	π/12 sr	π/6 sr

All of these are divisors of 4π, which is the number of steradians in a sphere. This is another reason that multiple Shashibos are candidates for three-dimensional tessellation. Among successful shapes are the cube and rhodo.

Although steradians can be converted to square degrees, doing so is not informative in this case.

§3F. Altitude between vertex and opposite edge:

vertex
& edge A & BC A & BD
A & CD
D & BC B & AC, B & AD
C & AB, C & AD
D & AB, D & AC B & CD
C & BD

exactly ½ ½√2 ⅓√6 1

numerically 0.5 0.7071 0.8165 1.0

vertex & edge	`A` & `BC`	`A` & `BD` `A` & `CD` `D` & `BC`	`B` & `AC`, `B` & `AD` `C` & `AB`, `C` & `AD` `D` & `AB`, `D` & `AC`	`B` & `CD` `C` & `BD`
exactly	½	½√2	⅓√6	1
numerically	0.5	0.7071	0.8165	1.0

§3G. Altitude between vertex and opposite face:

vertex
& face A & BCD B & ACD
C & ABD
D & ABC

exactly ½ ½√2

numerically 0.5 0.7071

vertex & face	`A` & `BCD`	`B` & `ACD` `C` & `ABD` `D` & `ABC`
exactly	½	½√2
numerically	0.5	0.7071

§3H. The present author's tetrahedron calculator produces additional measures. Note that it shows plane angles in radians rather than degrees. To use it, enter these numbers from the unit-size tetrahedron of §2 above:

§3I. Mentionworthy is an octahedron related to this atom.

§4. Any tetrahedron can be described using a polyhedral net. Here is a net suitable for the Shashibo atom:

these images are from group_0 in the download

On the left is the plain net; on the right a version with tabs for gluing in case the shape is cut out from cardboard.

Each face is drawn with a yellow background merely to contrast with the surrounding empty space. Shown in purple are both the one-letter and three-letter versions of the face names. The red letters denoting the vertices are printed on every face; this aids in studying the Shashibo when it is assembled.

When the atom is built, folds will be made on edges BC, BD, and CD. Then points A1, A2 and A3 will meet to form vertex A. If tabs are included, additional folds will be needed on edges A1D, A2B, and A3C.

This map can be described in a two-dimensional Cartesian coordinate system using the familiar format (x, y). Coordinates for the points in the maps above are:

exactly numerically numerically

x y x y x y

A1 +½ + ¼√2 −½ − ¼√2 +0.8536 −0.8536 T1 +0.35 +0.47

A2 −½ − ¼√2 −½ − ¼√2 −0.8536 −0.8536 T2 +0.57 +0.32

A3 0 +½ 0.0 +0.5 T3 +0.53 −0.97

B −½√2 0 −0.7071 0.0 T4 +0.27 −0.93

C +½√2 0 +0.7071 0.0 T5 −0.97 −0.53

D 0 −½√2 0.0 −0.7071 T6 −0.93 −0.27

	exactly	numerically			numerically
x	y	x	y	x	y
`A1`	+½ + ¼√2	−½ − ¼√2	+0.8536	−0.8536	`T1`	+0.35	+0.47
`A2`	−½ − ¼√2	−½ − ¼√2	−0.8536	−0.8536	`T2`	+0.57	+0.32
`A3`	0	+½	0.0	+0.5	`T3`	+0.53	−0.97
`B`	−½√2	0	−0.7071	0.0	`T4`	+0.27	−0.93
`C`	+½√2	0	+0.7071	0.0	`T5`	−0.97	−0.53
`D`	0	−½√2	0.0	−0.7071	`T6`	−0.93	−0.27

The locations of the tab vertices are not critical, and may need to be adjusted according to the material of which the atom is fabricated. Shown above are the rather arbitrary values chosen for this report.

Of course, these coordinates can be scaled, translated, rotated, and reflected for convenience and efficiency in printing.

§5. The download also contains .svg images of the Shashibo's twelve atoms, each customized with printed symbols for its position in the loop. Although all the atoms are of identical shape, they bear different markings which may aid in understanding. The markings appear in various colors for clarity, and are explained below the table.

these images are from group_12 in the download

Explanation
according to color of typography

In purple: The atoms are numbered 1 through 12. Each face is lettered A (for face BCD), B, C, D. These characters appear in a large font near the center of each face.

In red: On each face are marked the names of the vertices bounding that face. Each face has three out of the four letters A, B, C, and D. They are the same on all twelve atoms.

In blue: When the Shashibo is folded into a cube, the eight vertices of that cube are numbered 21 through 28. These numbers indicate which vertices of one atom meet which vertices of another atom. Numbers 21 through 28 will also serve to mark eight of the fourteen vertices of a rhodo.

In green: Numbers 31 through 36 denote the six vertices of the rhodo that do not have blue numbers. When the rhodo is folded into a cube, all six of these end up together, at the center of the cube.

In brown: Assembly instructions are marked with the letter H for "hinge". For example:

face 1A has the marking H^2A, and face 2A has the complementary marking H^1A;

face 1B has the marking H^2B, and face 2B has the complementary marking H^1B.

These are two notations for achieving the same result. Edge CD of atom 1 will be connected to edge CD of atom 2 with a hinge. Each of these atoms has one vertex marked 23 and another marked 24. Equal vertex numbers will adjoin when the Shashibo is correctly assembled.
The redundancy exists to help prevent mistakes, because assembling a Shashibo is tricky. One possible source of confusion is that, of the six edges of an atom, three of them are one length, while two others are of a second length.

In turquoise:

Cube: As an example, face 1A has the marking M^4A= while face 4A has the marking M^1A=. ("M" is for "meet".) These complementary symbols indicate that when the Shashibo is folded into a cube, those edges will meet. When an equal sign appears, the faces will be coplanar at the junction.

Rhodo: M^ markings with letters B and C have the corresponding purpose for when the Shashibo is folded into a rhodo, similarly indicating edges that will adjoin. These B-C meetings lack an equal sign because the faces will not be parallel.

When the Shashibo is folded into a rhodo, the M^nA= markings remain valid, applying to the A faces within the rhodo's hollow interior region. This is part of the reason for regarding the rhodo as an inside-out cube.
The M^ markings are provided to aid designers who seek an artistic pattern that is continuous at the meetings. Continuity can always be achieved at the hinges.

Face numbers (purple) are in the range 1-12; non-rhodo vertex numbers (blue) 21-28; rhodo-only vertex numbers (green) 31-36. Disjoint ranges were selected to reduce proneness to human error.

Suggestion: When assembling a Shashibo from the twelve atoms:

Make pairs, attaching atom 1 to atom 2 as indicated by the hinge markings. Then attach 3 to 4; 5 to 6; 7 to 8; 9 to 10; 11 to 12.
Make quartets, attaching atom 2 to atom 3; 6 to 7; 10 to 11.
Make one long chain of length twelve, attaching atom 4 to atom 5; 8 to 9.
Arrange all the atoms into the usual cube. After that, attach atom 1 to atom 12. The reason for forming the cube first is to prevent attaching the ends of the chain with a full twist, which is wrong.

§6. Sample coordinates.

§6A. The table in §5 above shows, by means of the blue numbers (21-28) and green numbers (31-36), which vertices of the cube and rhodo are associated with each vertex of an atom.

The table below shows Cartesian coordinates for all the vertices involved in a complete cube or rhodo formed of unit-sized atoms as introduced in §2 above. Each is placed at a convenient, but arbitrary, location in three-dimensional space.

An important geometrical observation is that negating an odd number of a figure's coordinates effects a reflection of the figure. For example, in three dimensions, changing (x, y, z) to (x, −y, z) for every point will reflect the figure. By contrast, negating an even number of coordinates does not cause reflection.

In the table below, there is negation of all three coordinates of the cube's corners (21-28) to obtain the corresponding coordinates of the rhodo corners (21-28). Negating exactly one would have been adequate, but less elegant. Given the inside-out relationship between the cube and rhodo, the presence of reflection is unsurprising.

cube rhodo

vertex ( x, y, z )

21 ( −½, −½, −½ )
22 ( +½, −½, −½ )
23 ( −½, +½, −½ )
24 ( +½, +½, −½ )

25 ( −½, −½, +½ )
26 ( +½, −½, +½ )
27 ( −½, +½, +½ )
28 ( +½, +½, +½ )

vertex ( x, y, z )

21 ( +½, +½, +½ )
22 ( −½, +½, +½ )
23 ( +½, −½, +½ )
24 ( −½, −½, +½ )

25 ( +½, +½, −½ )
26 ( −½, +½, −½ )
27 ( +½, −½, −½ )
28 ( −½, −½, −½ )

31 ( 0, 0, +1 )
32 ( 0, −1, 0 )
33 ( +1, 0, 0 )

34 ( −1, 0, 0 )
35 ( 0, +1, 0 )
36 ( 0, 0, −1 )

Vertices 21-28 of the cube equal vertices 28-21 respectively of the rhodo, which themselves define the empty cubic region therein. Meanwhile, rhodo vertices 31-36 form a regular octahedron.

§6B. Employing the coordinates of the table above, the table below gathers various metric information about three polyhedra:

the rhodo, which has fourteen vertices;
the cube formed by eight of the rhodo's vertices (blue numbers 21-28);
the regular octahedron formed by the six other vertices (green numbers 31-36).

In particular, the radii of these three spheres are given where they exist:

The measures are presented in a single table because many of the numbers appear in multiple places. Some of these numbers also turn up in the atom measures of §3 above.

value entire
rhodo cube
21-28 octahedron
31-36

exactly numerically

½ 0.5000 — insphere —

⅓√3 0.5774 — — insphere

½√2 0.7071 insphere midsphere midsphere

⅓√6 0.8165 midsphere — —

½√3 0.8660 edge circumsphere —

1 1.0000 — edge, volume circumsphere

⁴⁄₃ 1.3333 — — volume

√2 1.4142 — — edge

2 2.0000 volume — —

value	entire rhodo	cube `21`-`28`	octahedron `31`-`36`
exactly	numerically
½	0.5000	—	insphere	—
⅓√3	0.5774	—	—	insphere
½√2	0.7071	insphere	midsphere	midsphere
⅓√6	0.8165	midsphere	—	—
½√3	0.8660	edge	circumsphere	—
1	1.0000	—	edge, volume	circumsphere
⁴⁄₃	1.3333	—	—	volume
√2	1.4142	—	—	edge
2	2.0000	volume	—	—

When the circumsphere exists, its radius is each vertex's distance from the center of the polyhedron. The rhodo has no circumsphere, because no one sphere passes through all fourteen vertices. Six vertices of the rhodo are at distance 1 from the center, as the octohedron; eight vertices at distance ½√3, as the cube.

§7. Here are two axonometric views of each of the cube and rhodo, to help give an idea of the relationship of faces and vertices:

these images are from group_12 in the download

cube

rhodo

In the diagram, gray lines separate adjoining faces that are coplanar. The gray lines on the rhodo happen to form the edges of a cube.

On the cube, coplanar faces are never hinged to each other; on the rhodo, they always are.

When the rhodo is folded into a cube, vertices 31 through 36 all meet at the center of the cube.

Vertices 21 and 28 are where the cube can be opened to begin forming other shapes.

§8. There are three planes that will slice all the way through the Shashibo cube, separating some atoms from others, but not damaging any atom. Each of these planes can be defined by the four vertices it passes through. Any one plane separates the atoms into two groups of six; all three planes together produce six groups of two, each pair forming a diatom.

the plane through
these cube vertices separates the atoms
into these groups

21 - 23 - 26 - 28 1 - 2 - 3 - 4 - 5 - 6

7 - 8 - 9 - 10 - 11 - 12

21 - 22 - 27 - 28 1 - 2 - 3 - 4 - 11 - 12

5 - 6 - 7 - 8 - 9 - 10

21 - 24 - 25 - 28 1 - 2 - 9 - 10 - 11 - 12

3 - 4 - 5 - 6 - 7 - 8

all three planes 1 - 2 3 - 4 5 - 6

7 - 8 9 - 10 11 - 12

the plane through these cube vertices	separates the atoms into these groups
`21 - 23 - 26 - 28`	`1 - 2 - 3 - 4 - 5 - 6`
`7 - 8 - 9 - 10 - 11 - 12`
`21 - 22 - 27 - 28`	`1 - 2 - 3 - 4 - 11 - 12`
`5 - 6 - 7 - 8 - 9 - 10`
`21 - 24 - 25 - 28`	`1 - 2 - 9 - 10 - 11 - 12`
`3 - 4 - 5 - 6 - 7 - 8`
all three planes	`1 - 2`	`3 - 4`	`5 - 6`
`7 - 8`	`9 - 10`	`11 - 12`

A diatom is an asymmetric tetrahedron, and is not an atom. As an example, the diatom made of atoms 1 and 2 is defined by the coordinates of vertices 21, 23, 24, and 28. Using the coordinates of §6, these are:

vertex ( x, y, z )
21 ( −½, −½, −½ )
23 ( −½, +½, −½ )
24 ( +½, +½, −½ )
28 ( +½, +½, +½ )

vertex		(	x,	y,	z )
`21`	(	−½,	−½,	−½ )
`23`	(	−½,	+½,	−½ )
`24`	(	+½,	+½,	−½ )
`28`	(	+½,	+½,	+½ )

The same slicing planes apply to the rhodo, further supporting its interpretation as an inside-out cube.

the plane through
these cube vertices also passes through
these rhodo vertices

21 - 23 - 26 - 28 32 - 35

21 - 22 - 27 - 28 33 - 34

21 - 24 - 25 - 28 31 - 36

the plane through these cube vertices	also passes through these rhodo vertices
`21 - 23 - 26 - 28`	`32 - 35`
`21 - 22 - 27 - 28`	`33 - 34`
`21 - 24 - 25 - 28`	`31 - 36`

With the rhodo, no two atoms adjoin to form a diatom. However, the following pairs adjoin to make square pyramids pointing outward: 1 and 4; 2 and 11; 3 and 6; 5 and 8; 7 and 10; 9 and 12. When the Shashibo is folded into a cube, these same pyramids point inward.

§9. The pattern of numbers for the hinges is periodic, with a period of four; this suggests devising Shashibo variants with 4, 8, 16, 20, or more atoms. The pattern below shows a plan to produce atoms in groups of four, which may be repeated as desired:

these images are from group_4 in the download

P: → 1, 5, 9, 13... Q: → 2, 6, 10, 14...

R: → 3, 7, 11, 15... S: → 4, 8, 12, 16...

If the hinge between atoms 1 and 12 of a standard Shashibo is severed, they can be stretched out in a roughly linear configuration, as shown below. This helps convey the atom's periodicity. Vertex numbers are in blue.

this image is from group_12 in the download

The 4-atom Shashibo variant has little potential, but the 8-atom variant merits its own discussion. The 16-atom variant, although it is not pictured in this report, has been successfully constructed of cardboard by the present author. Many shapes are possible.

§10. A photograph of a Shashibo folded into an interesting shape does not give much indication as to how to fold the object into that shape; certainly not enough to mathematically define the figure. Also, a photograph of one side of the object gives no information about what the other side might look like. This problem is greatly compounded in multi-Shashibo creations.

A mitigation is to manufacture a Shashibo with the face name (e.g. 1A, 3C, 7D, 11B) printed directly on each face. A stopgap for a Shashibo that has already been produced with the usual artwork is to apply adhesive labels to the faces of the atoms.

Shashibos are sometimes sold in sets of two or four. A different letter can be assigned to each Shashibo, and printed in each of its faces. Because uppercase letters A, B, C, and D are heavily used in this report, the names of the respective Shashibos should, for contrast, be lowercase letters starting with e and continuing with f, g, etc. Using different letters and different cases will help prevent human error.

As examples, in the table below are the first three atoms of four Shashibos. A different background color, optional, has been added for each. Information such as vertex numbers and hinge connections may be added if desired.

these images are from group_3 in the download

Shashibo
e … nine more blue atoms …

Shashibo
f … nine more red atoms …

Shashibo
g … nine more green atoms …

Shashibo
h … nine more brown atoms …

The atoms of a Shashibo have a period of four, and the "roughly linear" image of the previous section can be turned upside down. Hence a Shashibo shape, even if asymmetrical, can be made with the loop of atoms placed in six different orientations. For an example, pictured below is the "Spaced Out" Shashibo followed by six (admittedly primitive) photographs of an (admittedly primitive) cardboard model. These images are from photo_12 in the download.

Within each row below, the three orientations are related by rotation of the loop; turning the loop upside down relates one row to the other.

The artwork on the typical Shashibo is repeated extensively from atom to atom, so the various loop orientations may not look different when implemented with a Shashibo as manufactured.

It is not clear if there can be a meaningful canonical criterion for selecting a preferred loop orientation when describing the implementation of a shape.

§11. The rhodo has 12 rhombic faces. The reader may be aware of another polyhedron with rhombic faces, namely the rhombic triacontahedron ("rhotria" for short). However, there is no simple way to construct a rhotria from Shashibos.

The acute angle between edges of each rhombus in a rhotria must be acos (1/√5) ≈ 63.43 deg, but that angle is not offered by any face of a Shashibo atom, nor any combination of faces. The table below shows what comes close, but not close enough.

angles between rhombus edges

acute obtuse

value ref. ltr. value ref. ltr.

rhotria acos (1/√5) ≈ 63.43 deg — acos (−1/√5) ≈ 116.57 deg —
rhodo
using face(s) B?C using vertex A:
acos (1/3) ≈ 70.53 deg h using two of vertices B, C, and D:
2 acos (1/√3) ≈ 109.47 deg 2g

rhodo
using face D using vertices B and C:
2 acos (2/√6) ≈ 70.53 deg 2e using vertex A:
acos (−1/3) ≈ 109.47 deg f

the reference letters are those in §3D

	angles between rhombus edges
	acute	obtuse
	value	ref. ltr.	value	ref. ltr.
rhotria	acos (1/√5) ≈ 63.43 deg	—	acos (−1/√5) ≈ 116.57 deg	—
rhodo using face(s) `B?C`	using vertex `A`: acos (1/3) ≈ 70.53 deg	h	using two of vertices `B`, `C`, and `D`: 2 acos (1/√3) ≈ 109.47 deg	2g
rhodo using face `D`	using vertices `B` and `C`: 2 acos (2/√6) ≈ 70.53 deg	2e	using vertex `A`: acos (−1/3) ≈ 109.47 deg	f
	the reference letters are those in §3D

§12. The drawings in this report are in the Scalable Vector Graphics (SVG) format. The reader may scale the images larger or smaller with negligible loss of accuracy. This contrasts with raster graphics, which generally does not scale well.

A consequence of SVG is that the exact appearance of the typography will depend on what fonts happen to be installed in the user's computer. The SVG files supplied here give merely the broad specification font-family="monospace", which does not demand any particular font. Because every SVG environment is required to have some sort of monospaced font, the specification font-family="monospace" is practically guaranteed to give results that are readable, even if not aesthetically satisfying. Different browsers on the same computer might select different monospaced fonts.

Incidentally, these SVG file are human-readable with an ordinary text editor, and can in fact be modified by the courageous. The files were generated with text output from the present author's homebrew C++ program.

§13. Here are some opinions held by the present author, with which reasonable people may well disagree.

I have great admiration for the geometric and magnetic design of the Shashibo; it is a major accomplishment.

However, I am puzzled by the artwork applied to the faces of the atom. In the case of the Shashibo titled "Spaced Out":

The A faces have one design;
The B and C faces share a second design;
Half the D faces have one design, while the other half have a different design.

In total, there are four different designs, and I cannot find any artistic relationship among them. I have a similar reaction to the other Shashibo I own, which is called "Disco Plaid". Indeed, if the D-face designs from one Shashibo were swapped with those of the other Shashibo, there would be no greater or lesser artistic unity.

An additional concern is that much advertising material shows only the design on the A faces, leaving the prospective purchaser in the dark about what might be on faces B, C, and D.

§14. Relationship to the Duomoto.

§14A. The Shashibo company has introduced a newer product, the Duomoto, a puzzle similar in character to the Shashibo. Although the two products are not compatible, it is worthwhile to compare their geometry.

A Shashibo is a loop of 12 tetrahedra, in this section termed "S-atoms". Meanwhile, a Duomoto contains two separate loops of 12 smaller tetrahedra, here called "D-atoms". The two Duomoto loops are equal in all regards.

Recall from §2 above: The Cartesian coordinates of the four vertices of a unit-size S-atom at a convenient location in space are:

A = ( ½, ½, ½ )

B = ( 0, 1, 0 )

C = ( 1, 0, 0 )

D = ( 0, 0, 0 )

With the definition of point E = (½, ½, 0), this tetrahedron is symmetric about the plane ADE.

Plane ADE can be used as a slicing plane, forming two equal tetrahedra (ABDE and ACDE), either of which can be used as the model for the D-atom. They are hinged at edge AE, and in some Duomoto shapes are folded together becoming congruent to an S-atom.

A Shashibo has 12 S-atoms, a Duomoto has 24 D-atoms, and a D-atom is half the size of an S-atom. Therefore, a Duomomo cube is exactly the same size as a Shashibo cube, and is shipped that way from the factory. Further, a Dumoto rhodo can be formed, and is congruent to the Shashibo rhodo. Indeed, a Shashibo cube fits precisely inside the cavity of a Duomoto rhodo, and vice versa.

From the geometry, it would seem that Duomotos and Shashibos are highly compatible; the pieces of the two can align cleanly when they are used together. The difference, however, is that the system of magnets in the Duomoto is utterly different from that in the Shashibo. The present author has been unable to find any attactive shape using one of each, if the magnets are to hold them together. Because magnets are fundamental to operation of both the Shashibo and Duomomo, the factory is wise to advertise the two puzzles as incompatible.

A third product from the same company, the Cubendi, bears a resemblance to the Shashibo and Duomoto. However, its geometry is so different from that of the Shashibo and Duomoto that meaningful comparison can hardly be made.

§14B. A long list of measures of the D-atom, in the manner of §3 above, is not included here. However, the tetrahedron calculator will produce numerical values for the D-atom when the numbers below are used. Note that the D-atom's B coordinates for calculator entry differ from the S-atom's.

Many of the D-atom's calculated numbers will found among the S-atom's, although often in different locations.

`A`	= ( 30½,	30½,	30½	)
`B`	= ( 0,	61,	0	)
`C`	= ( 61,	0,	0	)
`D`	= ( 0,	0,	0	)

2e + f = 2g + h = 2i + j = 180 deg
f + h = 180 deg	2g = f
e + g = 90 deg	2e = h

input

specifying vertices		x	y	z
	A
	B
	C
	D

specifying edge lengths		AB	AC	AD
specifying edge lengths		CD	BD	BC

these images are from `group_4` in the download
`P:` → `1, 5, 9, 13...`	`Q:` → `2, 6, 10, 14...`

`R:` → `3, 7, 11, 15...`	`S:` → `4, 8, 12, 16...`

	these images are from `group_3` in the download
Shashibo `e`		… nine more blue atoms …
Shashibo `f`		… nine more red atoms …
Shashibo `g`		… nine more green atoms …
Shashibo `h`		… nine more brown atoms …