Aid to understanding Shashibo. Aid to understanding Shashibo.
Version of Thursday 13 February 2025.
Dave Barber's other pages.

§1A. The Shashibo is a "patented magnetic puzzle cube", as described in the factory website; the present report covers many of its geometric properties. Not discussed here, however, are 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 great detail.

A Shashibo consists of twelve tetrahedra, each linked to two others to form a loop. All the tetrahedra are shaped the same, 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). 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:

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, in many of the images a face is denoted by one letter rather than three, indicating the sole corner 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 corners 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 "chunk" for any tetrahedron geometrically similar to these.


§3. Here are some measurements of a chunk, assuming unit size.


§3A. Lengths of edges:

edges AB, AC, AD BC BD, CD
exactly ½√3 √2 1
numerically 0.8660 1.4142 1.0

All four faces are isosceles triangles.


§3B. Areas of faces:

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


§3C. Volume is 112 ≈ 0.0833. This is consistent with the fact that the twelve chunks 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

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


§3E. Dihedral angles between faces:

faces ABC & ABD
ABC & ACD
ABD & ACD ABD & BCD
ACD & BCD
ABC & BCD
angle π/3 rad = 60 deg 2π/3 rad = 120 deg π/4 rad = 45 deg π/2 rad = 90 deg


§3F. Solid angles at corners, given in steradians:

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

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


§3G. Altitude between vertex and opposite face:

corner
& 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:

input
specifying vertices   x y z
A
B
C
D
specifying edge lengths AB
AC
AD
CD
BD
BC

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

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 corners are printed on every face; this aids in studying the Shashibo when it is assembled.

When the chunk is built, folds will be made on edges BC, BD, and CD. Then points A1, A2 and A3 will meet to form corner 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 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 corners are not critical, and may need to be adjusted according to the material of which the chunk is fabricated. Shown above are the rather arbitrary values chosen for this report.

Of course, these coordinates can be scaled, translated, rotated, and reflected.


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

these images are from group_12 in the download

Explanation
according to color of typography
In purple: The chunks 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 corners bounding that face. Each face has three out of the four letters A, B, C, and D. They are the same on all twelve chunks.
In blue: When the Shashibo is folded into a cube, the eight corners of that cube are numbered 21 through 28. These numbers indicate which corners of one chunk meet which corners of another chunk. Numbers 21 through 28 will also serve to mark eight of the fourteen corners of a rhodo.
In green: Numbers 31 through 36 denote the six corners 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 chunk 1 will be connected to edge CD of chunk 2 with a hinge. Each of these chunks has one corner marked 23 and another marked 24. Equal corner 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 a chunk, 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 corner numbers (blue) 21-28; rhodo-only corner numbers (green) 31-36. Disjoint ranges were selected to reduce proneness to human error.

Suggestion: When assembling a Shashibo from the twelve chunks:


§6. The chart in §5 above shows, by means of the blue numbers (21-28) and green numbers (31-36), which corners of the cube and rhodo are associated with each corner of a chunk.

The chart below shows Cartesian coordinates for all the corners involved in a complete cube or rhodo formed of unit-sized chunks as introduced in §2. Each is placed at a convenient, but arbitrary, location in three-dimensional space. Note that the coordinates for 21-28 in the rhodo are a negation of those for the cube, which is unsurprising given the inside-out relationship between the two.

cube rhodo
corner  ( 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 )
corner  ( 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 )

Corners 21-28 of the rhodo form a cube, i.e. regular hexahedron. They equal corners 28-21 respectively of the rhodo, which themselves define the cubic empty region therein. Meanwhile, rhodo corners 31-36 form a regular octahedron.


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

  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, corners 31 through 36 all meet at the center of the cube.

Corners 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 chunks from others, but not damaging any chunk. Each of these planes can be defined by the four corners it passes through. Any one plane separates the chunks into two groups of six; all three planes together produce six groups of two, each pair forming a bichunk.

the plane through
these cube corners
separates the chunks
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 - 23 - 45 - 6
7 - 89 - 1011 - 12

A bichunk is an asymmetric tetrahedron, and is not a chunk. As an example, the bichunk made of chunks 1 and 2 is defined by the coordinates of corners 21, 23, 24, and 28. Using the coordinates of §6, these are:

corner  ( 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 corners
also passes through
these rhodo corners
21 - 23 - 26 - 2832 - 35
21 - 22 - 27 - 2833 - 34
21 - 24 - 25 - 2831 - 36

With the rhodo, no two chunks adjoin to form a bichunk. 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 chunks. The pattern below shows a plan to produce chunks 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 chunks 1 and 12 is severed, the chunks can be stretched out in a roughly linear configuration. This helps convey the chunk's periodicity. Corner numbers are in blue.

this image is from group_12 in the download

The 8-chunk Shashibo variant is worthy of its own discussion.

Although it is not pictured here, a 16-chunk Shashibo variant 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 chunks.

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 chunks of four Shashibos. A different background color, optional, has been added for each. Information such as corner numbers and hinge connections may be added if desired.

  these images are from group_3 in the download
Shashibo
e
Shashibo
f
Shashibo
g
Shashibo
h

The chunks 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 chunks 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.

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 chunk to chunk, 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 polyherdron with rhombic faces, namely the rhombic triacontahedron ("rhotri" for short). However, there is no simple way to construct a rhotri from Shashibos.

The acute angle between edges of each rhombus in a rhotri must be acos (1/√5) ≈ 63.43 deg, but that angle is not offered by any face of a Shashibo chunk, 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.
rhotri acos (1/√5) ≈ 63.43 deg acos (−1/√5) ≈ 116.57 deg
rhodo
using faces B and C
using corner A:
acos (1/3) ≈ 70.53 deg
h using two of corners B, C, and/or D:
2 acos (1/√3) ≈ 109.47 deg
2g
rhodo
using face D
using two of corners B and/or C:
2 acos (2/√6) ≈ 70.53 deg
2e using corner 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 assured 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 chunk. In the case of the Shashibo titled "Spaced Out":

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.

Both of these matters have been addressed, at least in part, by a newer product from the Shashibo company. The Duomoto is similar in character to the Shashibo, although the two products are not compatible.