Matching Triangles on Solids


2020/06/30
If you color the edges of equilateral triangles using four colors you can get 24 different pieces. Long ago MacMahon suggested these pieces to construct lots of polygons with matching colors and same colored border. I wanted to cover solids with these pieces and took an octahedron augmented by 8 tetrahedrons or a cube with 6 augmented square pyramids. The icosahedron can be covered by 20 pieces of the set with the four same colored pieces left. But instead of matching equal colors you can only get matching pairs of edges because the number of equal edges is odd.

To find more solids to cover I started with convex solids add pyramids or divided triangular faces into four or more trianglar parts. Eleven tetrahedrons can also be joined to chains with 24 faces. There are two triangular faces for each tetrahedron and two additional faces at the ends. More than two thousands constructions seem to be possible. My count was 2606. It was easy to cover such a chain but it was difficult to prevent the constructions from breaking apart.

Furthermore I explored larger sets with more colors or restrictions like "no same colored pieces" or "all edges differently colored" . If the triangles aren't equilateral there is no rotational symmetry and the number of pieces for a given color is much larger. They can be used to cover solids with all corners on the surface of a sphere.

Click the number of pieces in the table to see some constructions.

Triangle Type Properties Colors
2 3 4 5 6 7 8 9 10 24 n
Equilateral Triangles All Combinations 4 11 24 45 76 119 176 249 340 4624 (n^3+2*n)/3
No Same Colored Pieces 2 8 20 40 70 112 168 240 330 4600 (n^3+2*n)/3 - n
All Sides Differently Colored 0 2 8 20 40 70 112 168 240 4048 n*(n-1)*(n-2)/3
Isosceles Triangles
being not equilateral
All Combinations 8 27 64 125 216 343 512 729 1000 13824 n^3
No Same Colored Pieces 6 24 60 120 210 336 504 720 990 13800 n^3 - n
All Sides Differently Colored 0 6 24 60 120 210 336 504 720 12144 n*(n-1)*(n-2)

Equilateral Triangles, 2 Colors, All Combinations, 4 Pieces

Let a and b the two given colors. If we cover a tetrahedron with this set the piece with aaa has one common edge with bbb. Therefore it's not possible to get a construction with matching a-a and b-b at all edges. But it is possible to have a-b matches at all edges. In the pictures the two colors are shown by combinations of circles grey-red and red-grey.

Equilateral Triangles, 4 Colors, All Combinations, 24 Pieces

Cube extended by
6 square pyramids

Triangular dipyramid
with divided faces

Octahedron extended by
8 tetrahedrons

Three chains of 11 tetrahedrons

Dodecadeltahedron
extended by 6 tetrahedrons

Tetracaidecadeltahedron
extended by 5 tetrahedrons

Hexadecadeltahedron
extended by 4 tetrahedrons

Hexagonal
Antiprism


To get a better idea how the last four solids are built click the pictures.

Equilateral Triangles, 6 Colors, All Combinations, 76 Pieces

Starting with a 16 faces deltahedron we first split each triangle into 4. Now we have 64 triangles. Then I wanted to add six tetrahedrons because this way we get 64-6+6*3 = 76 faces. You can divide the deltahedron into two square pyramids and one antiprism. Four tetrahedrons are attached at the centers of one square pyramid and two at the antiprism. A virtual object to turn around is here.

Equilateral Triangles, 8 Colors, All Combinations, 176 Pieces

This is a chain of 21 tetrahedrons with 3+19*2+3 = 44 faces. Each faces was divided into 4 triangles, so we have a total of 44*4=176 triangles. A virtual object to turn around is here.

Equilateral Triangles, 24 Colors, All Combinations, 4624 Pieces

A 16-faced deltahedron can be covered by 16 large triangles of size 17 each made from 289 pieces of the set. It's even possible to get same colored edges at the sides of the large triangles. I decided to use 8 colors at the edges of the pieces and added half a circle, half a rhomb or nothing to get 3*8 = 24 different edge types.

Let's number the colors from 0 to 23. A color shift is given, if each color c is replaced by (c+6) mod 24. For each piece there are always three other pieces which can be obtained by some color shifts. This way we get 1156 groups of 4 pieces which I used for the construction. Taking exactly one piece of each group for a quarter of the construction three other parts are given by color shifts. Predetermined borders must carefully be chosen to get matches between the first part and the shifted versions.

The net of the 16-faced deltahedron is shown, a SVG-file is also provided.

In the section about matching polygons a triangle of size 68 is shown which can be folded to get a tetrahedron, but there are three edges which are not same colored. Using the same method as above I found a net of 16 triangles for the tetrahedron with all edges same colored. The SVG-file is here.

Equilateral Triangles, 3 Colors, No Same Colored Pieces, 8 Pieces

The pieces are just enough to cover a octahedron. The octahedron in the picture is made from two square pyramids with iron foil attached. The printed pieces were glued to magnetic foil.
The three 'colors' of these pieces are gaps at the center, left from center and right from center. Left gaps match right gaps and vice versa whereas center gaps match center gaps.
There is only one chain of three tetrahedrons joined at their faces, which is often called boat. This chain can also be covered by the given set.

Equilateral Triangles, 4 Colors, No Same Colored Pieces, 20 Pieces

We have 20*3 = 60 edges with 60/4 = 15 edges of each color. Therefore a match a-a, b-b, c-c, d-d isn't possible. Instead I used gaps at four different positions: left from center, right from center, far left from center and far right from center. These gaps can match like r-l or R-L and we get an icosahedron fixed at the gaps. With colors we can use the pair red-green matching with green-red and the pair blue-yellow matching with yellow-blue. A construction is shown here.

Equilateral Triangles, 5 Colors, No Same Colored Pieces, 40 Pieces

Take a dicube with 10 quadratic faces and add a square pyramid to each face. Now we have 10*4 = 40 triangular faces, which can be covered by the given pieces. A solution is here. You can also cover some other solids which were shown using the set of pieces with 6 colors and all sides differently colored.

Equilateral Triangles, 6 Colors, No Same Colored Pieces, 70 Pieces

Take a pentagonal prism with a height three times the length of the pentagon sides. We can add pentagonal pyramids to bottom and top and square pyramids to the other faces. Then we have a total of 2*5+5*3*4 = 70 triangular faces. Since 70*3/6 = 35 is odd only matches of a-b, c-d and e-f are possible. A solution is here. You can also cover some other solids which were shown using the set of pieces with 7 colors and all sides differently colored.

Equilateral Triangles, 7 Colors, No Same Colored Pieces, 112 Pieces

Starting with a cube and adding six square pyramids to its faces we get 24 faces. After each face is divided into four triangles we have 96 faces. At least we add Tetrahedrons at 8 center triangles. You can turn the construction here.

Equilateral Triangles, 8 Colors, No Same Colored Pieces, 168 Pieces

A row of 10 cubes with 42 square faces is expanded by 42 square pyramids with a total of 42*4 = 168 triangular faces. Since 168*3/8 = 63 is odd matching a-a ... h-h isn't possible. Turn the row around here.

Equilateral Triangles, 9 Colors, No Same Colored Pieces, 240 Pieces

All 20 faces of an icosahedron are divided into 4 triangles. Now we can add 80 pyramids to these triangles and we get 240 faces. Here you can look at the construction from all directions.

Equilateral Triangles, 4 Colors, All Sides Differently Colored, 8 Pieces

The 8 pieces can be used to cover an octahedron. A net for a solution is shown.
The 'boat' composed of three tetrahedrons can also be covered, but I think it's easier to solve the problem using the 3-color set with no same colored pieces.

Equilateral Triangles, 5 Colors, All Sides Differently Colored, 20 Pieces

With 20*3/5 = 12 edges of each color matching of a-a, b-b, c-c, d-d and e-e is possible in contrast to the set of pieces with color repetition allowed but same colored pieces discarded. Here is the virtual icosahedron.

Equilateral Triangles, 6 Colors, All Sides Differently Colored, 40 Pieces

These are some solids, which can be covered by 40 triangles. On the left there is a pentagonal dipyramid where all faces are split into four parts. Below you can see a dicube with 10 square pyramids and examples of 4 joined tetrahedrons, whose faces are replaced by four triangles. Click the pictures to turn the constructions around.

Equilateral Triangles, 7 Colors, All Sides Differently Colored, 70 Pieces

Pentagonal prism with
17 augmented pyramids

Hexadecadeltahedron, faces split,
3 tetrahedron added

Chain of 31 tetrahedrons

Click the pictures to see hidden parts of the solutions.
The same solids can also be covered by a set with only 7 colors but allowed color repetion up to two sides.

Equilateral Triangles, 8 Colors, All Sides Differently Colored, 112 Pieces

The set with only 7 colors but allowed color repetion up to two sides has also 112 pieces and the same solid can be covered. You can turn the construction here.

Equilateral Triangles, 9 Colors, All Sides Differently Colored, 168 Pieces

The set with only 8 colors but allowed color repetion up to two sides has also 168 pieces and the same solid can be covered. But there is a difference. Since 168*3/9 = 56 is even we can match a-a, b-b and so on. You can turn the construction here.

Equilateral Triangles, 10 Colors, All Sides Differently Colored, 240 Pieces

The set with only 9 colors but allowed color repetion up to two sides has also 240 pieces and the same solid can be covered. You can turn the construction here.

Isosceles Triangles, 2 Colors, All Combinations, 8 Pieces

Take a square of size 1 and add two square pyramids of height 0.5 on both faces. You get an octahedron with 8 isosceles triangles. It's a shrunk version of the platonic octahedron, and it can be covered by the set. You can choose square pyramids of other heights because the construction with matching colors remains the same.

Isosceles Triangles, 4 Colors, All Combinations, 64 Pieces

We can stretch a 16-faced convex deltahedron, so that the faces aren't equilateral anymore but still isosceles. These triangles can be divided into 4 congruent parts giving 16*4 = 64 parts, just enough for the set. A version to turn the construction around is here.

Isosceles Triangles, 6 Colors, All Combinations, 216 Pieces

A column of 13 cubes augmented by 13*4 + 2 = 54 square pyramids has a total surface of 54*4 = 216 faces. The height of the pyramids is determined by the base and the legs of the given pieces. Here you can turn the construction around.

Isosceles Triangles, 3 Colors, No Same Colored Pieces, 24 Pieces

A cube augmented by 6 square pyramids and an octahedron augmented by 8 triangular pyramids is shown. It's rather easy to cover the faces by hand.

Isosceles Triangles, 4 Colors, No Same Colored Pieces, 60 Pieces

60 pieces are the perfect set to cover a dodecahedron with 12 pentagonal pyramids or a isocahedron with 20 trigonal pyramids. If all corners are on the surface of a sphere these solids are called pentakis dodecahedron or triakis icosahedron, respectively. Since there are 60*3/4 = 45 edges of each color a match of same colors isn't possibe but we can get matches of color pairs. If the colors are represented by gaps at different positions the match of a-b and c-d is natural in a real construction.
Here are virtual construction for the pentakis dodecahedron and the triakis icosahedron.

I used gaps at different positions instead of colors to print a real object.

Isosceles Triangles, 5 Colors, No Same Colored Pieces, 120 Pieces

The straight heptacube has a surface of 7*4 + 2 = 39 square faces. If we add square pyramids to these square faces, we get 30*4 = 120 triangular faces, which can be covered by the set. Here you can turn the construction around.

Isosceles Triangles, 6 Colors, No Same Colored Pieces, 210 Pieces

I chose a chain of 34 tetrahedrons with 32*2 + 6 = 70 faces and added triangular pyramids to get 210 faces. Due to the height of the pyramids the solids look quite different, but the structure of the matching edges remains the same. Because 210*3/6 = 105 is odd there must be a match of color pairs. Here is the virtual object.

Isosceles Triangles, 3 Colors, All Sides Differently Colored, 6 Pieces

Joining two triangular pyramids with equilateral base and congruent other faces we get a surface of 6 isosceles triangles, which ccan be covered by the set.

Isosceles Triangles, 4 Colors, All Sides Differently Colored, 24 Pieces

This is the third set with 24 pieces. The augmented cube and the augmented octahedron can also be covered by this set.

Isosceles Triangles, 5 Colors, All Sides Differently Colored, 60 Pieces

A pentakis dodecahedron and a trikatis icosahedron can be covered. Because 60*3/5 = 36 is even same colors can be matched at the edges, contrary to the set of 60 pieces with 4 colors and same colored pieces discarded.Here are virtual construction for the pentakis dodecahedron and the triakis icosahedron.

Isosceles Triangles, 6 Colors, All Sides Differently Colored, 120 Pieces

The 120 pieces with all edges differently colored can also cover the column of 7 cubes augmented by square pyramids. A virtual object to turn around is here.

Isosceles Triangles, 7 Colors, All Sides Differently Colored, 210 Pieces

I think the chain of 34 tetrahedrons with augmented pyramids looks better with matching colors at all edges, which now is possiblle since 210*3/7 = 90 is even. A virtual object to turn around is here.

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