The Convex Deltahedra

The Convex Deltahedra

And the Allowance of Coplanar Faces

And the Allowance of Coplanar Faces

Roger's Polyhedra, (c) 2006-2024, Roger Kaufman

The Convex Deltahedra are a set of polyhedra all built with triangular faces. The rules of convexity say that every dihedral angle of the outer structure must be less than 180 degrees. This puts certain limits on the set. What if instead we allow adjacent coplanar faces, areas of the polyhedron which are flat from one triangle to the next?

Deltahedra are polyhedra composed entirely of equilateral triangles. There are only eight convex deltahedra. They were enumerated in 1915 O. Rausenberger [Ref] and later in 1947 by H. Freudenthal and B. H. van der Waerden [Ref] The later paper was a formal proof that these eight are the complete set.

Here are models of the 8 convex deltahedra. Of these, only the Snub Disphenoid (J84) cannot be made by augmentation of simpler polyhedra and is elementary.

Each model includes the symmetry, face, edge and vertex counts, and the number of vertex orders occurring. e.g. for the Tetrahedron there are four 3-valence vertex connections.

Links below display various models

Clicking on pictures opens an Interactive OFF Viewer
OFF files open in an online browser. Downloaded files may be viewed with an application such as Antiprism
STEL files may be saved and opened with Stella
VRML files (wrl,switch,cyl) open in an online browser. Downloaded files may be viewed with a VRML browser such as FreeWRL

The Convex Deltahedra

The Convex Deltahedra

From the Euler Characteristic we know that V-E+F=2. For a polyhedron the number of edges and faces together subtracted from the vertices always equals 2. From that we can determine for a given number of faces how many vertices and edges are required. Each triangle has three edges and in the polyhedron each edge is shared with another face. For that reason the total number of edges required is fixed and is 3*F/2. The number of vertices is therefore also fixed and can be calculated directly from the Euler formula. Incidentally, this also proves that all deltahedra must have an even number of faces. If F is odd then 3*F/2 calculates to a fractional number of edges.

What we notice from the sequence above is that the number of faces is no higher than 20. If the number of faces is higher than 20, there is always the occurrence of six faces meeting at a vertex. When that happens, either all six faces are coplanar (a hexagon), two pairs of thee faces are coplanar (Triamonds), or some non-convex dihedral angles occur.

Notably missing from the sequence is 2 and 18. While there is often commentary on these, there usually are no models to show what happens when an attempt is made to constructing them. Let's investigate.

Exploring the Flat Case

Having only two faces which are connected front to back, the Triangular Dihedron can be considered a degenerate prism and it is not considered a valid polyhedron. It is 2 dimensional and has no volume but otherwise it has some of the characteristics of a polyhedron. There are two faces to every edge. The vertex connections are always to two other vertices.
This figure does not have coplanar faces since no two triangles are connected side to side on the same plane. But paradoxically it has two faces on the same plane!

From here on out we consider what the ramifications if we extend the set by allowing coplanar faces. It lets us fill in a gap, but it also allows for many more possibilities. An infinite number of possibilities.

The Missing Mystery Deltahedron

If we were to try making an 18 faced deltahedron from an icosahedron, one way would be to remove two faces. But this would create six faces meeting at a vertex. There are no non-convex dihedral angles in this model, but the coplanarism discounts it as completely convex. Like the Snub Disphenoid is elementary above, when we open up the set to include coplanar faces, this is the only deltahedron with faces less than 20 that cannot be made via augmentation.

Adrian Rossiter noted that an alternative form of the 18 faced deltahedra exists that can be produced through augmentation. (See below)

These are the only two forms of 18 faced deltahedra without non-convex dihedral angles possible.

Coplanarism or non-convexity is a certainty when the number of faces is greater than 20. Constraining the number of faces to 20 or less, what results if coplanar faces are allowed? It creates other cases for deltahedron other than the one with 18 faces. This is what we have to consider if we change the rules to allowing for coplanarism.

Restricting these polyhedra to having both convexity and coplanarism the number of forms is only 11. One is displayed above as the Mystery Deltahedron, and the other 10 are displayed below. These 10 are constructed when octahedra and tetrahedra are joined in various ways.

The first set of 5 result when starting with an octahedron and augmenting with one to four tetrahedra. This results in coplanar deltahedra with 10 to 16 faces. When augmenting with two tetrahedra there are two variants which possess 12 faces. When all 4 tetrahedra are attached, a tiled tetrahedron is the result.

Notice in the first two models, there are no occurrences of six faces meeting at a vertex but coplanarism is still present. When allowing for only convexity and coplanarism both, these are the only two deltahedra known to have this property.

Convex Deltahedra with Coplanar Faces Constructed from Augmentation

A second set of 4 results by starting with two octahedra connected on edge. Then the non-convex areas that result are filled in with two tetrahedra. The base form of this has 16 faces. When one tetrahedra is attached to the base form, an alternative 18 faced deltahedron is a result which was mentioned above. When two tetrahera are attached it can be done it two ways resulting in 20 faced forms.

There is one more form that is found by augmenting one octahedron on the face by another octahedron. Then filling in the non-convex area with 6 tetrahedra results in a third 20 faced coplanar deltahedra. Note that this form can also be created by connecting two copies of the above 14 faced coplanar deltahedron.

Convex Deltahedra with Coplanar Faces Constructed from Augmentation

Question or comments about the web page should be directed to PolyhedraSmith@gmail.com.

The deltahedra were created in Robert Webb's Stella application and Antiprism. The generation of OFF and VRML files were processed with Antiprism. The Hedron application by Jim McNeill was used to generate switch files. The image files were created with off2pov and POV-ray.

History:

2024-06-17 Use base.css
2024-06-13 Switch to Multi OFF Viewer
2023-09-03 Switch to Simple OFF Viewer and X_Ite VRML Viewer
2023-03-11 Open Interactive Viewer from model pictures
2019-03-12 Changed email address from defunct bigfoot.com
2013-05-15 Some more clarifications
2013-05-12 Some clarications
2013-05-10 Initial Release

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Link to this page as http://www.interocitors.com/polyhedra/Deltahedra/Convex

4 Faces - Tetrahedron

F=4 E=6 V=4

off stel wrl switch