Magic Triangular Pyramids

Why Magic Triangular Pyramids and not Magic Tetrahedra?

Because different edge lengths can only result in irregular and non-isosceles polyhedra, 4-faced perimeter-magic tetrahedra cannot exist. Nevertheless, it is possible to construct 3-faced Perimeter-Magic Triangular Pyramids.

Unlike a tetrahedron, whose four triangular faces can each be considered as its base, a Magic Triangular Pyramid is deemed to have a single triangular base and three triangular faces. The 6 connecting edges of a Magic Triangular Pyramid have different integer lengths, but the 3 triangular faces have the same perimeter.

Primitive Magic Triangular Pyramids using Consecutive Numbers

Using consecutive numbers from 1 to 6 for each of the edges, we can create 4 essentially different Primitive Magic Triangular Pyramids:

The perimeters of Solution A are the opposite vertices of Solution D, and vice versa. The same relationship exists between the Solution B and the Solution C.

Only two of the primitive solutions have consecutive numbers for their base and for one of their face edges. These are the Solution A with base edges (1, 2, 3) and a face with edges (3, 4, 5), and the Solution D with base edges (4, 5, 6) and a face with edges (2, 3, 4).

These four primitive pyramids are not just abstract numerical polyhedra (which display triangles that are either impossible, or too obtuse for 3D assembly), as they can also be extrapolated to construct viable consecutive-numbered Magic Triangular Pyramids.

For example, the Primitive Magic Triangular Pyramid Solution A only displays a single valid triangle which is the Pythagorean triple (3, 4, 5). The three other triangles (1, 2, 3), (1, 5, 6), and (2, 4, 6) do not exist! To obtain valid triangles we need to add 1 to each of the consecutive numbers. However, this is still insufficient, as some of the triangles are too obtuse and cannot be assembled correctly. To construct the smallest Magic Triangular Pyramid derived from Solution A, we need to add 2 and use the consecutive numbers 3 to 8, as shown in the net diagrams below:

Likewise, the Primitive Solutions B and C can only become Magic Triangular Pyramids after adding 5, and using the consecutive numbers from 6 to 11. 

On the other hand, for the Primitive Solution D to become a Magic Triangular Pyramid, it is only necessary to add 4, and use the consecutive numbers from 5 to 10.

Magic Triangular Pyramids using Consecutive Numbers

1/ The Smallest Magic Triangular Pyramid using Consecutive Numbers

As shown above, the smallest Magic Triangular Pyramid that can be constructed using consecutive numbers is the Primitive Solution A + 2. The resulting pyramid, which uses the numbers 3 to 8, is illustrated below:

The striking characteristic of this smallest Magic Triangular Pyramid is that it is based on the primitive Pythagorean Triangle (3, 4, 5). The volume V of a pyramid is 1/3 of the base area x the height of the apex.  Therefore, in this case, as the area of the base is 6, the volume V is the height of the pyramid x 2. 

With non-rational numbers rounded to five or six decimal places*, the other details of this Magic Triangular Pyramid are as follows:
The perimeter of the triangular base is 12. 

The perimeter of each of the three triangular faces is 18.
The total perimeters of the triangular base and faces is 66.
The area of the primitive Pythagorean triangle base (3, 4, 5) is 6.
The area of the triangular face (3, 7, 8) is 10.39230...*
The area of the triangular face (4, 6, 8) is 11.61895...*
The area of the triangular face (5, 6, 7) is 14.69694...*
The total areas of the triangular base and faces is 42.70819...*
The height of the apex is 4.213075...*
The volume V of the pyramid is 8.42615...*
The base vertices (3, 4, 8), (4, 5, 6) and (3, 5, 7) each total 15.
The apex (6, 7, 8) totals 21.

2/ The Smallest Magic Triangular Pyramid with an Integer Volume using Consecutive Numbers 

The smallest Magic Triangular Pyramid with an integer volume that can be constructed using consecutive numbers is the Primitive Solution D + 5. The resulting pyramid, which uses the numbers 6 to 11, is illustrated below:

The main characteristics of this Magic Triangular Pyramid is that it not only has a non-primitive Pythagorean triangle face (with area and perimeter both equal to 24), but also an integer volume of 48.

With non-rational numbers rounded to five or six decimal places*, the other details of this Magic Triangular Pyramid are as follows:

The perimeter of the triangular base is 30.

The perimeter of each of the three triangular faces is 24.

The total perimeters of the triangular base and faces is 102.

The area of the triangular base (9, 10, 11) is 42.42641...*

The area of the non-primitive Pythagorean triangle face (6, 8, 10) is 24.

The area of the triangular face (7, 8, 9) is 26.83282...*

The area of the triangular face (6, 7, 11) is 18.97367...*

The total areas of the triangular base and faces is 112.23289...*

The height of the apex is 3.39411...*

The volume V of the pyramid is 48.
The perpendicular distance from the Pythagorean triangle face (6, 8, 10) to the opposite vertex (7, 9, 11) is 3 x 48 / 24 = 6.
The base vertices (8, 9, 10), (6, 10, 11) and (7, 9, 11) each total 27.
The apex (6, 7, 8) totals 21.

This integer volume tetrahedron has already been described elsewhere, (for example in C. Chisholm and J.A. MacDougall's paper "Rational tetrahedra with edges in arithmetic progression" and in Wikipedia's article "Tetrahedron"), but no previous mention seems to have been made of the equal perimeter faces and the resulting magic pyramid aspect. C. Chisholm and J.A. MacDougall conjecture that "this is the only tetrahedron with edges in arithmetic progression having rational volume and a rational face area."

Magic Triangular Pyramids using Non-Consecutive Numbers

1/ A Magic Triangular Pyramid with 2 Primitive Pythagorean Triangles

The following Magic Triangular Pyramid has been constructed using two primitive Pythagorean triangles. The Pythagorean triangle used for the base (3, 4, 5) is the same as that used for the smallest Magic Triangular Pyramid (see Solution A + 2). The result is illustrated below:

The main characteristic of this Magic Triangular Pyramid is that the Pythagorean triangle (5, 12, 13) aligns the face edge 12 at right angles to the base edge 5. Also, as in the example of the smallest Magic Triangular Pyramid (see above), the volume is twice the height of the apex. We can see that this pyramid is equivalent to the Primitive Solution A + 8 for the 3 numbers that are greater than 3, and equivalent to the Primitive Solution A + 2 for the 3 numbers that are less than 4.

With non-rational numbers rounded to five or six decimal places*, the other details of this Magic Triangular Pyramid are as follows:
The perimeter of the triangular base is 12.
The perimeter of each of the three triangular faces is 30.
The total perimeters of the triangular base and faces is 102.
The area of the primitive Pythagorean triangular base (3, 4, 5) is 6.
The area of the primitive Pythagorean triangular face (5, 12, 13) is 30.
The area of the triangular face (3, 13, 14) is 18.97367...*
The area of the triangular face (4, 12, 14) is 22.24859...*
The total areas of the triangular base and faces is 77.22226...*
The height of the apex is 9.367497...*
The volume V of the pyramid is 18.734994...*
The base vertices (3, 5, 13), (4, 5, 12) and (3, 4, 14) each total 21.
The apex (12, 13, 14) totals 39.

2/ A Magic Triangular Pyramid with 2 Non-Primitive Pythagorean Triangle Faces

The following Magic Triangular Pyramid has been constructed using two non-primitive Pythagorean triangles that both have the same perimeter of 420, and share a same side length of 175. The resulting pyramid is illustrated below.

The main characteristic of this Magic Triangular Pyramid is that the Pythagorean triangles align the face edges 175 and 140 at right angles to the base edges 60 and 105.

With non-rational numbers rounded to five or six decimal places*, the other details of this Magic Triangular Pyramid are as follows:
The perimeter of the triangular base is 260.
The perimeter of each of the three triangular faces is 420.
The total perimeters of the triangular base and faces is 1520.
The area of the triangular base (60, 95, 105) is 2821.79021...*
The area of the non-primitive Pythagorean triangular face (105, 140, 175) is 7350.
The area of the non-primitive Pythagorean triangular face (60, 175, 185) is 5250.
The area of the triangular face (95, 140, 185) is 6500.96147...*
The total areas of the triangular base and faces is 21922.75168...*
The height of the apex is 129.94673...*
The volume V of the pyramid is 122227.474635...*
The base vertices (60, 95, 185), (95, 105, 140) and (60, 105, 175) each total 340.
The apex (140, 175, 185) totals 500.

3/ A Degenerate Heronian Magic Triangular Pyramid with Zero Volume

For the definition of a Heronian Magic Triangular Pyramid, we can refer to the description of a Heronian Tetrahedron (or Perfect Tetrahedron) as detailed by Eric Weisstein in MathWorld:
"A Heronian tetrahedron, also called a perfect tetrahedron, is a (not necessarily regular) tetrahedron whose sides, face areas, and volume are all rational numbers. It therefore is a tetrahedron all of whose faces are Heronian triangles and additionally that has rational volume."
We will impose two further constraints:
The 6 integer edge lengths must be different (with no isosceles triangles). The perimeters of 3 out of the 4 faces must be equal.

This degenerate Heronian Magic Triangular Pyramid was found when searching for a solution using Heronian triangles. The Pyramid can be correctly assembled, but the resulting apex has zero altitude! The following net diagram shows the plan view of the components of the pyramid together with the construction lines for the apex intersection, prior to assembly:

The 3D view that follows shows the degenerate Heronian Magic Triangular Pyramid after it is assembled. Although the faces (10, 35, 39) and (17, 28, 39) can fold over the base, the third face (21, 35, 28) has to remain in a horizontal position - thereby imposing the coplanarity of the four vertices:

The example is particularly interesting because Heronian triangles are used throughout the construction, and this produces totally rational areas and volume, (albeit zero for the latter).

The details of this degenerate Heronian Magic Triangular Pyramid are as follows:
The perimeter of the Heronian triangular base is 48.
The perimeter of each of the three Heronian triangular faces is 84.
The total perimeters of the Heronian triangular base and faces is 300.
The area of the primitive Heronian triangular base (10, 17, 21) is 84.
The area of the Heronian and non-primitive Pythagorean triangular face (21, 28, 35) is 294.
The area of the primitive Heronian triangular face (10, 35, 39) is 168.
The area of the primitive Heronian triangular face (17, 28, 35) is 210.
The total areas of the triangular base and faces is 756.
The height of the apex is 0.
The volume V of the pyramid is 0.
The base vertices (10, 21, 35), (10, 17, 39) and (17, 21, 28) each total 66.
The apex (28, 35, 39) totals 102.

Degenerate examples, (with a volume, that although rational, is equal to zero), should not be neglected: Their perfect geometry makes them worthy of interest. Both Heronian and degenerate Heronian Magic Triangular Pyramids merit to become the subject of future research.

4/ An Integer Heronian Magic Triangular Pyramid

On the 26th August 2018, a first Heronian Magic Triangular Pyramid was jointly discovered by Walter Trump and William Walkington. This example was found in a list of Heronian Tetrahedra that had previously been calculated by Sascha Kurz during his research "On the Generation of Heronian Triangles." (At the time, Sascha Kurz was not looking for solutions with 3 equal face perimeters).

The net of this Integer Heronian Magic Triangular Pyramid is shown below:

The 3D view that follows shows the Integer Heronian Magic Triangular Pyramid after it is assembled:

The example is particularly interesting because Heronian triangles are used throughout the construction, and in this case, totally rational integer areas and volume are produced.

The details of this Integer Heronian Magic Triangular Pyramid are as follows:
The perimeter of the primitive Heronian triangular base W is 2587706.
The perimeter of each of the three Heronian triangular faces is 2004956.
The total perimeters of the Heronian triangular base and faces is 8602574.
The area of the primitive Heronian triangular base W (773053, 860349, 954304) is 314938488480.
The area of the Heronian triangular face X (569974, 662929, 773053) is 183994812120.
The area of the primitive Heronian triangular face Y (481678, 662929, 860349) is 158732366520.
The area of the Heronian triangular face Z (481678, 568974, 954304) is 104418108480.
The total areas of the Heronian triangular base and faces is 762083775600.
The non-integer, but rational height of the apex is 214201+83/85.
The volume V of the pyramid is 22486815566358528.
The base vertices (662929, 773053, 860349), (481678, 860349, 954304) and (568974, 773053, 954304) each total 2296331.
The apex (481678, 568974, 662929) totals 1713581.

5/ A Prime Perimeter Magic Triangular Pyramid

Two months after the first publication of this post, the following Magic Triangular Pyramid was constructed using prime number perimeters:

With non-rational numbers rounded to five or six decimal places*, the details of this Prime Perimeter Magic Triangular Pyramid are as follows:
The perimeter of the triangular base is 47.
The perimeter of each of the three triangular faces is 71.
The total of the perimeters of the triangular base and faces is 254.
The area of the triangular base (11, 17, 19) is 92.69405…*
The area of the triangular face (11, 29, 31) is 159.49980…*
The area of the triangular face (17, 23, 31) is 192.20351...*
The area of the triangular face (19, 23, 29) is 218.15634...*
The total of the areas of the triangular base and faces is 662.55370...*
The height of the apex is 21.05451...*
The volume V of the pyramid is 650.54247...*
The base vertices (11, 17, 31), (17, 19, 23) and (11, 19, 29) each total 59.
The apex (23, 29, 31) totals 83.

Acknowledgements

Alan Grogono has written an improved version of my proof concerning the impossibility of having equal perimeters for the four faces of a tetrahedron (except if the triangles are fully congruent), and I wish to thank him for this contribution.

Walter Trump has kindly provided a list of "all Heronian non isosceles triangles up to perimeter p=10000" which should prove to be an invaluable tool when searching for Heronian solutions. In addition, he has simplified my initial proof as to why the totals of the edge lengths of each of the base vertices of a MTP are always equal, and he has discovered another interesting property of all Magic Triangular Pyramids (MTP): 

a - a' = b - b' = c - c'

My thanks also go to Walter Trump for his large contribution to the definition of the edge length notation schema for the Magic Triangular Pyramids, which is explained in the observations that follow.

Observations

In order to facilitate any correspondence and exchanges it will be useful to adopt the same way of writing the edge lengths of the Magic Triangular Pyramids. It is proposed that we always use the following schema:

In this case the edges ( a, b, c ) are base to face edges, and the edges ( a', b', c' ) are face to face edges. The letters ( a, a' ), ( b, b' ), ( c, c' ) represent pairs of opposite edges which do not share a same vertex. Without needing to illustrate the tetrahedron, the edge lengths of a Magic Triangular Pyramid can be expressed as ( a, b, c, a', b', c' ) where a < b < c. Taking for example, the edge lengths of some of the different cases of Magic Triangular Pyramids illustrated at the beginning, these should be conventionally expressed as follows:

The edges of the Primitive Magic Triangular Pyramid Solution A are (1, 2, 3, 4, 5, 6).
The edges of the Primitive Magic Triangular Pyramid Solution B are (1, 3, 5, 2, 4, 6).
The edges of the Primitive Magic Triangular Pyramid Solution C are (2, 4, 6, 1, 3, 5).
The edges of the Primitive Magic Triangular Pyramid Solution D are (4, 5, 6, 1, 2, 3).

The list of Magic Triangular Pyramids (MTP) found since the 27th August 2018, uses the conventional edge length expressions described above. It is intended to update this list from time to time to include the findings of others (should they agree to seeing their results published here). Any contributions will be much appreciated.

Further Developments

In February 2021 Steve Wait informed me of his "Harmonious Pythagorean Triangular Pyramid Conjecture" and asked if I had any ideas concerning a degenerate pyramid threshold that he had observed. As I am not particularly skilled at trigonometry I took the liberty of forwarding Steve's question to Walter Trump. Please refer to Steve Wait's site "The Cognitive Condition" for more information on his very interesting "Precursory Study of Pythagorean Theorem Generalization as Consequence of Three-Dimensions" and also for the details of Walter Trump's contribution to the understanding of the degenerate pyramid limits of the Pythagorean Pyramids.

As for the Magic Triangular Pyramids, there is plenty of scope for further research: For those who are programmers there may be some interesting sequences to find in the evolution of the consecutive numbers of the 4 primitive Magic Triangular Pyramids. Other aspects, such as for example palindromic number solutions, remain to be investigated.