# Tetrahedral Pyramid **URL:** \ **Category:** Factual Questions\ **Created:** [April 4, 2025, 12:08pm UTC](https://boards.straightdope.com/t/tetrahedral-pyramid/1016608 "2025-04-04T12:08:35Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![CalMeacham](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/calmeacham/32/35_2.png) [@CalMeacham](https://boards.straightdope.com/u/CalMeacham)\ **Post date:** [April 4, 2025, 12:08pm UTC](https://boards.straightdope.com/t/tetrahedral-pyramid/1016608/1 "2025-04-04T12:08:35Z")
There are pyramid-shaped buildings all over the world – most famously the Pyramids of Egypt (and most famously, the three at Giza, although there are plenty of others elsewhere). There are pyramids in Mexico (in Teotihuacan and Cholula and Chichen-Itza, among others), in China (Xi’an). There’s the modern pyramid at Memphis, Tennessee, echoing the one in Memphis, Egypt, and the Pyramid of Luxor in Las Vegas. And the Pyramid at the entrance to the Louvre. And lots and lots of smaller ones elsewhere (here’s a list of pyramidal mausoleums in the US: > **[List of pyramid mausoleums in North America](https://en.wikipedia.org/wiki/List_of_pyramid_mausoleums_in_North_America)** > > This is a list of pyramid mausoleums in North America. This Egyptian Revival funerary architecture was generally an extravagance of American tycoons who wanted themselves remembered as long and as well as the ancient pharaohs. Many of these date from the 1890s to the 1920s, when older, more modest expressions of "Egyptomania" gave way to "Egyptian temples or pyramids replete with guardian sphinxes, lotus-bedecked urns and columns, and Egyptian- themed stained glass windows." ) Every one of these, as far as I can tell, has a square base, making them sort of like a regular octahedron cut in half. The shape, it has been claimed, was inspired by natural eroded rock structures throughout the Middle East. But there’s no reason for such an eroded structure to have a square cross-section. Why do all human-built pyramids have square horizontal cross-sections, no matter where they’re built? In particular, why aren’t there any tetrahedral pyramids, with triangular horizontal cross-sections? You’d expect there to be some natural eroded tetrahedral-shaped structures. The Tetrahedron is one of the five Platonic Solids, and was certainly familiar to the ancient world (_Caltrops_ were common in the ancient world, and have tetrahedral symmetry). Why don’t we see tetrahedral pyramids? One discussion board on the internet suggested it was hard to build such a structure – “What would the bricks look like?” Well, you can build a pretty good approximation of a tetrahedral pyramid using square or rectangular blocks. But if you want the bricks to fit the shape, you can easily build a perfect tetrahedral pyramid out of courses of equilateral triangular bricks, or a combination of triangular and hexagonal bricks (each hexagon made up of six of the equilateral triangles), or out of trapezoidal bricks and triangular bricks (each trapezoid being made up of three equilateral triangles, or half a hexagon). Or a combination of all three.) Heck, if you insist on rectangular bricks, you can relax the requirement about sixty degree angles at the apex and create a three-sided pyramid with right angles at the apex made up of rectangular bricks – they just won’t be lying horizontally. A three-sided pyramid is what you get if you have one point of a cube pointing upwards. I’ve looked through the internet, and can find no tetrahedral structures or buildings. Tetrahedrons HAVE been used as the basis for construction. As the simplest three-dimensional form, it’s also the most rigid, as a triangle is the most rigid 2D form. Alexander Graham Bell made structures made out of linked tetrahedra and claimed they were sturdy and simple to produce – but none of them looked like a simple tetrahedron itself. The Tetrahedron Skyscraper in San Diego similarly exploits the structural rigidity of the tetrahedral form, but it doesn’t look like simple tetrahedron, either. > **[Tetrahedron Skyscraper in San Diego - eVolo | Architecture Magazine](https://www.evolo.us/tetrahedron-skyscraper-in-san-diego/)** > > Architecture and Design Magazine for the 21st Century. Organizer of the Annual Skyscraper Architectural Competition. Via 57 West in New York City claims to be a “tetrahedron”, but doesn’t look it. > **[VIA 57 West](https://en.wikipedia.org/wiki/VIA_57_West)** > > VIA 57 West (marketed as VIΛ 57WEST) is a residential building at 625 West 57th Street, between 11th and 12th Avenues, in the Hell's Kitchen neighborhood of Manhattan in New York City. The pyramid shaped tower block or "tetrahedron", designed by the Danish architecture firm Bjarke Ingels Group (BIG), rises 467 ft (142 m) and is 35-stories tall. > Bjarke Ingels met the New York developer Douglas Durst Chairman of The Durst Organization in the early 2000s when he was in Denmark. Durst, who visite... This paper suggests building tetrahedral structures on other planets because it’s so structurally stable > **[2483.pdf](https://www.hou.usra.edu/meetings/lpsc2014/pdf/2483.pdf)** > > 462.64 KB ..but if it’s so simple, so elegant, and so stable, why doesn’t anybody already do it here on earth? Wouldn’t a stable building be a shoo-in in earthquake zones? ---
**Author:** ![TriPolar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tripolar/32/3008_2.png) [@TriPolar](https://boards.straightdope.com/u/TriPolar)\ **Post date:** [April 4, 2025, 12:32pm UTC](https://boards.straightdope.com/t/tetrahedral-pyramid/1016608/2 "2025-04-04T12:32:47Z")
Is a tetrahedral pyramid made from stacked blocks or piled earth any stronger than a square based pyramid? ---
**Author:** ![engineer\_comp\_geek](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/engineer_comp_geek/32/504_2.png) [@engineer\_comp\_geek](https://boards.straightdope.com/u/engineer_comp_geek)\ **Post date:** [April 4, 2025, 12:38pm UTC](https://boards.straightdope.com/t/tetrahedral-pyramid/1016608/3 "2025-04-04T12:38:42Z")
I can’t answer the main question in the OP, but I did want to comment on this part: > [@CalMeacham](#): > > The shape, it has been claimed, was inspired by natural eroded rock structures throughout the Middle East. I don’t know about the inspiration for the shape, but if you want to build something really big, and you don’t have an understanding of arches yet, you’re going to get a pyramid. If you try to build a giant cube with ancient construction methods, it’s going to collapse. ---
**Author:** ![Cervaise](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cervaise/32/16693_2.png) [@Cervaise](https://boards.straightdope.com/u/Cervaise)\ **Post date:** [April 4, 2025, 12:52pm UTC](https://boards.straightdope.com/t/tetrahedral-pyramid/1016608/4 "2025-04-04T12:52:24Z")
Extremely vague memory from thirty-plus-year-ago anthropology class: Ancient peoples ascribed great importance to the orientation of their buildings against the compass. The Egyptian pyramids, for example, have been noted for their careful placement, being aligned almost perfectly with respect to north, south, east, and west. A tetrahedral structure, with three base corners, doesn’t allow this. You need to come up with some other basis for laying down the foundation. And no civilization I know has a compass with anything but four primary/cardinal directions. (Of course, based on this being an old vague memory, I welcome elaboration on this point.) Ergo, any civilization that values building alignment to the compass (which is many, if not most) will naturally choose a square-footed design. ---
**Author:** ![Francis\_Vaughan](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/francis_vaughan/32/3093_2.png) [@Francis\_Vaughan](https://boards.straightdope.com/u/Francis_Vaughan)\ **Post date:** [April 4, 2025, 12:55pm UTC](https://boards.straightdope.com/t/tetrahedral-pyramid/1016608/5 "2025-04-04T12:55:36Z")
One problem is the steepest angle of repose. Or its equivalent when trying to stack blocks of stone. There is good evidence that 45 degrees is about the steepest ancient builders were happy building. The [Bent Pyramid](https://en.m.wikipedia.org/wiki/Bent_Pyramid) was initially built at an angle of about 54 degrees (with even steeper possibly intended), but the angle was shallowed out part way up to 43 degrees. The current wisdom is that the builders were getting worried about stability. Possibly because its predecessor (the Meidum pyramid) likely failed. 43 degrees seems to be the preferred angle. That gets you pretty much all the later pyramids. Usefully 45 degrees gets you a pyramid as half a regular octahedron. So 45 isn’t far off. It isn’t just the pyramid structure itself. You need ramps to build it. And they need to be stable and viable as you go. A regular tetrahedron has sides of over 70 degrees. That isn’t going to be a happy construction project. Appealing to regularly of geometry is a bit thin, but not unreasonable. If you visit the Valley of the Kings, the hill above where all the tombs are dug has a very clear four sided pyramid shape. However entombing in the valley came long after the era of pyramids, and it is more likely the valley was chosen for the hill’s shape, rather than inspiring pyramids. There are the odd tetrahedral pyramids in modern architecture. But no doubt, mirroring the Egyptian pyramids is the common theme. The pyramids of the Louvre being a great example. Square bases are much easier to lay out. Architecture likes right angles. Triangles are a nuisance. ---
**Author:** ![We\_re\_wolves\_not\_werewolves](https://avatars.discourse-cdn.com/v4/letter/w/838e76/32.png) [@We\_re\_wolves\_not\_werewolves](https://boards.straightdope.com/u/We_re_wolves_not_werewolves)\ **Post date:** [April 4, 2025, 1:23pm UTC](https://boards.straightdope.com/t/tetrahedral-pyramid/1016608/6 "2025-04-04T13:23:04Z")
> [@Francis\_Vaughan](#): > > There is good evidence that 45 degrees is about the steepest ancient builders were happy building. The [Bent Pyramid](https://en.m.wikipedia.org/wiki/Bent_Pyramid) was initially built at an angle of about 54 degrees (with even steeper possibly intended), but the angle was shallowed out part way up to 43 degrees. The current wisdom is that the builders were getting worried about stability. Possibly because its predecessor (the Meidum pyramid) likely failed. 43 degrees seems to be the preferred angle. That gets you pretty much all the later pyramids. I think you may be getting face angles and edge angles of the pyramids mixed up. Specifically, according to this site: [The Slopes of the Egyptian Pyramids](https://sites.math.washington.edu//~greenber/pyressay.html) the bent pyramid initially had the _face angle_ you cite of 54 degrees, but the upper portion had the shallower 43 degrees. This corresponds to _edge angles_ of about 44 and 33 degrees, respectively. The pyramids at Giza seem to have angles that are more closely aligned to the LOWER part of the bent pyramid, with the pyramid of Khufu having a face angle of 51 degrees and an edge angle of 41 degrees. ---
**Author:** ![CalMeacham](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/calmeacham/32/35_2.png) [@CalMeacham](https://boards.straightdope.com/u/CalMeacham)\ **Post date:** [April 4, 2025, 1:28pm UTC](https://boards.straightdope.com/t/tetrahedral-pyramid/1016608/7 "2025-04-04T13:28:25Z")
> [@Francis\_Vaughan](#): > > There are the odd tetrahedral pyramids in modern architecture. Name one. I haven’t been able to find any. Regarding the tetrahedral sides, the point about the angle of repose is a good one, but I’d settle for any three-sided pyramid, whether it’s a true tetrahedron or not. Its not having four sides isn’t a problem – if you’re concerned about directions, just point one corner north and the opposite line running east-west. Easy enough to do if you find north by using the shadow of a stick marked off throughout the day. The shortest shadow gives you true North. As for laying it out, it was early on learned that you can use a compass to inscribe a hexagon in a circle. So make your circle, inscribe a hexagon with one point to True North. Then take every other point to get a perfect equilateral triangle. It wouldn’t have to be their first construction – it could come after centuries of building four-sided pyramids. My point is, no one seems to have done this at any era in history. ---
**Author:** ![Horatius](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/horatius/32/3572_2.png) [@Horatius](https://boards.straightdope.com/u/Horatius)\ **Post date:** [April 4, 2025, 1:55pm UTC](https://boards.straightdope.com/t/tetrahedral-pyramid/1016608/8 "2025-04-04T13:55:07Z")
> [@Francis\_Vaughan](#): > > Square bases are much easier to lay out. Architecture likes right angles. Triangles are a nuisance. This would be my answer. How many buildings do we build with triangular footprints? Not too many, because it makes using the building harder. People don’t like weird angles, and you often end up with at least some unusable space. There are exceptions where the piece of land has a weird shape, so you either build a triangle, waste some real estate, or don’t build, but those are exceptional cases. ---
**Author:** ![Pleonast](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pleonast/32/1183_2.png) [@Pleonast](https://boards.straightdope.com/u/Pleonast)\ **Post date:** [April 4, 2025, 2:34pm UTC](https://boards.straightdope.com/t/tetrahedral-pyramid/1016608/9 "2025-04-04T14:34:58Z")
> [@CalMeacham](#): > > Regarding the tetrahedral sides, the point about the angle of repose is a good one, but I’d settle for any three-sided pyramid, whether it’s a true tetrahedron or not. Note that any four-sided solid is a tetrahedron. I think what you intend to say is “regular tetrahedron”. ---
**Author:** ![Lumpy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lumpy/32/446_2.png) [@Lumpy](https://boards.straightdope.com/u/Lumpy)\ **Post date:** [April 4, 2025, 2:50pm UTC](https://boards.straightdope.com/t/tetrahedral-pyramid/1016608/10 "2025-04-04T14:50:57Z")
AFAIK, none of the classic Egyptian pyramids are geometrically precise half-octahedrons, so that wasn’t the design goal. A later pyramid tradition, that of Nubia, commonly featured pyramids much steeper than the classic Egyptian ones, so clearly the challenge of making a steeper structure got solved. Early drafts of the story that became “2001: A Space Odyssey” had the enigmatic alien structure be a tetrahedron, but Clarke decided he didn’t want people to ascribe any “Chariots of the Gods” associations with it. ---
**Author:** ![TriPolar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tripolar/32/3008_2.png) [@TriPolar](https://boards.straightdope.com/u/TriPolar)\ **Post date:** [April 4, 2025, 4:46pm UTC](https://boards.straightdope.com/t/tetrahedral-pyramid/1016608/11 "2025-04-04T16:46:34Z")
I don’t see any strength advantage to a non-rigid tetrahedral structure. Perhaps if interlocking blocks were used but I don’t know if that’s been done with anything of size. Cutting cubic blocks going to be easier and miscut blocks can be recut into more useful orthogonal sides, although the bulk of a tetrahedral pyramid would probably use mostly square cut blocks on the interior anyway. Miscut edge pieces are going to be less useful. And of course it’s very easy to measure a block with square and rectangular sides for cutting. > [@CalMeacham](#): > > My point is, no one seems to have done this at any era in history. This is more of a question. Something the size of the Great Pyramid would be easier to construct with a square base but the Egyptians had built many smaller pyramids before that massive undertaking. Why aren’t there tetrahedral pyramids found anywhere of smaller size though? ---
**Author:** ![Pleonast](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pleonast/32/1183_2.png) [@Pleonast](https://boards.straightdope.com/u/Pleonast)\ **Post date:** [April 4, 2025, 4:50pm UTC](https://boards.straightdope.com/t/tetrahedral-pyramid/1016608/12 "2025-04-04T16:50:55Z")
> [@CalMeacham](#): > > As for laying it out, it was early on learned that you can use a compass to inscribe a hexagon in a circle. So make your circle, inscribe a hexagon with one point to True North. Then take every other point to get a perfect equilateral triangle. Thinking about this more, it seems to me that building a pyramid with a hexagonal base, using hexagonal tiles, is the next simplest case after a pyramid with a rectangular base. Did any ancient cultures make use of hexagonal tiles? ---
**Author:** ![Lumpy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lumpy/32/446_2.png) [@Lumpy](https://boards.straightdope.com/u/Lumpy)\ **Post date:** [April 4, 2025, 5:07pm UTC](https://boards.straightdope.com/t/tetrahedral-pyramid/1016608/13 "2025-04-04T17:07:47Z")
> [@TriPolar](#): > > Why aren’t there tetrahedral pyramids found anywhere of smaller size though? If the Egyptians even knew of regular tetrahedrons, maybe they simply didn’t think they were particularly nifty or significant. ---
**Author:** ![TriPolar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tripolar/32/3008_2.png) [@TriPolar](https://boards.straightdope.com/u/TriPolar)\ **Post date:** [April 4, 2025, 5:20pm UTC](https://boards.straightdope.com/t/tetrahedral-pyramid/1016608/14 "2025-04-04T17:20:09Z")
That may be the case but plenty of other pyramids were built around the world. I do think a square base pyramid would be more attractive to the cultures who created them because the cardinal directions were significant in dividing up space. What explanation would be offered for the purpose of other polygonal shapes? ---
**Author:** ![MikeS](https://avatars.discourse-cdn.com/v4/letter/m/919ad9/32.png) [@MikeS](https://boards.straightdope.com/u/MikeS)\ **Post date:** [April 4, 2025, 6:52pm UTC](https://boards.straightdope.com/t/tetrahedral-pyramid/1016608/15 "2025-04-04T18:52:40Z")
> [@CalMeacham](#): > > One discussion board on the internet suggested it was hard to build such a structure – “What would the bricks look like?” Well, you can build a pretty good approximation of a tetrahedral pyramid using square or rectangular blocks. But if you want the bricks to fit the shape, you can easily build a perfect tetrahedral pyramid out of courses of equilateral triangular bricks, or a combination of triangular and hexagonal bricks (each hexagon made up of six of the equilateral triangles), or out of trapezoidal bricks and triangular bricks (each trapezoid being made up of three equilateral triangles, or half a hexagon). Or a combination of all three.) The ancient Egyptians knew that a triangle with side lengths of 3, 4, and 5 had a right angle, so they could use that knowledge to make cuts at 90° to each other with a high degree of precision. I don’t think that they had the same level of knowledge they’d need to construct the 60° or 120° angles needed for triangular or hexagonal blocks to the same degree of precision. > [@](#): > > Heck, if you insist on rectangular bricks, you can relax the requirement about sixty degree angles at the apex and create a three-sided pyramid with right angles at the apex made up of rectangular bricks – they just won’t be lying horizontally. A three-sided pyramid is what you get if you have one point of a cube pointing upwards. This sort of arrangement would strike me as much less stable than a set of courses laid horizontally, one on top of the other. If [Wikipedia is to be believed,](https://en.wikipedia.org/wiki/Construction_of_the_Egyptian_pyramids#Third_through_Fifth_dynasties) early Egyptian pyramids were built with courses that sloped inwards; but that sort of arrangement was found to be less stable than simply stacking the courses atop one another. And you’d have to very carefully cut the stones for the base of the pyramid at weird angles if you wanted to use the stacking method you propose; it’s almost certainly easier to just level the area for the base and then lay horizontal courses of rectangular blocks on top of that. None of this explains why more modern pyramidal structures couldn’t be triangular (or pentagonal or nonagonal or whatever), of course. That may just be more of a function of inertia and a cultural construction of “this is what a pyramid is supposed to look like.” ---
**Author:** ![Pleonast](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pleonast/32/1183_2.png) [@Pleonast](https://boards.straightdope.com/u/Pleonast)\ **Post date:** [April 4, 2025, 6:59pm UTC](https://boards.straightdope.com/t/tetrahedral-pyramid/1016608/16 "2025-04-04T18:59:45Z")
> [@MikeS](#): > > I don’t think that they had the same level of knowledge they’d need to construct the 60° or 120° angles needed for triangular or hexagonal blocks to the same degree of precision. Angles of 60° or 120° are easy with the method: > [@CalMeacham](#): > > As for laying it out, it was early on learned that you can use a compass to inscribe a hexagon in a circle. Use a compass to make a circle. Without changing its length, mark off six arcs along the circle. ---
**Author:** ![MikeS](https://avatars.discourse-cdn.com/v4/letter/m/919ad9/32.png) [@MikeS](https://boards.straightdope.com/u/MikeS)\ **Post date:** [April 4, 2025, 7:40pm UTC](https://boards.straightdope.com/t/tetrahedral-pyramid/1016608/17 "2025-04-04T19:40:10Z")
You know that and I know that, but did the ancient Egyptians necessarily know that? It seems obvious to us now, but it had to be discovered at some point, and the question is whether that point happened early enough on for the Egyptians to use it. I’ll concede that they might have known that fact, though; it’d be interesting to ask a historian this question. ---
**Author:** ![Pleonast](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pleonast/32/1183_2.png) [@Pleonast](https://boards.straightdope.com/u/Pleonast)\ **Post date:** [April 4, 2025, 9:20pm UTC](https://boards.straightdope.com/t/tetrahedral-pyramid/1016608/18 "2025-04-04T21:20:04Z")
I think the Egyptian cultures are well enough documented that we’d know if they use ever use triangles or hexagons in their constructions. I think the wider question is if any ancient cultures used triangles or hexagons. It’s quite possible, since the geometry is straightforward. It is those hypothetical cultures that would’ve built tetrahedral pyramids. ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [April 4, 2025, 9:22pm UTC](https://boards.straightdope.com/t/tetrahedral-pyramid/1016608/19 "2025-04-04T21:22:22Z")
> [@Pleonast](#): > > Use a compass to make a circle. Without changing its length, mark off six arcs along the circle. It’s easier than that. Just take 3 sticks of equal lengths and make a triangle. Or lengths of rope and stretch out into a triangle. ---
**Author:** ![Pleonast](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pleonast/32/1183_2.png) [@Pleonast](https://boards.straightdope.com/u/Pleonast)\ **Post date:** [April 4, 2025, 9:24pm UTC](https://boards.straightdope.com/t/tetrahedral-pyramid/1016608/20 "2025-04-04T21:24:22Z")
Right, that covers a regular triangle. But not a regular hexagon. [Next page](https://boards.straightdope.com/t/tetrahedral-pyramid/1016608.md?page=2)