Eulers Polyhedra Math Magic

Super fact 64 : V – E + F = 2 is true for all convex polyhedra, where V is the number of vertices (corners), E is the number of edges, and F is the number of faces of a convex polyhedron. This is called Euler’s formula for polyhedra.

Stick drawings of a tetrahedron on the left and a cube on the right | Eulers Polyhedra Math Magic | Euler's Polyhedron Formula
Polyhedron Euler’s formula. The cube has 6 faces, 8 corners (blue dots), and 12 edges (grey lines) and 8 – 12 + 6 = 2. Popular shapes Asset id: 2362684465 by ramonparaiba.

Convex polyhedra are polyhedra without holes, or it has no internal angles larger than 180 degrees. To express that exactly; A convex polyhedron is a three-dimensional solid with flat polygonal faces where a line segment connecting any two points on its surface or interior remains entirely within the solid. I can add that convex polyhedra consists of convex polygons as their faces. All its faces are convex polygons.

On the left are three convex polygons and on the left are three concave polygons.
The difference between convex and concave polygons. Notice that the concave polygons on the right have “holes” in them, or inner angles larger than 180 degrees, and that you can draw a line starting and ending inside the polygon that goes outside the polygon.

It is easy to understand that there are an infinite number of convex polyhedra. All pyramids (with convex polygons for bottoms) are convex polyhedra whether the bottom is a square/rectangle, or a convex pentagon, hexagon, heptagon, octagon, or any kind of convex polygon, even with trillions.

A hexagonal pyramid with differently colored faces | Eulers Polyhedra Math Magic | Euler's Polyhedron Formula
A hexagonal pyramid (it is convex). Tomruen at en.wikipedia, Public domain, via Wikimedia Commons.

Examples of convex polyhedra are cubes, boxes, a tetrahedron, pyramids of various sorts, an octahedron, dodecahedron, icosahedron, but there are infinitely many.

As an example, take a cube (like a dice). It has six faces (F = 6), eight vertices (corners V = 8), and 12 edges (V – E + F = 2 or 8 – 12 + 6 = 2). A tetrahedron has four faces (F = 4), four vertices (V =4), and six edges (E = 6) (V – E + F = 2 or 4 – 6 + 4 = 2). A pyramid (with a square bottom) has five faces (F = 5), five vertices (V = 5) and eight edges (E = 8) (V – E + F = 2 or 5 – 8 + 5 = 2). An octahedron has eight faces (F = 8), and six vertices (V = 6), and twelve edges (E = 12) (V – E + F = 2 or 6 – 12 + 8 = 2). A dodecahedron has twelve faces (F = 12), and twenty vertices (V = 20), and thirty edges (E = 30) (V – E + F = 2 or 20 – 30 + 12 = 2). An icosahedron has twenty faces (F = 20), and twelve vertices (V = 20), and thirty edges (E = 30) (V – E + F = 2 or 12 – 30 + 20 = 2). Euler’s Polyhedral Formula works for all of them and for all the infinite number of convex polyhedra that exists.

Line of colorful Platonic solids on a grey surface (polyhedra - tetrahedron, cube, octahedron, dodecahedron and icosahedron, 3d render, banner.
Tetrahedron, cube, octahedron, dodecahedron and icosahedron, Shutterstock Asset id: 2138211027 by Dotted Yeti

I consider this a super fact because it is quite amazing and surprising that the number of faces, vertices, and edges follow the same formula for all convex polyhedra. At least it is surprising if you haven’t seen it before. I should say that Leonhard Euler’s most celebrated discovery is probably his discovery of the Euler Number and the associated Euler’s Formula.

More Polyhedra Examples

The three first examples below are convex polyhedra so Euler’s Polyhedral Formula apply. The last two examples are not so Euler’s Polyhedral Formula does not apply.

Blue truncated icosahedrons. Geometric soccer ball or football shape. Archimedean solid. Regular polygon outline with pentagonal and hexagonal faces. Blue gradient polygonal figure | Eulers Polyhedra Math Magic | Euler's Polyhedron Formula
This is a truncated icosahedron, which is a convex polyhedra, so Euler’s Polyhedral Formula apply. Shutterstock Asset id: 2309812551 by Mima Subota.
A green Rhombicuboctahedron with 8 triangles and 18 squares as faces.
This Rhombicuboctahedron is an example of convex polyhedron, so Euler’s Polyhedral Formula applies. Cyp, CC BY 4.0 <https://creativecommons.org/licenses/by/4.0&gt;, via Wikimedia Commons.
Icosidodecahedron with faces that are triangles and pentagons.
This Icosidodecahedron is another example of a convex polyhedron, so Euler’s Polyhedral Formula applies. Attribution must be given to Robert Webb’s Stella software as the creator of this image along with a link to the website: http://www.software3d.com/Stella.php. A complimentary copy of any book or poster using images from the Software would also be appreciated where feasible., Attribution, via Wikimedia Commons
This polyhedron is donut shaped | Eulers Polyhedra Math Magic | Euler's Polyhedron Formula
This hexagonal torus / toroidal polyhedron is not a convex polyhedra, so Euler’s Polyhedral Formula does not apply in this case. Tom Ruen, SVG version from de:User:Antonsusi, Public domain, via Wikimedia Commons
3d polyhedron Merkaba, esoteric bronze crystal, sacral geometry shape, volume david star, mesh form, abstract vector object
This Merkaba Polyhedron is not a convex polyhedra, so Euler’s Polyhedral Formula does not apply in this case. Shutterstock Asset id: 1296063079 by Panimoni
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Author: thomasstigwikman

My name is Thomas Wikman. I am a software/robotics engineer with a background in physics. I am currently retired. I took early retirement. I am a dog lover, and especially a Leonberger lover, a home brewer, craft beer enthusiast, I’m learning French, and I am an avid reader. I live in Dallas, Texas, but I am originally from Sweden. I am married to Claudia, and we have three children. I have two blogs. The first feature the crazy adventures of our Leonberger Le Bronco von der Löwenhöhle as well as information on Leonbergers. The second blog, superfactful, feature information and facts I think are very interesting. With this blog I would like to create a list of facts that are accepted as true among the experts of the field and yet disputed amongst the public or highly surprising. These facts are special and in lieu of a better word I call them super-facts.

40 thoughts on “Eulers Polyhedra Math Magic”

  7. I remember the old bouncing ball animation they used to show off the Amiga’s graphics capabilities back in the 80s. That was one of these. I don’t remember how many sides it had, but we were all pretty impressed that you could do that with a computer, back in the day.

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    1. Yes I remember something similar. Perhaps not on an Amiga but other computers. In the 1990’s, a bit later, I used to create graphics for simulating industrial robots. I used vrml created from CAD drawings and then I attached robot control systems to the graphics, and kinematics between the simulated motors and joints/links. It was an early factory floor simulation program and I won an award for it.

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  9. “Wow! 🌟 Euler’s formula is really fascinating. It’s amazing how V – E + F = 2 holds true for all convex polyhedra, no matter how simple or complex. I also liked the examples of different polyhedra and how you showed which ones it applies to and which ones it doesn’t. The visuals make it super easy to understand. Thanks for sharing this super fact! ❤️”

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    1. Yes I agree that is fascinating, and today I discovered it exists for multiple dimensions as well. It wasn’t Euler who invented those extensions, that is modern math, but it makes it even more fascinating. It turns out that the right hand side is 2 for all odd dimensions and 0 for all even dimensions.

      // 3D
      V – E + F = 2  (V = vertices/corners, E = edges, F = faces)

      // 4D
      V – E + F – C = 0 (V = vertices/corners, E = edges, F = faces, C = cells (3D polyhedra))

      // 5D
      V – E + F – C + N5= 2 (V = vertices/corners, E = edges, F = faces, C = cells, N5 = hyper-cells)

      // 6D
      V – E + F – C + N5 + N6= 0 (V = vertices/corners, E = edges, F = faces, C = cells , N5 = hyper-cells, N6 = five dimensional cells)

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          1. Oh my, thank you for the gentle correction, Thomas! 😅 My apologies for the mix-up—and I truly appreciate your kindness and understanding. Names matter, and I’ll be sure to remember yours with gratitude. 🌿

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  11. Leonhard Euler introduced this formula in the 18th century, and it’s considered one of the earliest results in topological mathematics. His insight laid the groundwork for modern geometry and even influenced fields like computer graphics and architecture.

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