The Magic of Euler’s Equation: V-E+F=2. An Eye Opener.
If you have not yet studied topology before, you are in for some very nice surprises when you read David S. Richeson’s book “Euler’s Gem, The Polyhedron Formula and the Birth of Topology” . These revelations come not only in the shape of pretty pictures, but you’ll also get deeper insights and see new connections between mathematical disciplines.
You can of course pick up the book yourself if you want to avoid me spoiling your surprises. If you prefer to ride the fast lane, here follows my personal account of what I discovered while reading the book.
1 Regular Polyhedra or Platonic Solids
1.1 What are Regular Polyhedra?
From reading some popular maths books back in the mid 1980s, I was happy to discover that there was something called the Platonic solids. Their definition can be simplified to that they are 3 dimensional structures, consisting of faces that are all identical regular polygons of the same type and size such that all the vertices lie on a perfect sphere.
The most well known is the cube that consists of 6 equally sized squares. The tetraeder is likely the second-most well known. It consists of 4 regular triangles. The octaeder consists of two equal pyramids with a square base oppositely attached to each other at their bases. It has 8 regular triangular faces.
In today’s public knowledge there are two lesser known solids that are also platonic; the dodecahedron consisting of 12 regular pentagonal faces and the icosahedron consisting of 20 regular triangles for faces.
Wolfram has made some very nice visualisations of these solids via their software package Mathematica. They look like this.
Richeson’s account of their chronological discovery is that according to a reading of Euclid’s Elements, only the cube, tetrahedron and dodecahedron were known to the Pythagoreans (Pythagoras (c. 560-480 BC))
One tale suggest that Hippasus of Metapontum (c. 530 -c. 450 BC), while being a member of the Pythagorean club, discovered the dodecahedron but failed to give credit to Pythagoras. Hippasus also discovered that there are non-rational numbers by studying the length of a diagonal of a unit square (root of 2) or/and the length of an edge of the pentagram star (root of 5 plus 1 over 2, also called the golden ratio) that is inscribed in a regular pentagon with sides of length one. The Pythagoreans, on the contrary, discovered that many numbers occurring in nature were ratios of integer numbers. For example in music, halving or dividing by 3, the length of a string resulting in two tones that went well together. So they attributed some magic to rational numbers and liked to believe that all numbers were rational. The tale continuous saying that both of Hippasus’ discoveries upset Pythagoras so much so that Hippasus got banned from the club. Ironically, the proof of the existence of irrational numbers is one of the most significant and lasting contributions of the Pythagoreans to mathematics.
Richeson mentions that the octahedron and icosahedron were only later discovered by Theaetetus of Athens (c. 417–369 BC).
It surprised me that the octahedron, which seems much more easy to construct with sticks or even imagine in ones head than the dodecahedron would not have been discovered before it.
Thinking about that, I wondered if maybe the Greeks would have thought: “Well we have one with triangles already, so let’s focus on polyhedra with other types of faces, like squares and pentagons.” Likely, right?
Or maybe there was still an utilitarian bias in that a dodecahedron was a better primitive football than the way too pointy octahedron. Hmm, those long-robe-wearing thinkers were probably not so much into football, so that seems less likely, right?
Still, it got me thinking …
1.2 How Did the Old Greeks Discover the Regular Polyhedra?
How would you have done it? In your head?
I think it’s likely that they may have taken a practical approach by constructing some regular triangles by collecting or chopping some equal length wooden sticks or as I did with toothpicks. Would toothpicks have existed? They would then have fitted the corners of 3 triangles together in one point and then bend the faces so that one edge of the first coincides with one edge the third. That directly ends up in a tetraeder. Imagine the wonder the first person ever to do that may have felt when the fourth triangle appearing at the bottom appeared to have exactly the same shape as the three others.
When taking 4 triangles together one ends up with a pyramid with a square base. It should have been easy from there to double it and construct the octaeder, right? Or were they at first not convinced that the vertices all lie on a sphere? Maybe because they had the tetraeder already?
When taking 5 triangles, one ends up with a ‘pyramid with a pentagonal base. Note that one cannot just mirror this pyramid to end up with a 10 triangle ‘platonic solid’, since that would not satisfy the requirement that the vertices all lie on a sphere. So one has to realise that an extra row of 10 triangles (in up- down-up-down…-configuration) has to be put in between those two pentagonal pyramids to make up an icosahedron.
By fitting 6 regular triangles together, since all triangles have 60 degree angles, we need not bend them for an edge of the first to coincide with the edge of the last, since that already happens when they are all flat on the table in 1 plane. So no regular solid can be made out of it.
With 3 squares we get the cube and with 3 pentagons we get the dodecahedron.
There are no other regular solids constructable with squares nor pentagons since 4 of each type put together already make 4*90=360 degrees respectively 4*108=432 degrees which does not satisfy the ≤360 degrees requirement to make a chance on obtaining a solid fitting in a sphere.
1.3 Are there only 5 Regular Polyhedra?
There is the question if there are more regular polyhedra than these five, and in fact there are not. Did the old Greeks know it?
We can easily prove that there cannot be Platonic solids constructed out of regular n-gons with n≥6, since the internal angle of a hexagon is 120 degrees (divide the hexagon into 6 triangles to see that) and when you shift 3 hexagons together they will fill the plane without gaps like in a beehive, meaning that you cannot build a sphere-like structure with it.
Regular n-gons have internal angles of each (n-2)*180/n. So for larger n, also the internal angles get larger, so surely for n>6, there cannot exist regular polyhedra constructed with such n-gons.
Also for n-gons with 3≤n≤5, from the above practical construction procedure one can get convinced that there are no polyhedra, other than the 5 pictured above.
Euclid (435 BC.-365 BC) in his final book XIII of ‘The Elements’ gives geometric proof along the lines of the above argument of the conjecture that there are only 5 regular polyhedra. In fact this proof was already known by Theaetetus (c. 417–369 BC). We know this from the books Sophist and Theaetetus written by Plato (427–347 BC).
Since topology did simply not exist until some time after Euler (1707–1783) came up with his formula, we have to wait about 2000 years longer for a topological proof of the same conjecture.
Topologists say that for example in a cube, in each vertex, the 3*90 degrees angles leads to an angle deficit of 360-(3*90)=90 degrees and in general with 3 n-gons, there is an angle deficit of 360-(3*(n-2)*180/n). For n = 3,4,5,6 and 7 this gives respectively: 180, 90, 36, 0 and -25.7 degrees. The angle excess is defined as minus the angle deficit. One needs an angle deficit larger than 0 to have any chance at constructing a polyhedron.
1.3 Mysticism: Some Saw a Little Too Much in Regular Polyhedra
1.3.1 The Primal Elements: Earth, Air, Fire and Water
Plato learned about the 5 regular polyhedra from Theaetetus and, like some other figures much later, found that there was some cosmic beauty to it. Plato was also familiar with the writings from Empedocles (c. 492–432 BC) stating that all matter was created from four primal elements: earth, air fire and water.
Nowadays that view seems naive. But imagine you lived back then. Fewer man made things existed. There was no plastic, bubble gum or nuclear plants. When you looked around, the man made things that existed were simpler to guess were they came from. Earthwork can be seen as coming from earth and water and then baking it in an oven with fire. Wine comes from grapes, which grows on grapevines that grow in earth and ‘inhale’ air. Clothes consisted of animal or plant derived material, which in turn come from plants which grow in earth and inhale air. So in fact the view could be considered correct in the sense of tracking back things to their four primitive origins. Of course they do not literally consist of these 4 elements anymore.
In Timaeus Plato wrote up a fictional account of a discussion between Socrates, Hermocrates, Critias and Timaeus, where he explains his mapping as follows.
“To earth let us give the cube because of the four kinds of bodies earth is the most immobile and the most pliable”. With arguments of ‘stability or pliability of faces” (which makes little sense to me) he then assigns to fire the tetrahedron, to air the octahedron and to water the icosahedron. As such, there is no element left for the dodecahedron. Timaeus, in Plato’s story, then says “this one the god used for the whole universe, embroidering figures on it”.
In so doing, one can of course make a mapping from any list to any other list even if the number of items in both lists differ. :)
Later, Aristotle adopted and expanded the belief with a 5th element, called the aether or the quintessence. The term aether has been used by medieval alchemists for a medium similar or identical to that thought to make up the heavenly bodies (stars and planets). Later able scientists like Christiaan Huygens (1629–1695) also felt the need for a substance to exists, so that light could move through a vacuum. Huygens’ theory was replaced by subsequent theories proposed by Maxwell (1831–1879), Einstein (1879–1955) and de Broglie (1892–1987), which rejected the existence and necessity of aether to explain the various optical phenomena.
This belief in the existence of an aether to make up for phenomenons one understands yet may seem silly now, but will we not think similarly of dark matter and dark energy in the future when we get more insight into those? After all aren’t they just placeholder-names for what we do not comprehend fully yet today? As an extension of this thought, is/was god such a placeholder name too?
1.3.2 Johannes Kepler and his ‘Cosmic Mystery’
The astronomer Johannes Kepler (1571–1630), very capably, came up with his three laws of planetary motion that accurately described the trajectories of the planets.
However, he also obsessed about the Platonic solids and attributed more astrological meaning to them than they had (none). He assigned to each of the 5 then-known planets (Saturn, Jupiter, Mars, Venus and Mercury, so excluding planet Earth) one of the platonic solids. A picture of this pseudo-relation is shown here. Both ‘revelations’ were written up in the same book called ‘Cosmic Mystery’.
1.3.3 Mystical Mappings
So the old Greeks tried to map 4 concepts to 5 and declare it somehow as ‘natural’ and Kepler tried to defend a 6 to 5 mapping. Nowadays we programmers would call this a ‘one off error’. It seems that even the greatest minds sometimes read more into some construction or discovery than there is to it.
Yet, as we will see later, sometimes it’s the opposite, that they do not dig into it enough, and that they leave some aspect undiscovered, unknowingly, sometimes for centuries.
2 Semiregular Polyhedra or Archimedean Solids
Richeson does not dive deeply into it but Archimedes of Syracuse (c. 287-212 BC) discovered that there are a limited set of semi regular polyhedra as well;13 to be exact. They consist of more than one type of regular polygons as faces. Their exact definition is a bit involved but given here. Wolfram Mathworld gives a nice visualisation.
Johannes Kepler had completed the rediscovery of the 13 polyhedra in 1620.
Up to this day, the truncated icosahedron is better known as the structure used to knit a leather football from. This is likely because the regular polyhedra are not spherical enough in shape yet and because of the sem-regular ones it is the one where the faces are all of approximately the same size, also giving it shape that deviates the least in different places from the spherical shape. This should make its trajectory more predictable when you kick it.
To show how advanced Archimedes’ understanding of geometry already was we quote Wikipedia. “Archimedes is considered to be the greatest mathematician of ancient history, and one of the greatest of all time. He anticipated modern calculus and analysis by applying concepts of infinitesimals and the method of exhaustion to derive and rigorously prove a range of geometrical theorems, including: the area of a circle; the surface area and volume of a sphere; area of an ellipse; the area under a parabola; the volume of a segment of a paraboloid of revolution; the volume of a segment of a hyperboloid of revolution; and the area of a spiral.”
Yet, he missed to remark or publish some simple rule about the Platonic solids, which in fact even applies to ‘his own’ Archimedean solids.
3 Euler’s Formula
Richeson’s tale is a chronological one in the order of events as mathematical discoveries were becoming public knowledge to the world by publication time. In this sense, Euler’s made the discovery first.
3.1 Euler’s Formulation
When one tabulates the number of vertices V , edges E and faces F for each of the Platonic solids we get this table.
The old Greeks as well as Kepler noticed that some polyhedra are the dual of another in the sense that the number of vertices of the first equals the number of vertices of the second and vice versa. The octahedron is the dual of the cube and the icosahedron is the dual of the dodecahedron. The tetrahedron is the dual of itself. Thanks to that, the vertices of a regular polyhedron fit nicely into the centre of the faces of its dual regular polyhedron.
Richeson’s book starts with the sentence “They all missed it.”, meaning that when Euler discovered a relation between the number of vertices, edges and faces and wrote it up in 2 papers in 1750 and 1751, but only published in 1758, nobody else had noticed or published this before. Maybe some realised it but thought it was an irrelevant or accidental curiosity and judged it as no material not worthy for publication.
But indeed, see for yourself, in the table above one can clearly see that V-E+F=2 for each of the 5 solids. Is it meaningful? To know that, we need to see for which other classes of polyhdra it also holds, if any and where it does not hold anymore.
It turns out that the Euler equality also holds for the Archimedean solids. One can easily check this for each of them from the table mentioning the numbers of V, E and F for each of them here.
3.2 Descartes Formulation
Later, in 1860 evidence emerged that René Descartes (1596–1650), in 1630, so preceding Euler by 120 years, already discovered a relationship which he wrote down as P=2F+2V-4. As for Euler later, he called F faces and V solid angles instead of vertices. However, Descartes counted P as plane angles where Euler counted E for edges. Plane angles are the interior angles of all the faces, so there are F * n of them for polyhedra consisting of faces that are n-gons. For example, in a cube there are 6*4=24 of them. 2F+2V-4= 2*6+2*8–4 is 24 as well indeed.
Because in any polyhedron, it is a general truth that an edge connects two face angles, it follows that P=2E. So Descartes formula is equivalent to 2E=2F+2V-4 or to V-E+F=2 which is Euler’s formula. Because of that some argue that this equation should be called Descartes formula or the Descartes-Euler formula.
Some mathematicians claim that if the notion edge would have existed at Decarte’s time, he would have formulated it the Euler way. It is Euler who came up with the notion edge.
I find it interesting to note that over the course of history, features that have in some dimensions, a size equal to 0 tend to take more time in history to receive a proper name. An edge, apart from the dimension along the line it is coincident with is in other dimensions of size 0. A vertex, has size 0 in all dimensions and used to be called solid angle. Similarly, the number 0 was only introduced into the decimal system in use in Europe by Fibonacci (c. 1170–1250). He used the term zephyrum. Judging from the given names solid angle and plane angle it seems that the lower dimensional features edge and vertex were mainly seen as resulting from intersections of two and more faces respectively.
This leads us to the realisation that the formula V-E+F=2 relates zero-, one- and two-dimensional properties of polyhedra, in that order. Later generalisations will show that this is quite fundamental for this relation to hold.
4 Categories for which Euler’s Equation holds and Proofs
4.3 Euler’s Proof for Convex Polyhedra
A year after his publication of the discovery that his formula applies on Platonic solids, so in 1759, Euler gave proof of the fact that V-E+F=2 is applicable to general polyhedra. The proof starts with a general (convex) polyhedron and repeatedly removes vertex at a time, keeping track of V, E, F across different cases and ending up with a triangular pyramid for which V-E+F=2 holds. He then correctly concludes that V-E+F=2 holds for the original polyhedron.
However (1) he made the only implicit assumption that polyhedra are convex and (2) in his method of reducing polyhedra by subsequent removal of vertices, he ignores some possible cases of ending up with degenerate polyhedra (non-convex ones, disconnected polyhedra, even a combo of two polyhedra joined by an edge or vertex — so non-polyhedra). This proof has later been completed for these cases, but more simple proofs have been given later as well.
4.4 Legendre’s Proof for Convex Polyhedra
Adrien-Marie Legendre (1752-1833) proved that the Euler equation holds for convex polyhedra, starting from an entirely different angle: by calculating the sum of the areas of all faces of the polyhedra as projected on a unit sphere contained in/around the polyhedron on equating it to 4 Pi, which was already known from the old Greeks.
The English astronomer, mathematician Thomas Harriot (c. 1560–1621) and the French mathematician Albert Girard (1595–1632) had proven two useful formula for the area of a geodesic triangle and a general (even non-convex) geodesic polygon on a sphere. In both cases the generalised formula is A = a1+a2+..+an — (2-n)Pi, where A is the area, the a1, a2, ..., an are the polygon interior angles in radians and n is the number of vertices of the polygon.
So Legendre ‘simply’ noticed that per n-edged face of a polyhedron, we have a contribution of area when projected from a center point, onto the unit sphere with the same center point of a1+a2+..+an -(2-n)Pi. One can visualise this by assigning a contribution of Pi to each of the face edges and one 2 Pi contribution to the face itself. The sum of all geodesic areas for all faces should be 4 Pi r*r with r = 1 so 4 Pi. From the visualisation, this sum clearly is 2 F Pi - 2 E Pi + 2 V Pi. The -2 E Pi results from noting that each edge occurs in two faces. The 2 V Pi results from the fact that we know that at each vertex all geodesic angles add up to 2 Pi on the geodesic sphere. So 2 F Pi-2 E Pi+2 V Pi=4 Pi or after division by 2 pi, this leads to V-E+F=2, which proves Euler’s formula for convex polyhedra. How short, smart and beautiful this proof, isn’t it?
Does the proof rely on the convexity of the polyhedron? Not explicitly, but in fact, the idea of the projection on the sphere from some not further specified central point to result into non overlapping faces on the sphere, some condition must hold on the polyhedron with start with. Clearly this will not work for all polyhedra. For convex ones, such a point can always be found. For example the central point of the polyhedron always works. Are there non-convex polyhedra for which Legendre’s method of proof will hold, and so Euler’s relation will also hold?
4.5 Poinsot’s Discovery of Star-Convex Polyhedra
Must the polyhedra be convex to satisfy the Euler equality? In fact it turns out that no, this is not a necessary requirement since some non-convex polyhedra like two of the four Kepler-Poinsot’s polyhedra also satisfy the Euler equality. There are four Kepler-Poinsot polyhedra, and they look like this.
The list of V, E, F number for these polyhedra in the same order is as follows.
From this table, it is clear that the second and third polyhedron satisfy V-E+F=2 while the first and last do not. What is so different between the two categories? Can you see it?
It turns out that for the first and the last, no point can be found from which to project the polyhedron faces on a sphere such that no two of them overlap each other on the sphere.
4.6 Cauchy’s Proof for Flattened/Projected Convex Polyhedra
By using the idea of the projection of convex Polyhedra, Cauchy attempted to prove the Euler formula directly on the 2D picture of the projection obtained. He used a method of subsequent removal of first triangulating each face in the 2D picture and then, removing one vertex (and the connected edges and enclosed face) at a time. By keeping track of some different cases, he ends up with a triangle for which V-E+F=2 still holds and can then reason that this must have been the case for the original 2D picture as well. (Note that the exterior face around the triangle counts for a face as well.) This method can be seen as the 2D version of the 3D method of proof from Euler.
4.7 Hamilton’s Insight that V-E+F=2 holds for all Planar Graphs
William Rowan Hamilton (1805–1865) noticed, while Cauchy himself did not, that Cauchy’s proof was in fact applicable to any graph also with non-straight edges as long as it does not have crossing edges.
Some graphs have crossing edges, but can be redrawn in a way without crossing edges, just by lengthening, shortening or bending edges, and displacing vertices, but still taking care that vertices and edges are not disconnected and edges are not broken up nor removed. We call such a graph a planar graph. So one must in fact read planar graph a as graph that is planarisable without crossing edges.
Examples of planar graphs are the ones for the Platonic solids.
Note that if you want to count faces of these graphs, you also have to count the single ‘outer face’ that surrounds the graph as one. You will then see that the Euler formula still holds on these planar graphs.
Again, note that even though a graph is planar, one can still draw the same graph in a way that some edges do cross. For example here are some equivalent graphs of the graph for the dodecahedron where the first two demonstrate planarity, but the next 10 do not, even though they are equivalent graphs.
Also the Archimedean polyhedra, necessarily because they satisfy the Euler formula, have planar graph representations, for example as generated by Mathematica below.
So since Euler’s relation has been proved to hold for convex polyhedra, we know that all convex polyhedra (and some more, like the 2 of the Kepler-Poinsot polyhedra satisfying the Euler formula) are represented in 2D by a planar graph.
5 The Connection to Graph Theory
Graph theory has become a separate discipline within mathematics and computer science.
5.1 Euler Walks on Graphs
Euler defined a walk as a tracing of a graph starting at one vertex, following edges and ending at another vertex. A walk that has the same begin and end vertex is called a circuit. A walk that visits every edge just one is called an Euler walk. When beginning and ending vertex of the walk are the same we call the walk an Euler circuit. The degree of a vertex is the number of incident edges in that vertex.
Euler proved that, a graph has an Euler walk precisely when the graph is connected and there are zero or two vertices of odd degree. When there are two, the walk starts at a vertex with odd degree, otherwise the walk may begin anywhere.
5.2 Hamilton Walks on Graphs
Apart from Euler walks on graphs, there are also Hamilton walks, where all vertices rather than all edges should be visited.
Applied on the Platonic Solids this gives the following Hamilton cycles in 3D or 2D as pictured by Wolfram.
‘The’ necessary and sufficient condition for a graph to possess a Hamilton cycle is not as simple as the condition for a graph to have a Euler cycle. As an example, a sufficient condition for a simple graph with n vertices to have a Hamilton cycle is is that n≥3, and d(v1)+d(v2)≥n whenever v1 and v2 are not adjacent, where d(v1), d(v2) are the degrees of vertex v1 and v2.
So the connection of Euler’s formula to graph theory is that the formula holds for planar graphs, even though Euler did not know that when he discovered his formula holding for some classes of polyhedra and even when he proved it for some (implicitly assumed) convex polyhedra a year later.
5.3 Applications
Algorithms on computers to efficiently find walks that visit every vertex or edge in large graphs, with or without extra conditions, are still an important research area in computer science today. Applications are for example, how can a travelling salesman most efficiently plan to visit his clients in a day and in what order, and related, if he has 5 days to visit 25 of them, how to subdivide them into the 5 days of the week?
6 The Connection to Topology
When we look back at how the category of structures for which the Euler formula holds expanded from Platonic solids, Archimedean solids, convex polyhedra, some star-convex polyhedra expanded in 3D and how the category was then ‘projected down to 2D’ and planar graphs in the end, one may realise that there was a centuries long ‘unnecessary detour’ in determining the crux to it all. It would have been better to start with drawing some arbitrary planar graphs on paper and discovering that V-E+F=2 holds for all of these. Imagine in a parallel universe Theaetetus would have gotten up one day and made some simple 2D planar graphs and counted V, E and F. How much earlier would V-E+F=2 have been discovered then? ¹
Yet, apart from graph theory, there is even more that came out of the V-E+F=2 formula and it’s called topology, another mathematical discipline.
For all polyhedra we considered up to now, they are imagined on a sphere-like shape. For example consider cutting an apple in 8 parts in three perpendicular cuts (one along each of the 3 axis-planes: xy, xz, yz planes). We then have 8 apples parts and so 8 skin faces, 3 * 4=12 edges and 6 vertices. V-E+F=6–12+8=2. The Euler relation holds! If we consider a division directly on the sphere, for example of the worlds globe in latitude and longitude lines, no matter how many lines we choose, we could also count vertices, edges and faces and it turns out the Euler relation still holds!
So this makes one realise that maybe, the Euler relation has more to do with the sphere’s nature than with (convex or non-convex) polyhedra or the particular division of a globe itself.
Say we make such a similar 3-cut or surface division on a torus. What results? V-E+F=8–16+8=0, so 0 instead of 2, so the Euler relation does not hold anymore? It turns out that the Euler relation can be generalised to V-E+F=X(S), where X(S) is the Euler characteristic of the surface S. X(sphere)=2 and X(torus)=0. And an g-holed torus has X(S)=2–2g.
Topology goes much further with cross caps, mobius bands and klein bottles, but suffice it to say that the Euler formula still holds but can be generalised for surfaces and even in multiple dimensions and that the Euler characteristic is still one of the invariants used to classify topological surfaces.
Richeson dedicates a chapter to showing in another way how fundamentally different spheres and tori are. If one imagines a continuous vector field, say describing the wind on planet Earth. There will always be at least two places on earth where there will be no wind. Try it. As for a different field, the magnetic field on earth, this is similar. There will be the magnetic north pole and the magnetic south pole. On a torus, imagine putting it as a doughnut, with the hole in front of you and dripping chocolate on it. You will have a ‘north pole’ where — locally — you see the chocolate going downwards, spreading out to all sides (a source point). You will have a ‘south pole’ where — locally — you will see all chocolate coming together (a sink point). But you will also have an upper and a lower point lying on the inner hole where for both, from some sides chocolate glides down into it and to other sides chocolate glides out to. These two are called saddle points.
When we then defined the (Poincare-Hopf) indices on these different types of points. We will derive a surprising fact. This index for a point in a field is defined as the number of times the incoming field vector turns by -2 Pi when going around clockwise on a small circle around the point in anti-clockwise direction. The figure below shows four cases. For a source and a sink this can be see to be 1. For a saddle point with two incoming chocolate streams and two outgoing ones interleaved, this can be seen to be -1. The fourth case in the figure shows a saddle point that has 3 incoming and 3 outgoing vectors, which leads to an index of -2.
When we add the indices for the ‘zero’ points of the field on a sphere, we get (+1)+(+1)=2, while for the torus, we get from top to toe: (+1)+(-1 + -1) +(+1)=0. Both sums are exactly equal to the Euler characteristic of the respective surfaces. So the zeroes of the fields defined over a surface reveal some intrinsic properties of that surface, just like polyhedra defined over surfaces reveal fundamental properties of that surface.
Richeson describes many more topics like knot theory, the four color problem (it takes at most 4 colors to color any map on a sphere, but it takes at most 7 for a map on a torus (as demonstrated for this map here)
and generally p=floor((7+sqrt(49–24 X))/2)) for a map on any surface with Euler number X, extensions to n dimensions where Poincare defined an Euler number X(M) on a manifold M, and it turns out that in a generalised formula for X(M), you still have the + and -signs alternating as in V-E+F over more and more higher simplices. Amazing how much structure is present! But this would lead us too far here.
7 Conclusions
In the introduction to this article we mentioned that with the story of Euler’s formula, you were in for some nice surprises. As for me, these surprises were the following.
- how it took about 2000 years until someone discovered that ‘simple formula’ on polyhedra that were out there already for so long,
- how something from one field, geometry (a formula on polyhedra with discrete rigid features) does actually have a connection with topology (the number of holes in smooth surfaces a polyhedron can be imagined to be constructed on),
- how something from one field, geometry (that same formula) does actually have another connection (or manifestation of the same connection) with topology (the sum of the indices of all the ‘zeroes’ in a continuous field constructed on surfaces a polyhedron can be imagined to be constructed on),
- how 2 and 3 can give rise to an entirely new field in Mathematics (topology),
- related to the previous surprise, how only very few people over these centuries know where to look, how to think, what to investigate to find this deeper structure (For all the others, that’s probably called intuition or genius.),
- how much beautiful structure there is to discover (by yourself or from reading) in Mathematics and so in nature but at the same time how one has to pay attention not to imagine things that are not really there, called mystic at best,
- how even something that abstract as topology still has some applications (though Richeson does not focus on that).
It’s also surprising to me that this fun story of discovery that does not require more than primary school arithmetic to understand, does not make part of the secondary school mathematics education. It was not even part of my full engineering curriculum.
Footnotes
- I like the idea of changing one single small thing that would have happened earlier in history and imagining the effect it could have happened on later development. Spoiler alert … For one, McEwan’s novel ‘Machines Like Me’ is the story of Alan Turing supposedly having proven that P=NP in the 60s and the subsequent accelerated development of science and technology that (could have) changed history, with the actual story playing in the 80s. So it’s called a retro-futurist story. It’s a bit like steampunk or diesel-punk are based on the idea that steam or diesel would have persisted as general energy source. For another, the series ‘For All Mankind’ available on Apple TV+, makes the assumption that in ’69, the Russians beat the Americans as the first ones to set foot on the Moon. Subsequently everything could indeed have been very different. Would a black sheep have been searched for? Would it have been Wernher von Braun? Would the US have become more aggressive in its space exploration or rather have given up? Would the space race never have ended/been paused?
Peter Sels, July 12th 2020. Written for my nephew Edward, to show him that there is a lot of fun in continuing self-discovery in Maths or Science after secondary school. :)
Copyright © 2020 Logically Yours BV.
