"Explanation" between concrete and abstract

I realized that an apparently well-understood word, "explanation", may lead to controversies in discussions about the foundations of physics. The foundations are already controversial enough, but this adds even more to the confusion. It gives you a double featured feeling: on the one hand, of being misunderstood, and on the other hand, that you don't understand where the interlocutor is going on.

What is an "explanation"? Probably the most usual meaning is that explanation is to reduce the unknown to the known, the unfamiliar to the familiar. When this happens, we get the sense of understanding.

Even since childhood, we had so many questions, and the grown ups explained them - reduced the unfamiliar to more familiar notions. In school, the teachers continued to provide us explanations, and we appreciated most the teachers who managed to make the unclear things more intuitive for us. When reading about the foundations of physics, we usually start with popular physics books. The most recommended such books are those providing the feeling of understanding, appealing to our intuition. When we try to read something more advanced, even if it is recommended by our favorite pop-sci books, we find ourselves in a totally different situation. Instead of finding the deeper explanations we are looking for, we find ourselves thrown in the turbulent torrents of the abstract mathematics, drifting without an apparent purpose. And what is most annoying, these textbooks and articles full of equations actually claim to explain things!

Why is this happening? I think that they are guided by another meaning of the term "explanation": "to give an explanation to a phenomenon is to deduce the existence of that phenomenon from hypotheses considered more fundamental. For example, when from the principles of General Relativity was deduced the correct value from the perihelion precession of Mercury, it was considered that GR explained this precession. On the other hand, the deflection of light by the Sun was considered a prediction. After the full experimental confirmation, it became an explanation. I consider that "prediction" is just a temporary status of a scientific explanation, and that the fact that many explanations are first predictions is a historical accident.

There seem to be a similarity between principles/phenomena and axioms/theorems. This similarity suggests the reason why mathematics plays such an important role in the explanation of phenomena. To deduce more from less, complicated from simple, diverse from universal, this means to use logic and mathematics. And there is no limit of the difficulty of the needed mathematics, even if the principles are not that difficult.

This notion of explanation, I understand now, it is not shared by all of us. The reason is simple: because "explanation" usually means to reduce the unfamiliar to familiar. When somebody claimed to explain a phenomenon, we expect him to show how this strange phenomenon can be described in more familiar, concrete terms. Instead, we find that he or she starts describing it in more abstract terms. How come that such more and more abstract terms are shamelessly named "more basic principles", "more elementary principles" and so on? Isn't this a lie?

Maybe the explanation by "reducing to concrete things" has pedagogical reasons, and the explanation by "reducing to universal principles" is in fact foundational research. But does this means that the gap between pedagogical and scientific explanation should grow as it does nowadays? Wouldn't be much, much better to have a mechanistic explanation? After all, Maxwell sought for such an explanation of the electromagnetic waves, even though he had the equations! The ether theorists of the XIXth century tried to reduce electromagnetism to vibrations in a medium. This tradition still continues, and we encounter on a daily basis renowned scientists trying to explain things which other renowned scientists consider to be already explained: electromagnetism, wave-particle duality, gravity, entropy, the Unruh effect, spacetime, time, black holes and so on.

Probably it would be better to have a mechanistic explanation of everything. This would definitely help the public outreach of physics, and will help physics to advance faster. This may have a huge impact on technology, and on our lives. But who can bet that God, when created the world, bothered about our need to reduce the things to what we know? Why would the universe care about our limited understanding, when decided what principles to follow? Who are we, why would we be so important? I think that, although it would be desirable to find concrete, familiar universal principles behind this complex and diverse world, we have no guarantee that this will ever happen. "You shall not make for yourself a carved image, or any likeness of anything that is in heaven above, or that is in the earth beneath, or that is in the water under the earth."

The definition of "explanation" as a reduction to universal principles has its own advantages, given that we do not take these principles as ultimate truths, but just as hypotheses. One of these advantages is that it allows us to equally appreciate theories which seem to contradict each other. We can appreciate its explanatory power in the sense stated above: as its efficient encapsulation of a wide variety of phenomena in fewer, simpler, and more general principles. This doesn't mean that we should consider these principles as being "true". It is not about being "true", just about encapsulating as much phenomena as possible in as few principles as possible, even if these principles are more abstract. If we insist to become fans of one theory or another as the ultimate "truth", we may reduce our capacity to grasp other explanations. This would not be a problem, if we could prove our theories beyond any doubt, but the truth is that we cannot, no matter how convincing they may look to us.

The Essence of Quantum Theory

The purpose of this short post is to provide a very brief presentation of Quantum Theory.

Short:

In quantum theory, particles are waves of various shapes. You cannot directly observe the waves, only some of their properties. Each property is well defined only for some of the possible shapes. There is no shape for which the properties "position" and "momentum" are simultaneously well defined (Heisenberg's principle). When you observe a property, you find the wave in a shape corresponding to that property (like magic!), without regard of its previous shape. Entanglement: n particles are a single wave in a space with n x 3 dimensions, they don't have individual shapes.

Details:

In classical physics, particles are points moving on well-defined trajectories. This picture turned out to be an approximation: a particle is in fact a wave (although there is no waving medium for this wave). We know it is a wave, because it interferes, it can be diffracted, its allowed states in an atom are those corresponding to an integral number of wavelengths, and it is governed by a wave equation. As a wave, it has no definite trajectory, and insisting in discussing in terms of position and momentum as for point particles leads to problems.

But you can't observe the wave directly, only classical properties, like position or momentum. Each property you observe is well defined only for a particular set of possible shapes of the wave. When you observe its position, the wave appears to be concentrated at a point, but it has an undefined momentum. Conversely, the possible shapes that have well defined momentum have no well defined position – they are spread in all the space. Similar things happen when you want to observe any other classical property.

The first strangest thing about quanta is that when you look at them, they take precisely one of those shapes corresponding to the property you observe, without regard of their previously known shape. If further you try to observe another property, which is not well defined for the previously observed shape, you will find the new kind of shape, allowed by the new property. Knowing its shape before an observation, you can not predict which of the allowed shapes you obtain, but only the probability for each allowed shape.

The second strangest thing is the entanglement. When dealing with more particles, say n, they are not described by individual waves, but by a single wave on a space obtained by multiplying the usual three-dimensional space with itself n times. This means that after two particles interact, they have no individual shape, but a common shape on this space with 6 dimensions. We can still observe one of the particles, and obtain a particular 3-dimensional shape for it, but if we try to observe both particles, the shape of one is dependent on the shape of the other. The strangest part is that their shapes are correlated even if the particles are separated by very large distances.

Are vector bundles fundamental in Physics?

Vector bundles and gauge theory

The idea in Gauge Theory is that the fields of the known forces can be expressed starting with some principal bundles and their associated vector bundles. To be more precise, let's consider Maxwell's electromagnetic field $F_{ab}$. It can be represented with the help of a principal bundle of group $U(1)$, and a connection on this bundle. The connection corresponds to the electromagnetic potential, and the curvature to the electromagnetic field. It is known that we can modify the potential to $A_a(t,x)\mapsto A_a(t,x) + \partial_a \theta(t,x)$, and obtain the same $F_{ab}$. In terms of bundles, this transformation corresponds to a gauge transformation of $\mathbb C$ by the action of $e^{i\theta(t,x)}$. The connection will appear to depend on the gauge, but the curvature is gauge invariant.


A bundle is just another manifold

Both principal bundles and vector bundles are differential manifolds (that is, topological spaces which looks locally, from topological viewpoint, like a vector space with with a fixed number of dimensions, and on which we can define partial derivatives). A fiber bundle over spacetime looks locally like the cartesian product between the spacetime and a fixed manifold named fiber. For the vector bundles the fiber is a vector space, for the principal bundle it is a Lie group. The $U(1)$ bundle looks locally like a cartesian product between the spacetime and a circle. This space is 5-dimensional, and it was used by Kaluza and Klein in their attempt to unify electromagnetism with gravity by using a 5-dimensional version of general relativity.

After the electromagnetic force was understood as a gauge field, Yang and Mills provided a generalization which allowed us to see as gauge fields also the strong and electroweak forces. It seemed as easy as replacing the $U(1)$ group with a non-abelian group like $U(2)$ for the electroweak force, and $SU(3)$ for the color force. New bundles resulted, and they can be viewed as well as spacetimes with more dimensions, from which some are compactified.

The obvious problem with these extra dimensions is that we cannot "see" them. What explanation is that we cannot test? To avoid this questions, these dimensions are referred as corresponding to "internal spaces", and the Kaluza-Klein interpretation is in general avoided, being preferred that in terms of bundles.


What is more fundamental, the field or the connection?

It was believed that the potential is only a mathematical trick to simplify Maxwell's equations, and that it has no correspondent in reality. There are some reasons to change this view.

One is, as I explained here, in chapter III., the following. Maxwell's equations contain constraints imposed on the field for equal time, that is, between the values of the field at spacelike separated points (Gauss' law). This may seem a little bit acausal, because requires the initial conditions at two spacelike separated points to be related. Of course, the separation between the two points is infinitesimal, but it still exists, and has non-local consequences. In terms of the potential, these constraints are no longer needed. If we consider the connection as fundamental, then the curvature will be a derived field. It will still obey Gauss' law, but this time just as a consequence of being associated to the connection, which is the true fundamental field. And the connection is not constrained.

Taking a charged field, such as the Dirac electron field, under a gauge transformation it is multiplied by $e^{i\theta(t,x)}$. The Dirac-Maxwell equations maintain their form, if we apply the corresponding gauge transformation to the potential. This allows us to perform an experiment to see whether the potential is a real field, or just a mathematical trick. This experiment was imagined by Werner Ehrenberg and Raymond E. Siday, and Aharonov and Bohm, a decade later. It was verified experimentally by S. Olariu and I. Iovitzu Popescu, and confirmed two years later by Osakabe et. al..

Basically, this effect shows that the electromagnetic potential has a fundamental nature. But how can a potential be the fundamental quantity? Which potential, considering that there can be an infinity such choices, related by $A_a(t,x)\mapsto A_a(t,x) + \partial_a \theta(t,x)$? The only way known for this is if it represents a connection on a $U(1)$-bundle. This way, the potential is just the expression of the connection, in a particular frame on the bundle. Gauge transformations are just changes of that frame.

The Aharonov-Bohm effect is interpreted topologically as an effect of the holonomy of a connection on this bundle (which is the electromagnetic potential). These properties are captured by Wilson’s loops.


Are those "internal spaces" real?

It is easy to check the number of dimensions of our space: it is the number of coordinates required to indicate the position of a point, that is, 3. The number of numbers needed to express a rotation, 3(3-1)/2=3, indicates also that we live in a 3-dimensional space. How can we check the extra, "internal" dimensions? We just count the numbers needed to represent them. Since the electromagnetic potential can be changed in a way indicating a rotation of a circle, we conclude that the internal space has one dimension. It is the same as in the case of the 3-dimensional space. The only difference is that we can actually move in this space, and this is why we consider it real. We cannot move in the internal dimensions. But can we, at least, send particles to move in those dimensions?

In fact we can. The Aharonov-Bohm effect shows that we can rotate the wavefunction of an electron. We can compare the rotation of a part of the wavefunction of an electron with that of another part. To do this, we just make them interfere, and see the relative rotation between them. Isn't this remind us of comparing the speed of light in two arms of the Michaelson-Morley interferometer? Only that the Aharonov-Bohm effect succeded, and showed that there is an "internal rotation".

Now, it is time to remember the notion of existence as it is used by mathematicians. Something exists from a mathematical viewpoint if it is logically consistent. The 5-dimensional spacetime (3+1+1) of the electromagnetism exists, in this respect. Did the rotation verified by the Aharonov-Bohm effect confirm its physical existence? In fact, we can take for the internal space, instead of a circle, the complex space $\mathbb C$. The group $U(1)$ acts as well on this space, and we can think that the physical spacetime is in fact 6-dimensional (3+1+2). What is the true number of dimensions? I would say that this number is given by the number of dimensions of the $U(1)$-bundle, that is, 3+1+1. And the internal space happens to be a circle because the $U(1)$ group itself is, topologically, a circle. It has one dimension too. And both the circle bundle and the $\mathbb C$ bundle are associated to this principal bundle, that is, they are obtained from representations of the $U(1)$ group.

OK, so the space dimensions are more real for us, because we can move almost freely in these dimensions. Time is the fourth, at least mathematically, and some people can accept that it is the fourth physically too. They think that this is true, because of the great beauty and symmetry of the Lorentz group. But the internal dimensions, have they more than a mathematical existence? We can ask as well whether the three space dimensions are true or not. What if the real number of space dimensions is two, as the holographic principle suggests?

Do we have a criterion to distinguish between real dimensions and simple mathematical constructions in physics? Can this criterion be the experiment?

Why are vector bundles natural in Physics?

Aren't usual vector fields enough?

When we work with a space $M$ (a differentiable manifold in fact), we may need to consider fields on that space. The fields can be scalar, vector, tensor, spinor fields, depending on the possible values they can take - scalars, vectors, tensors, spinors. But all these can be considered vectors in some spaces, so in general the fields will be considered to be vector fields.

We can think that, considering vector fields on a space, it is as simple as considering functions on that space $M$, valued in a vector space $V$. Unfortunately, this is not the case, and there is a very good reason for this. I will explain it here.

When working with a function $f:M\to V$, we can represent it by its graph, which is in fact a subset of the cartesian product, $\{(x,f(x))|x\in M\}\subset M\times V$. Therefore, we may hope that all the vector fields on $M$ valued in $V$ are subsets of $M\times V$. If the base manifold is the sphere $S^2\subset \mathbb{R}^3$, its tangent vector fields cannot be, in general, represented as subsets of the cartesian product $S^2\times\mathbb{R}^2$ (we say that $S^2$ is not parallelizable). This and other simple examples force us to consider a more general definition of vector fields.

On the other hand, there are spaces with which this representation works always. For example, we can take $M=\mathbb{R}^n$ or a simply connected open subset of it, $M\subset\mathbb{R}^n$. All possible vector fields of such an $M$ can be represented as subsets of $M\times V$.

The idea behind the vector bundles was to consider the base space $M$ as being covered by open sets like above. The restriction of a vector fields to such an open set $U\subset M$ can be represented as a subset of $U\times V$. But the way they are glued together can vary very much, because when they are glued together, the vector space $V$ can be transformed relatively to $V$ on another open set. Take for example a circle as the basis manifold, and consider as a vector space the Euclidean one-dimensional space. We can glue it to each point o the circle in two ways: as a cylinder, and as the Mobius strip. The idea is that we can cover the base manifold with opens such that the way we associate vector spaces to its points is trivial on each open set from the covering.

The basic point is that, in order to have fields of any kind on a manifold, you need bundles. The fields are "sections" in the bundles. Now, these fields can be combined as we do with the vectors. In fact, what we can do with the vector spaces, we can do with vector bundles as well. We can construct direct sums, duals, tensor products.

Vector bundles and quantum entanglement

There is an important difference between two types of tensor products. The fields which are sections of a given vector bundle $E\to M$ form themselves a vector space $\Gamma(E\to M)$. Two such vector spaces of sections can as well be tensored. The tensor product $\Gamma(E_1)\otimes \Gamma(E_2)$ of two vector spaces of sections of two bundles over the same base manifold $M$ is larger than the vector space defined by the sections of the tensor product of the two bundles, $\Gamma(E_1\otimes E_2)$. The first contains nonlocal fields of the form $\phi(x,y)$, while the second contains only local fields, of the form $\varphi(x)=\phi(x,x)$. $\Gamma(E_1)\otimes \Gamma(E_2)$ are no longer sections of a vector bundle. The entangled states in Quantum Mechanics are represented by such nonlocal fields.

One of the most important applications of vector bundles in Physics is related to the Gauge Theory. We will discuss more about this other time.

Polyhedra and Groups

November 21, 2004

The central idea of this article is a direct, geometric method to multiply/compose permutations, by using polygons and regular polyhedra.

I begin with a short review of the properties of regular polyhedra and permutation groups, as well as of the relations between these two areas. These relations led me to the geometric method of multiplying permutations.

Introduction
Regular polyhedra

The regular polyhedra, also known as platonic solids, are represented below:

Inscribing one regular polyhedron in another

There are many ways to inscribe one regular polyhedron in another. One useful case is that when each vertex of the first polyhedron lies at the center of a face of the second, and each face of the second polyhedron contains one vertex of the first.

The cube can be inscribed in this way in the octahedron, but also reciprocally. The same reciprocity holds between the icosahedron and the dodecahedron. Such polyhedra are said to be dual. The tetrahedron is self-dual.

The following ways to inscribe regular polyhedra will be useful too: one or two tetrahedra in a cube, and cube in a dodecahedron.

Permutations

You can permute the set {1, 2} in two ways: (1, 2) and (2, 1). What about the set {1, 2, 3}? It is easy to see that we have six possibilities: (1, 2, 3), (2, 3, 1), (3, 1, 2), (2, 1, 3), (3, 2, 1), (1, 3, 2). The result is the same, for any sets of three objects, not only numbers. The number of permutations depends only on the number of elements of the set, and not on their nature. For a set with n elements, this number is 1·2·...·n, and is named the factorial n! of the number n.

An ordered set of the first n positive integers can also be used to express the permutation, as a specific operation to be applied to another ordered set. For example, (2, 1) shows that the elements of an ordered set of two elements are reverted. (2, 3, 1) shows that an ordered set of three, for example (1, 2, 3), changes its order to (2, 3, 1). Such transforms are named permutations of the set {1, 2, ... , n}.

We can multiply two permutations, which means that we apply them successively to the ordered set. All the permutations of a set with n elements forms a grup, because the operation of multiplication of permutations is associative, has a neutral element (which is the identical permutation (1, 2, ... , n), leaving the order unchanged), and every permutation has an inverse which cancels its effect. The group of permutations of the set {1, 2, ... , n} is named the symmetric group of degree n, denoted by $S_n$, and having n! elements.

A transposition is a permutation which interchanges only two elements. For example, (2, 1, 3), (3, 2, 1), (1, 3, 2) are transpositions of the group $S_3$. Any permutation can be decomposed as a product of transpositions in a non unique way. Yet, there is something independent of the way we decompose the permutation as a product of transpositions. If the number of transpositions in such a decomposition is even, it will be even for any other decomposition of the same permutation, and we name such a permutation even. Otherwise, we call it odd. The set of even permutations of a set with n elements forms a subgroup of $S_n$, named the alternating group, denoted by $A_n$, and having n!/2 elements.

The symmetries of a regular polyhedron

By rotating the regular tetrahedron around one of its heights with 120º or 240º, this one remains unchanged. We say that the regular tetrahedron is unchanged by these transformations. We can rotate a regular polyhedron so that, after this transformation, it occupies exactly the same position, but having the faces not necessarily in the same positions. Also, they admit symmetry planes. In fact, this is the idea about the regular polyhedra – their rich symmetry.

The transformations leaving unchanged a polyhedron are named symmetry transformations. One can multiply the transformations. Because each symmetry transformation interchanges the faces of the polyhedron, we can associate to the transformation a permutation from $S_n$, n being the number of faces of the polyhedron. The symmetry transformations leaving invariant a polyhedron form a subgroup of $S_n$, named the automorphism group of the polyhedron.

How many symmetry transformations have each of the regular polyhedra? One easy method to count them is the following. Choose a face a. After a transformation it will take the place of another face a’. Since the polyhedron has n faces, we have n possibilities. Let’s choose a face b, neighbor to the first face before transformation, a. After the transformation, b goes into a face b’, neighbor to the face a’. Each face has the same number of edges k. The face b’ can be one of the k faces neighbor to a’. Therefore, we have nk possibilities. These transforms are the rotations of the polyhedron. But the face a is a polygon, therefore it has two sides. Consequently, when we move the face a in a’, this can flip. In this case, the transformation is no longer a rotation. It can no longer be obtain simply by moving the polyhedron, but by taking its mirror image.

Thus, among all transformations of the polyhedron, a special subgroup is formed by the nk rotations, but the total number of transformations is 2nk. Both these groups are subgroups of $S_n$. For the regular tetrahedron we obtain 2·4·3 = 24 automorphisms, from which the rotations are 4·3 = 12. The cube’s automorphisms group contains 2·6·4 = 48 automorfisms, from which 6·4 = 24 are rotations. The regular octahedron has 2·8·3 = 48 automorfisms, from which 8·3 = 24 rotations, like the cube. Both the icosahedron and the dodecahedron have 2·4·3 = 24 automorfisms, 4·3 = 12 rotations.

We see that the dual polyhedra have the same symmetries.

The symmetries of regular polyhedra and the permutations

Let’s label the vertices of a regular tetrahedron with the numbers {1, 2, 3, 4}:

At a symmetry transformation, the vertex 1 can go in any of the four vertices. The vertices 2, 3 and 4 are all neighbor with the vertex 1, and they will remain so after the transformation too. Their order around the vertex 1 is preserved in the case of rotations, otherwise it is reverted. Therefore, the vertices’ permutation is even if and only if the transformation is a rotation. The regular tetrahedron having 24 automorphisms, they coincide with the elements of the group $S_4$. The rotations correspond to the elements of the alternating group $A_4$.

Let us now label the cube’s vertices such that the ends of each large diagonal have identical labels from the set {1, 2, 3, 4}, like below:

Let’s choose one of the two regular tetrahedra inscribed in the cube. Its vertices are labeled with the numbers {1, 2, 3, 4}. A rotation of the cube rotates also the tetrahedron. Each cube rotation which let the tetrahedron in place corresponds to a rotation of the tetrahedron, and it is an even permutation. The even permutations of the cube’s labels correspond to transformations which interchange the two tetrahedra. Thus, the cube’s rotations correspond to the symmetric group $S_4$.

By labeling the faces of the regular octahedron, we obtain that it has the same symmetry groups as the cube (being its dual).

To obtain the symmetry group of the regular dodecahedron, let’s label its edges with the numbers {1, 2, 3, 4, 5}, like in the image:

For each edge of the dodecahedron, we take the four adjacent edges, and the other vertices of these edges form a square (hint: the four edges are diagonals in identical regular pentagons). These edges form five cubes. Each of the five cubes highlights six of the dodecahedron’s edges. It is easy to see that for each face we obtain a different ordering for the labels. Each rotation of the dodecahedron will take the face labeled by the ordered set (1, 2, 3, 4, 5) (we count starting with the vertex 1) in another face, also labeled with a permutation of the five numbers, so that 1 goes to one number, 2 to another etc. Thus, to the permutation (1, 2, 3, 4, 5) we can associate, as a result of the rotation, another permutation. Because we limit ourselves to rotations, we can choose one orientation (for example clockwise). The rotations will keep this orientation. Because we can start from any vertex of a face when we read the permutation, we will have five permutations for each face. This way, each corner of a face represents a permutation. We can check that these permutations are always even. Each dodecahedron’s rotation corresponds to an even permutation of the face labeled by (1, 2, 3, 4, 5). The dodecahedron’s rotations group is isomorphic with the alternating group $A_5$.

The regular icosahedron being dual to the regular dodecahedron, its 30 edges can be labeled like the ones of the dodecahedron. In this case, we will use cubes with the vertices in the centers of the icosahedron’s faces.


Group operations with polygons

"Calculator" for the Klein group

The Klein’s group has four elements {E, A, B, C}, and its multiplication is given by the label:

This group is isomorphic with the automorphism group of a rectangle. We can construct a „calculator” for multiplying elements of the Klein group by using to identical cards:

The first card will be used as witness card. For obtaining all the multiplications with the element B, hence the „multiplication with B table”. For doing this, we rotate the second card (the result card) so that the element B is moved in the position corresponding to the neutral element E of the witness card:

Now, to obtain the product of any element X from the group with B, we just read from the result card the element from the position corresponding to the position of X in the witness card. For example, to see the result of the operation C·B, we look for the element C in the witness card. It is in the lower right corner. We look in the same position in the result card. The corresponding element is A. We can check in the multiplicative table that indeed C·B = A.

Calculator for the groups $A_3$ and $S_3$

We start with the group $A_3$ of the even permutations of a set with 3 elements. Its elements are the permutations (1, 2, 3), (2, 3, 1), (3, 1, 2). We construct a card shaped as an equilateral triangle and we label its vertices with the numbers 1, 2 and 3:

The rule is: we label the vertices with the numbers {1, 2, 3} clockwise, and the edges with the permutations given by the order in which we met the vertices by starting from that edge and go clockwise.

We see that the rotations with 120º, 240º or 360º preserve this triangle. To find the multiplications of the elements of the $A_3$ with one element of the group, say (3, 1, 2), we keep the witness card in the normal position, and rotate the result card such that the permutation (3, 1, 2) corresponds to the position of the identical permutation (1, 2, 3) from the witness card. In order to find the result of the multiplication of a permutation with the permutation (3, 1, 2), we search on the witness card the position of the desired permutation, and on the result card we just read the permutation on the corresponding position. For example, to find the result of multiplying the permutations (3, 1, 2) and (2, 3, 1), we look on the result card for the permutation corresponding to the position occupied by the permutation (2, 3, 1) on the witness card. We see that the result is the permutation (1, 2, 3):

The group $S_3$ contains, in addition to the even permutations (1, 2, 3), (2, 3, 1), (3, 1, 2) from the group $A_3$, the odd permutations (2, 1, 3), (3, 2, 1), (1, 3, 2). This is why we will allow, besides the rotations maintaining the triangle in plane, transformations obtainable by lifting the triangle from the table and flipping it. On its back face we will mark the odd permutations:


Calculator for the groups $C_n$ and $D_n$

A finite group such that all its elements can be obtained by multiplying one element (named generator of the group) with itself, is named cyclic group. We denote the cyclic group with n elements by $C_n$. The cyclic group $C_n$ is isomorphic with the group of integers modulo n, $\mathbb{Z}_n$. The cyclic group C3 is isomorphic with the alternating group $A_3$, but this doesn’t hold for n > 3. We can consider the group $C_n$ as representing the plane rotations of a regular polygon with n edges. If we allow mirror symmetries, obtained by taking the polygon outside the plane and flipping, the number of the possible symmetries doubles, and their group is named the dihedral group, $D_n$. We observe from the definition that for n = 3 the dihedral group is isomorphic to the permutation group of a set of three elements: $D_3$ ~ $S_3$, but, as for the cyclic group, we can’t generalize for n > 3.

The calculators for the cyclic and dihedral groups can be made from regular polyhedra. They can be used similarly to those described for the groups $A_3$ and $S_3$, with a witness polygon and a result polygon.

Polyhedral calculators of permutations

Calculator for the group $S_4$

Let’s consider again the cube with vertices labeled like this:

Each face has four edges, and we label each of them, on that face, with the permutations obtained by reading the vertices’ labels, starting from that edge and walking clockwise. We obtain a cube labeled on each side of each edge with permutations. The reader can construct her own cube by printing this image:

After assembling it in 3D, it will look like this:

For finding all the multiplications of the permutations from $S_4$ with a particular permutation, say (4132), we put the witness cube with the face containing the identical permutation (1234) in front, such that the identical permutation is below:

Then we put the resulting cube in a similar position, only that on the position of the identical permutation we put the permutation (4132):

Let’s suppose we want the result of the multiplication of the permutation (4132) with another permutation from $S_4$, for example (3124). We look in the witness cube the position of the permutation (3124), and in the result cube we read the permutation from the corresponding position. Because the permutation (3124) occurs in the witness cube on the left edge of the right face, we read the permutation from the left edge of the right face of the resulting cube. This is (3412).

Once we position the cubes, we can simply read all the results of the multiplications with the chosen permutation (4132) with any element of $S_4$ on the result cube.

The reader is invited to study the symmetries of the permutations written on the cube.

For example, two permutations associated to the same edge differ by a transposition between the elements labeling the edge’s ends. The permutation (4123) is neighbor of the permutation (3124) – they are on the same edge of the cube. Their common edge has the vertices labeled with the numbers 4 and 3. These numbers are transposed in one of the two permutations to obtain the other. In fact, any two permutations associated to the same edge differ by inverting the positions 1 and 4.

Another interesting property of this cube is that, applying a rotation, we obtain on the initial position of the identical permutation a permutation showing how the four large diagonals of the cube were permuted.

Calculator for the group $A_5$

As we remember, the group of rotations of the dodecahedron is isomorphic with the alternating group $A_5$. We can label the dodecahedron by using even permutations of the set {1, 2, 3, 4, 5}, like this:

By printing it and gluing the edges, you can construct a dodecahedron. By constructing two identical such dodecahedra, we can use them to multiply permutations from the alternating group $A_5$, in a similar way we used two cubes to multiply elements of $S_4$.