The spinorial ball: A macroscopic object of spin-1/2

We propose here to describe and introduce spin-1/2 objects without the help of quantum mechanics but rather through geometry and group theory. We believe that this approach will help students and interested readers to get better insight on the properties of such systems. We will not discuss here the particular (and important) manifestations of spin-1/2 objects in quantum mechanics, instead referring the interested reader to Ref. 1 for a compact introduction and to Ref. 2 for a more technical approach. Although we have tried to make this presentation self-contained, basic knowledge about rotations and matrix theory could help for some technical aspects, such as found in Ref. 3. Whenever possible, we focus on the underlying concepts and physical intuitions, at times omitting technical details, and rather provide references for proofs that are omitted. Lastly, we would like to point out that the name of spinorial ball initially originates from an article from Baez and Huerta dedicated to visualization of other Lie groups.⁴ 

Rotating an object, say a favorite quantum mechanics book, by a full turn around any axis will return it to its original state, as shown in Fig. 1(a). On the contrary, spin-1/2 objects are not invariant under this operation and two complete turns are needed to bring them back to their initial state. This unusual feature makes them puzzling objects. As many educators know, it is possible to create analogs of spin-1/2 objects at a macroscopic scale by adding some constraints. For instance, one can tie the book to one end of a ribbon, the other end being clamped to a wall as shown in Fig. 1(b). After one complete rotation of the book, the ribbon acquires a twist that cannot be removed without rotating the object back. If one applies a second turn to the book, however, those twists can be removed without rotating the book by using a combination of translations (see, for instance, videos^5,6). This is nothing but the famous Dirac belt trick (cf. Ref. 7, pp. 43–44) that is widely used in quantum mechanics lectures to help students understand how spin-1/2 objects behave. Indeed, a $2 π$-rotation does not leave spin-1/2 objects, nor the system {book + ribbon}, invariant, while a $4 π$ rotation does, at least up to translations.

There are nevertheless several issues with this picture. The translations needed to restore the original state of the system have no direct analog in quantum mechanics (and even seem like a bit of cheating). One can also wonder if there is anything analogous to the constraints introduced in this trick for free quantum particles. Although useful, this picture is, therefore, somewhat limited as a way to provide better understanding of spin-1/2 objects. Another macroscopic spin-1/2 system based on coupled mechanical pendulums has also been described,^8,9 but the effective spin-1/2 object evolves in an abstract space of parameters. Therefore, a direct and unconstrained physical realization of a half-integer spin at macroscopic scale is still missing. We now propose to fill this gap by introducing the spinorial ball.

The spinorial ball is a polyhedron of roughly spherical shape with pentagonal and hexagonal faces (see picture in Fig. 1(c)). Each face is illuminated with colored LEDs. In this section, we explain the main observations likely to result from casual experimentation with the ball by non-experts. While we primarily focus on its implementation as a manipulable electronic device, a mobile version¹⁰ (which works by moving the phone) and a browser-based simulation¹¹ are available, and we encourage the reader to use them as an experimental apparatus while reading this introduction. A supplementary video showing its manipulation is also available.

If one starts to move the ball, the colors of the faces change smoothly. After some time, one notices two things:

• All faces of the same shape (either pentagons or hexagons) are illuminated in the same color, which we denote $C P$ or $C H$⁠.

• The colors displayed do not change when the ball is translated, but they do when the ball is rotated.

Further experimentation will reveal that:

• A specific set of colors corresponds to a single orientation of the ball.

• The converse of the previous property fails: for each physical orientation of the ball, two different sets of colors $( C P 1 , C H 1 )$ and $( C P 2 , C H 2 )$ can be displayed, depending on the history of rotations applied to the ball.

• For a given orientation, one can go from one set of colors to another by performing a full rotation about any axis. Further full rotations cause the colors to alternate, thus showing whether an odd or an even number of turns have occurred.

Of course, the system {ball} (ignoring illumination) is back in its initial state after one turn. However, its very symmetric shape prevents keeping track of its orientation by any other mean than through the displayed colors. On the other hand, as the full system {ball + colors} does not come back to its initial state after one turn but does after two turns, the illuminated ball acts as a free macroscopic spin-1/2 object. From this, several questions can be raised. What do the colors represent? How exactly are the colors related to the rotations of the ball? Why do the colors behave like a spin-1/2? Could one create a spin-1/3 in a similar way, encoding each orientation in three possible sets of colors?

In Secs. II and III, we answer these questions by carefully explaining how the spinorial ball works. We first introduce the rotation group $SO ( 3 )$ to describe the possible rotations of the physical ball. We then introduce the spinors and the group $SU ( 2 )$⁠, which will correspond to the different illumination states of the ball. We then define a group homomorphism between $SU ( 2 )$ and $SO ( 3 )$ that will provide the link between the physical rotation of the ball (an element of $SO ( 3 )$⁠) and the colors it displays (an element of $SU ( 2 )$⁠). Having introduced all the needed concepts, we come back in the last section to the spinorial ball itself and answer all the questions raised above.

Note that Secs. II and III are, although elementary, quite technical and easier to understand for a reader who is already familiar with the basic properties of the rotation groups SU(2) and SO(3). Those sections also introduce some advanced mathematical concepts (such as path lifting and homotopy classes) through concrete examples, making it of interest for more advanced lectures in quantum mechanics. The reader unfamiliar with group theory or interested only in manipulating the spinorial ball can go directly to Sec. IV for some practical applications and uses of the spinorial ball.

The possible rigid motions of an object that preserve handedness and do not move the origin are elements of the rotation group in $ℝ 3$ that will be denoted $SO ( 3 )$⁠. Here, the origin is taken as the center of the spinorial ball and the elements of $SO ( 3 )$ will correspond to all the possible rotations of the ball that we will now describe mathematically. Formally, the rotation group is defined as the set of $3 × 3$ real matrices R that

• (P1) preserve the length of all vectors: $R † R = I 3$⁠, where $R †$ denotes the conjugate-transpose of R.

• (P2) preserve the handedness of space: $det R = 1$⁠.

The rotation group provides a clear mathematical description of the ball's physical motion. We will now formally introduce the “color space” of the ball, and see the analog of rotations in this space.

The commutation relations (4) and (9), combined with the formulas (3) and (7), suggest a strong connection between $SO ( 3 )$ and $SU ( 2 )$⁠. Section III C describes the link between the two groups, which is at the heart of the spinorial ball.

Obtaining a non-invertible map is more than just an artifact of our construction; it reflects a genuine difference between the groups $SU ( 2 )$ and $SO ( 3 )$⁠. These two spaces are topologically different, and there is no continuous bijection between them. This is discussed further in Sec. III E.

As each element of $SO ( 3 )$ admits two preimages in $SU ( 2 )$⁠, it is impossible to invert the map T. Therefore, acting on the spinor (that is, the colors) by rotating the ball seems to be ill-defined: if one rotates the physical ball by applying an element $R n → ( Ψ )$ of $SO ( 3 )$⁠, we do not know whether one should apply $− S n → ( Ψ )$ or $S n → ( Ψ )$ to the color state. This is correct if one considers rotations as operators that can be applied instantaneously. However, in the physical world of the spinorial ball, we do not apply a given rotation instantaneously but rather a continuous family of rotations, which we refer to as a path. Formally, it is defined as a continuous function $Γ : t ∈ [ 0 , 1 ] ↦ R ( t ) ∈ SO ( 3 )$⁠, i.e., a continuous one-parameter family of rotation matrices with $Γ ( 0 )$ and $Γ ( 1 )$ the endpoints of that path. One can, for instance, consider $Γ ( t ) = R n → ( Ψ t )$ that goes continuously from $Γ ( 0 ) = I 3$ to $Γ ( 1 ) = R n → ( Ψ )$⁠, which corresponds physically to continuously rotating the ball about $n →$ until the total rotation angle reaches $Ψ$⁠. The ending points $Γ ( 0 )$ and $Γ ( 1 )$ contain the information on the initial and final orientation of the ball, respectively, while the path $Γ ( t )$ keeps track of all intermediate orientations. As we will see now, it is possible to “invert” T if one uses the full information contained in the continuous path rather than only the endpoints.

As discussed before, each element of such a path has two preimages $± S n → ( Ψ t )$ in $SU ( 2 )$⁠. Moreover, there exist exactly two continuous paths in $SU ( 2 )$⁠, namely, ${ S n → ( Ψ t ) } 0 ≤ t ≤ 1$ and ${ − S n → ( Ψ t ) } 0 ≤ t ≤ 1$⁠, that map to $Γ$ under T. This is shown in Fig. 2(b). Those paths go from $I 2$ to $S n → ( Ψ )$ and from $− I 2$ to $− S n → ( Ψ )$⁠, respectively. If one specifies $I 2$ as the starting point in $SU ( 2 )$⁠, then there exists only one way to continuously lift a path from $SO ( 3 )$ to $SU ( 2 )$⁠. In other words, there is a unique continuous way to associate a preimage of each element of this path once the preimage of one element of the path is chosen.

One can generalize this result to show that the same holds for any continuous path in $SO ( 3 )$⁠. Namely, such a path in $SO ( 3 )$ has two associated continuous paths (lifts) in $SU ( 2 )$⁠, and choosing a preimage for one point along the path in $SO ( 3 )$ suffices to distinguish one of them. The path lifting property will be the key ingredient to realize a spinorial ball, as it provides a recipe to go from the continuous physical rotation of the ball to a continuous action on the colors.

Before coming back to the spinorial ball, we briefly discuss how the lifting of paths from $SO ( 3 )$ to $SU ( 2 )$ gives some insight into why these spaces are topologically different. For this, we will consider loops, i.e., paths with the same starting and ending point: $Γ ( 0 ) = Γ ( 1 )$⁠. We then say that two loops are in the same homotopy class if one loop can be continuously deformed into the other while keeping the ending points untouched. The homotopy classes of loops are, therefore, defined as collections of loops that can be continuously deformed one to another. Note that in the rigorous definition of deformation, the length of the loop is irrelevant and should be thought of intuitively as an infinitely elastic rope that can be contracted or extended at will and which can furthermore pass through itself without getting snagged or tangled. For instance, in the $ℝ 2$ plane, any loop can be continuously deformed into any another by suitable pushing and stretching, as the plane offers no obstacles to such movement. There is, therefore, only one homotopy class of loops in $ℝ 2$⁠.

Let us now focus on the homotopy classes in $SO ( 3 )$⁠. Consider as in Fig. 2(c) two loops $Γ 1$ and $Γ 2$ in $SO ( 3 )$ going from $I 3$ to $I 3$⁠. When those loops are lifted in $SU ( 2 )$ as $γ 1$ and $γ 2$ starting from $I 2 = γ 1 ( 0 ) = γ 2 ( 0 )$⁠, all we can conclude from the construction is that the endpoints $γ 1 ( 1 )$ and $γ 2 ( 1 )$ are both T-preimages of $I 3$⁠. It may happen that one of these lifted paths is a loop, i.e., $γ 1 ( 1 ) = I 2 = γ 1 ( 0 )$⁠, while the other is an arc joining $I 2$ to $I 2$⁠, i.e., $γ 2 ( 1 ) = − I 2 = − γ 2 ( 0 )$⁠. Both of these possibilities actually occur for loops in $SO ( 3 )$⁠, and making a continuous deformation of a loop in $SO ( 3 )$ cannot change the type of path (loop or arc) in $SU ( 2 )$⁠. Thus, there are at least two types of loops in $SO ( 3 )$ that start at $I 3$⁠: the ones that remain loops when lifted to $SU ( 2 )$⁠, and the ones that open up to arcs from $I 2$ to $− I 2$⁠. In particular, the path $Γ 1$ and $Γ 2$ represented in Fig. 2(c) cannot be continuously deformed one into another. To see this, the reader is invited to imagine that such deformation exists, to consider how the pre-image of the ending point $I 3$ evolves and to finally conclude that a discontinuity must occur to go from one to the other.

Therefore, $SO ( 3 )$ admits at least two homotopy classes of loops starting at $I 3$⁠, and one can show that there are in fact exactly two. On the other hand, $SU ( 2 )$ is the unit sphere of $ℂ 2$ (see parametrization in Eq. (6)), which is equivalently the unit sphere of $ℝ 4$⁠. Arguments similar to the one sketched above for $ℝ 2$ can be used to show that the unit sphere in $ℝ 4$ has a single homotopy class of loops. From here, it can be seen that there is no bijective continuous map from $SU ( 2 )$ to $SO ( 3 )$⁠, as this would imply that they have the same number of homotopy classes of loops. More details and comprehensive proofs can be found in Ref. 3.

The Dirac belt trick presented in the introduction can also be understood through the viewpoint of homotopy classes, and for completeness, we explain the mechanism briefly. The ribbon provides a continuous path in $SO ( 3 )$ that joins the identity element to the current rotational state of the object. To see this, one can form an orthonormal frame of $ℝ 3$ at any point along the ribbon using one vector tangent to the ribbon and pointing toward the end where it is fixed to the wall, a second vector orthogonal to this one but still tangent to the ribbon, and a third vector that is orthogonal to the ribbon. When moving from the wall to the object, the orthonormal frame thus defined undergoes a continuous family of rotations, ending at the net rotation applied to the object. From this, one sees that there exists a direct mapping between a configuration of the ribbon and a path in $SO ( 3 )$⁠. Moreover, translating the object at the end of the ribbon corresponds to continuously deforming the associated path in $SO ( 3 )$ while keeping its endpoints fixed. Indeed, a translation will continuously affect the orientation of the orthonormal frames attached to the ribbon while leaving the frames associated with the ends of the ribbon unaltered.

If we now consider the particular case of an object that has returned to its original orientation, the path defined by the ribbon starts and ends at the identity element of $SO ( 3 )$ and thus forms a loop. In the case of Fig. 1(b), the associated loops $I 3 → I 3$ are, respectively, $Γ 0 : R z → ( 0 ) → R z → ( 0 )$ for the untwisted ribbon, $Γ 1 : R z → ( 0 ) → R z → ( 2 π )$ for the ribbon with one twist, and $Γ 2 : R z → ( 0 ) → R z → ( 4 π )$ for the ribbon with two twists. Translations of the object will deform the ribbon continuously, hence continuously modify the associated loop in $SO ( 3 )$⁠. Therefore, one can go from one configuration of the ribbon to another through translations only if the corresponding loops in $SO ( 3 )$ are in the same homotopy class. As we have seen before, $Γ 0$ and $Γ 2$ belong to the same homotopy class of loops, while $Γ 1$ belongs to another one. This explains why the twists can be undone using translations after a $4 π$ rotation, but not after $2 π$⁠.

We now have all the tools needed to fully describe what is displayed on the spinorial ball. Each complex component of a spinor $( C P , C H ) ∈ ℂ 2$ is encoded as a color, with polar coordinates $( r , θ )$ corresponding to color saturation and hue, respectively (see color map in Fig. 3(a)). These colors are shown on the faces of the ball, with $C P$ as the color of the pentagons and $C H$ as the color of the hexagons. The ball is equipped with an electronic gyroscope to continuously track changes to its orientation. It, therefore, monitors the path of the ball's rotations in $SO ( 3 )$⁠, and lifts that path continuously to $SU ( 2 )$ in real time. The effect of such rotations on the spinor is displayed through changes of faces color.

While motivated by spin-1/2 physics, the spinorial ball can equivalently be seen as a physical model of path-lifting from $SO ( 3 )$ to $SU ( 2 )$⁠. The LED panels show the evolution of the spinor along the path in $SU ( 2 )$ when the physical ball is transformed according to a path in $SO ( 3 )$⁠. The rotation of the ball is the image by T of the element of $SU ( 2 )$ displayed by the LED, but this rotation cannot be found easily without the panel's colors due to the very symmetrical shape of the object. That design element is what makes the spinorial ball behave as spin-1/2: its directly evident features do not show its rotation, but rather a spinor that continuously lifts the ball's rotation in $SU ( 2 )$⁠.

The ball's panels are 3D printed in translucent plastic and glued together in order to form a truncated icosahedron. An RGB LED is placed on the inner side of each panel. The inner cavity of the ball contains a battery, an orientation sensor module (Bosch BNO055), and an Arduino-compatible microcontroller board to which the sensor and all of the LEDs are connected. As is common in designs with many RGB LEDs, we use LED modules with integrated logic (WS8211 ICs), allowing them to receive color data over a serial bus. All LEDs can, therefore, be controlled with just three microcontroller's pins.

It turns out that for practical reasons (computation speed or interpolation purposes, for instance), many commercial gyroscopes and orientation sensors encode rotations directly in elements of $SU ( 2 )$⁠, or equivalently through quaternions. Our gyroscope does not return the rotation matrix R but one of the two associated elements $± S$ of $SU ( 2 )$⁠. Depending on the previous state, the firmware of the ball in the Arduino will flip or not the sign of this raw spinor in order to keep the evolution of $S ( t )$ continuous. The updated spinor is then used to compute new color data for the LEDs.

All details, including the schematics, code, and 3D printing models needed to build a spinorial ball, are available at Ref. 12 and are free to use for any research or teaching activity.

We now give several examples of the spinorial ball evolution, the starting point always being $S 0 = I 2$ corresponding to spinor $s 0 = | ↑ ⟩$⁠. The corresponding pictures of the spinorial ball after each transformation are shown in Fig. 3(c).

• A $2 π$ rotation around the $z →$ axis (or any other axis) has the effect of moving in $SU ( 2 )$ from $I 2$ to $− I 2$⁠, corresponding to the spinor transformation $| ↑ ⟩ → − | ↑ ⟩$⁠. The colors of the pentagonal and hexagonal faces are each replaced by their opposite in the complex plane, i.e. each replaced with opposite hues of equal saturation (see colormap in Fig. 3(a)).

• A $4 π$ rotation around $z →$ (or any other axis) restores the colors to their original state.

• A rotation of $ϕ$ around the $z →$ axis performs the transformation $| ↑ ⟩ → e − i ϕ / 2 | ↑ ⟩$⁠. The phase factor $e − i ϕ / 2$ can be observed on the spinorial ball by a color change of the pentagons while the hexagons remains dark.

• A $π$ rotation around the $y →$ axis corresponds to a path from $I 2$ to $− 2 i S y$⁠, corresponding to $| ↑ ⟩ → | ↓ ⟩$⁠. In other words, when the ball is turned upside down, the spinor goes from up-state to down-state.

• A $π / 2$ rotation around the $x →$ axis corresponds to a path in $SU ( 2 )$ from $I 2$ to $1 / 2 ( I 2 − 2 i S x )$⁠, corresponding to $| ↑ ⟩ → 1 / 2 ( | ↑ ⟩ − i | ↓ ⟩ )$⁠.

As mentioned earlier, the interested reader can manipulate a virtual version of the ball in our browser-based simulation that uses WebGL 3D graphics.¹¹ A mobile version that uses the smartphone's accelerometer to sense orientation and displays the spinor as a pair of colors on the screen is also available,¹⁰ and a supplementary video shows some manipulations of the ball.

The homotopy classes of loops in $SO ( 3 )$ can also be seen with the spinorial ball. A loop in $SO ( 3 )$ is a rotation process that leaves the ball's orientation unchanged, such as the $2 π$ and $4 π$ rotations around $z →$ discussed above. If the initial and final colors are the same (as for the $4 π$ rotation), then the path in $SO ( 3 )$ lifts as a loop $I 2 → I 2$ in $SU ( 2 )$⁠. However, if the colors have changed for their opposite (as for the $2 π$ rotation), the path lifts as an arc $I 2 → − I 2$⁠. The spinorial ball can, therefore, be used as a homotopy class detector, similar to the Dirac belt but without any tether or constraint on its motion. In particular, we recover the fact that the $2 π$ and $4 π$ rotation do not belong to the same homotopy classes.

Note that the previous remark also explains why a spin-1/2 object must come back to its initial state after two turns. After one turn, the path in $SU ( 2 )$ is $I 2 → − I 2$⁠. However, if one performs a second turn, the corresponding path will be another arc $− I 2 → I 2$⁠. When chained, those two arcs, therefore, form a closed loop $I 2 → I 2$ and the spin comes back to its initial state.

The spinorial ball can also be used to multiply elements of $SU ( 2 )$⁠. To see this, let us assume that at a given time, the rotation matrix is $R c$ in $SO ( 3 )$⁠, associated with $S c$ in $SU ( 2 )$ displayed by the LED panels. We will now apply a rotation R to the ball, thus generating a path $R c → R × R c$ in SO(3). This path will be lifted as $S c → S × S c$⁠, where S is one of two the preimage of R by T. Thus, we see that rotating the ball by R corresponds to multiplying the current state $S c$ by S in $SU ( 2 )$⁠. The fact that T is a group homomorphism is crucial here: if it was not the case, lifting $R × R c$ might differ from lifting R as S and $R c$ as $S c$ separately and then multiplying them in $SU ( 2 )$⁠. Note also that the pre-image S that is chosen depends on the path chosen in $SO ( 3 )$⁠.

Let us consider, for instance, the cases where we rotate the ball by $π$ or $3 π$ around the $x →$ axis. In both cases, this corresponds to a path $I 3 → R x → ( π )$ in $SO ( 3 )$⁠. In the first case, the initial spinor $S 0$ is multiplied by $S = − 2 i S x$⁠, while in the second case, it is multiplied by $S = 2 i S x$⁠. Despite having the same ending point, those two paths in $SO ( 3 )$ belong to different homotopy classes and are, therefore, associated with different multiplication operations in $SU ( 2 )$⁠. We can also see the group structure at work here by decomposing $R x → ( 3 π ) = R x → ( 2 π ) R x → ( π )$⁠, which is consistent with the decomposition of $S = − I 2 × − 2 i S x = 2 i S x$⁠. More generally, to multiply two elements of $S 1$ and $S 2 ∈ S U ( 2 )$⁠, one needs to know which physical motion to apply on the ball to perform $I 2 → S 1$ and $I 2 → S 2$ in $SU ( 2 )$⁠. The result of $S 2 × S 1$ is then directly displayed by the LED after applying these motions successively.

Finally, let us mention that $SU ( 2 )$ can also be identified with the set of unit elements of the field of quaternions.¹³ Therefore, the ball can also be used to multiply unitary quaternions. In particular, the multiplication table of the quaternion group (composed of 8 unit quaternions¹⁴) can be visualized by applying sequences of $π$-rotations around the axes $x → , y → , z →$⁠.

As a last question, one may wonder whether the ball could display a “spin-1/3” behavior (or that of any other fractional integer spin) by using a different color encoding, or for instance by inserting a $1 / 3$ instead of a 1/2 in Eq. (12). If one could do so, then $SO ( 3 )$ would admit more than two different homotopy classes of loops (since $I 2$ would admit more than two pre-images). As it is not the case, only integer or half-integer spins can exist in three dimensions. This is of course closely related to the fact that $SU ( 2 )$ has only one homotopy class. The interested reader is invited to perform the experiment and explore the consequences of using $1 / 3$ in place of 1/2 in Eq. (12) to see what happens.

This set of matrices is homeomorphic to the unit circle in the plane, which can, for example, be seen by taking the first row of the matrix as the 2D coordinates of a point. Two loops in $SO ( 2 )$ can then be deformed into one to another if and only if the associated loops in the unit circle have the same algebraic winding number around the origin (with each full counterclockwise turn counting as $+ 1$ and each clockwise turn as $− 1$⁠). The homotopy classes of loops are, therefore, labeled by integers.

Fractional spin of any integer denominator $1 / N$ is, therefore, possible in this context. Such fractional two-dimensional spin can be made at a macroscopic scale, for instance, by placing a flat demonstration screen on a table. Its possible motions are rotations around the axis normal to the table, that is $SO ( 2 )$⁠. The screen displays a single color hue chosen from the circular edge of the colored disc in Fig. 3(a). It is programmed so that when it is rotated by an angle $α$⁠, the hue changes by angle $α / N$⁠. Such an object will behave as one with spin- $1 / N$ in the two-dimensional world of the table's plane. In particular, it will return to its original color after N full turns. More generally, replacing $1 / N$ with any fractional number $p / q$ in the screen's program would give rise to a macroscopic visualization of an object with spin $p / q$⁠. Such demonstration device could be done practically by placing a mobile phone on a table and using a slightly modified version of our online mobile app.

In this article, we have described a macroscopic object that is able to move freely while exhibiting all the characteristics of a spin-1/2 particle. In particular, it exhibits the connection between $SU ( 2 )$ and $SO ( 3 )$ in a visual way and shows how paths can be lifted from the latter to the former once an initial point is fixed. It also demonstrates that spin-1/2 is not a purely quantum feature but rather a topological property of rotation groups that can also be fully visualized at the human scale with the spinorial ball. It could, therefore, be used by students to understand how quantum spin-1/2 particles behave at the microscopic scale.

It would also be interesting to extend this work to the case of two macroscopic spin-1/2 objects. Communication between their driving electronics might be used to simulate entanglement, in a way that is still to be defined, and one could also simulate the failure in Bell's inequalities in this way. We also believe that this device could be used to popularize group theory or quantum mechanics for undergraduate students, as a practical exercise for students in electronics or engineering, or simply as an interesting toy that offers the possibility of opening a gateway to interesting mathematics and physics for any careful observer.

See the supplementary material for the manipulation of the spinorial ball.

The authors would like to acknowledge Emmanuel Fort, Tony Jin, David Martin, and Marc Abboud for insightful discussions and feedback. Parts of this project were developed during a semester program at the Institute for Computational and Experimental Research in Mathematics (ICERM) at Brown University, and at “Les Gustins” Summer School with support of Jean Baud and Ingénieurs et Scientifiques de France – Sillon Alpin (IESF-SA). The authors thank these organizations and acknowledge attendees of these programs for stimulating discussions.

The authors have no conflicts to disclose.