Synthetic construction of the Hopf fibration in a double orthogonal projection of 4-space

Hopf fibration, Hopf tori, four-dimensional visualization, stereographic projection, synthetic construction

Highlights

A Hopf fiber on a 3-sphere corresponding to a point on a 2-sphere is constructed synthetically with only elementary constructions.
Double orthogonal projection is used to visualize the Hopf fibration.
Connection between three-dimensional (3D) orthogonal and stereographic images is constructed.
Interactive 3D models of Hopf fibration and Hopf tori are presented.

1 Introduction

Mathematical visualization is an important instrument for understanding mathematical concepts. While analytical representations are convenient for proofs and analyses of properties, visualizations are essential for intuitive exploration and hypothesis making. Four-dimensional (4D) mathematical objects may lie beyond the reach of our three-dimensional (3D) imagination, but this is not an obstacle to their mathematical description and study. Furthermore, using the modeling tools of computer graphics, we are able to construct image representations of 4D objects to enhance their broader understanding. While many higher dimensional mathematical objects are natural generalizations of lower dimensional ones, the object of our study – the Hopf fibration – does not have this property. The Hopf fibration, introduced by Hopf (1931, 1935), defines a mapping between spheres of different dimensions. In this paper, we restrict ourselves to the correspondence it gives between spheres embedded in 4D and 3D spaces. To each point on a 2-sphere in 3D space is assigned a circular fiber on a 3-sphere in 4D space. The standard method of visualizing the fibers on the 3-sphere is to project them onto a 3D space via stereographic projection, by which we can also grasp the topological properties of the Hopf fibration. Two distinct points on the 2-sphere correspond to disjoint circular fibers on the 3-sphere and their stereographic images are linked circles (Fig. 1). To the points of a circle on the 2-sphere correspond circles on the 3-sphere that form a torus (Fig. 2).

$Illustration of the Hopf fibration. Two distinct points Q and Q′ on the 2-sphere $\mathcal {B}^2$ correspond to circular fibers c and c′.$

Figure 1:

Illustration of the Hopf fibration. Two distinct points Q and Q′ on the 2-sphere |$\mathcal {B}^2$| correspond to circular fibers c and c′.

$The stereographic images of circular fibers corresponding to points on a circle on the base 2-sphere $\mathcal {B}^2$ are linked Villarceau circles on a torus.$

Figure 2:

The stereographic images of circular fibers corresponding to points on a circle on the base 2-sphere |$\mathcal {B}^2$| are linked Villarceau circles on a torus.

Another facet of our interpretation is the method of visualization itself. Instead of visualizing an analytical representation of a given object, we create the object directly in a graphical environment constructively point by point. To do so, we use generalized techniques of classical descriptive geometry to create 3D images of 4D objects. Our constructions or models being technically impossible to create by hand without computer graphics, we use interactive (or dynamic) 3D geometry software in which the images are placed in a virtual 3D modeling space, and the observer is able to reach any point of this space and rotate views by manipulating the viewpoint in the graphical interface. Furthermore, in the models we shall present, the user can interactively manipulate (or animate) some parts or parameters of the model and immediately observe changes in the dependent objects. (Furthermore, by interactive models or visualizations we mean models with dynamically manipulable elements. This should not be confused with time-varying interactivity, often used in 4D visualization.)

Two (not necessarily disjoint) groups might benefit from our paper: computer scientists interested in computer graphics, for whom we present a method of constructive visualization applied to the example of the Hopf fibration; and mathematicians, or students of mathematics, for whom we give a synthetic construction of the Hopf fibration and interactive graphical models to help better understand and intuitively explore it.

1.1 Related work

Construction of the Hopf fibration is usually performed in abstract algebraic language as it involves the 3-sphere embedded in |$\mathbb {R}^4$| or |$\mathbb {C}^2$|⁠. However, recent research in computational geometry and graphics based on analytical representations has made possible partial video animations of the Hopf fibration along with models of stereographic images and visualizations in various software. The front covers of Mathematical Intelligencer, vol. 8, no. 3 and vol. 9, no. 1 featured stereographic images of the Hopf fibration by Koçak and Laidlaw (1987), taken from their pioneering computer-generated film projects with Banchoff, Bisshopp, and Margolis. Banchoff (1990) wrote a comprehensive illustrated book on the fourth dimension, and its front cover has another inspiring picture of the Hopf fibration from his film. Moreover, moving toward the computer visualization of the Hopf fibration, Banchoff (1988) constructed stereographic images of Pinkall’s tori of given conformal type. The primary inspiration for writing this paper was another film – “Dimensions” by Alvarez et al. (2008), in which the Hopf fibration is well explained and visualized in a variety of separate models. Coincidentally, another front cover – of Notices of the AMS, vol. 44, no. 5 – was inspired by explanatory illustrations created in Wolfram Mathematica published by Kreminski (1997) in the context of the structure of the projective extension of real 3-space, |$\mathbb {RP}^3$|⁠. A popular visualization of the Hopf fibration showing points on the base 2-sphere and the corresponding stereographic images of the fibers was created by Johnson (2011) in the mathematics software Sage. Johnson’s code was modified by Chinyere (2012) to visualize a similar fibration with trefoil knots as fibers instead of circles.

The visual aspect is foregrounded by Hanson (2006) (pp. 80–85 and 386–392) in visualizing quaternions on a 3-sphere, and the author also describes them using Hopf fibrations. Visualizations of Hopf fibrations have been beneficial in topology in relation to the Heegaard splitting of a 3-sphere by Canlubo (2017), in the use of quaternions in physics by O’Sullivan (2015), and in the description of motion in robotics by Yershova et al. (2010). Based on analytical representations, Black (2010) in chapter 6 of his dissertation depicts similar orthogonal projections of tori as do we, and supplemented them with animations.

Interactive tools for the visualization of 2D images of 4D objects from different viewpoints in 4-space were developed and applied by Zhou (1991). In another early-stage thesis in the area, Heng (1992) (supervised by Hanson) wrote on the use of interactive mathematical visualization techniques in computer graphics applied to the exploration of 3-manifolds. The continuous development of interactive frameworks and methods of 4D visualization is also apparent in subsequent work co-authored by Hanson (e.g. Hanson et al., 1999; Thakur & Hanson, 2007; Zhang & Hanson, 2007; Chu et al., 2009, the last two papers including visualizations of a flat torus embedded in 4D from different viewpoints).

In this exposition, we use the language and elementary constructions of the double orthogonal projection described in Zamboj (2018a), and the constructions of sections of 4D polytopes, cones, and spheres published in the series of articles in Zamboj (2018b, 2019a, b).

1.2 Contribution

We present an application of a method of visualization using computer graphics for constructing and examining a phenomenon of 4-space that has no analogy in lower dimensions. We use the method of double orthogonal projection to create a purely synthetic graphical construction of the Hopf fibration. This paper contributes to the field in two directions: a novel graphical construction of the Hopf fibration and an application of the double orthogonal projection. In contrast to previous work on visualization of the Hopf fibration, in which separate illustrations created from analytical representations were graphical results or explanatory additions, we use mathematical visualization via the double orthogonal projection as a tool to synthetically construct the fibration. For this purpose, we revisit the analytical definition of the Hopf fibration (points on a circle on a 3-sphere map to a point on a 2-sphere, given by expression 5), which gives us the solved puzzle, and initiate our synthetic approach by decomposing the fibration into pieces. After this, we propose an elementary step-by-step construction of the inverse process. From points on the 2-sphere, we construct fibers of the 3-sphere with the use of only elementary (constructive) geometric tools. On top of that, we construct the resulting stereographic images of the fibers in one complex graphical interpretation. Even though stereographic images are common in visualization of the Hopf fibration, the difference here lies in our synthetic construction of them in order to confirm our results and observations. Finally, the method of visualization is applied to provide a graphical analysis of the properties of cyclic surfaces on a 3-sphere, 4D modulations, and filament packings. Our constructions (Figs 8–12, 14–17) are supplemented by interactive 3D models in GeoGebra 5 [see the online GeoGebra Book; Zamboj, 2019c; the software GeoGebra 5 is used due to its general accessibility, but any other 3D interactive (dynamic) geometry software may be used with the same outcomes. For overall understanding, we strongly recommend following the 3D models simultaneously with the text]. The final visualizations (Figs 18–24) and videos (Suppl. Files 8 and 9) are created in Wolfram Mathematica 11 for better graphical results. Throughout the paper, we give the relevant analytical; background to our synthetic visual approach to enrich overall understanding of the Hopf fibration.

1.3 Paper organization

In Section 2, we introduce the analytical definition and basic properties of the Hopf fibration, which is followed in Section 3 by a brief description of how to depict a 3-sphere in its double orthogonal projection onto two mutually perpendicular 3-spaces and how to construct its stereographic image. In the main part of the paper, Section 4, we give a synthetic construction of a Hopf fiber on a 3-sphere, corresponding to a point on a 2-sphere, using only elementary tools. The resulting double orthogonal projection and stereographic images of tori on the 3-sphere, corresponding to two families of circles on the 2-sphere, are given in Section 5. In Section 6, we apply the method to constructions of cyclic surfaces, visualization of 4D modulations with respect to polyhedral arrangements of vertices on the 2-sphere, and filament packings of the 3-sphere. After concluding with perspectives on future work, an appendix gives the parametrizations used in the figures.

2 Mathematical Background

In algebraic topology, a fibration is a certain type of projection from one topological space (the total space) onto another (the base space) that decomposes the total space into fibers. We proceed to formally define these and other terms that are needed in the sequel.

A topological space is an ordered pair (X, τ), where X is a set and τ is a collection of subsets of X satisfying the following conditions:

The empty set and X belong to τ.
Arbitrary unions of sets in τ belong to τ.
The intersection of any finite number of sets in τ belongs to τ.

τ is called the topology on X of the topological space (X, τ).

Let |$\mathcal {T}$| and |$\mathcal {B}$| be topological spaces and |$f: \mathcal {T} \rightarrow \mathcal {B}$| a continuous surjective mapping. The ordered triple |$(\mathcal {T}, f ,\mathcal {B})$| is called the fiber space or fibration, |$\mathcal {T}$| is the total space, |$\mathcal {B}$| is the base space, and f is the projection [or fibration (The term fibration is sometimes used for the whole triple and sometimes for the component mapping of the triple. For the Hopf fibration, we freely use convention throughout the paper as it should not cause any confusion.)] of the fiber space. The inverse image of a point (element) of the base space |$\mathcal {B}$| in the projection f is the fiber above this point.

A 3-sphere is the 3D boundary of a 4-ball in 4D Euclidean space. It is a natural generalization of a 2-sphere defined as the 2D boundary of a ball in 3D Euclidean space.

Some elementary examples may help illustrate the above definition of a fibration. Hanson (2006), p. 386, gives a shag rug as an informal example of a trivial fibration, where the backing of the rug is the base space and the threads are fibers. Similarly, a spiky massage ball has the 2-sphere as the base space and its spikes are fibers above points of the sphere. The Hopf fibration, defined shortly, will also have the 2-sphere as the base space, the fibers will be circles (not necessarily glued to the base space), but the topology of the total space will be less trivial. An important nontrivial example of a fibration is a Möbius band as the total space (see also Shoemake, 1994), with the central circle as the base space, and fibers being a collection of line segments twisting around the circle (see Fig. 3). The projection maps each point of the Möbius band, which belongs to exactly one of the line segments, to its point on the central circle, and the inverse image of a point on the central circle is the whole line segment.

Figure 3:

Möbius band with central circle as the base space and segments as fibers. Each segment is projected onto its point of intersection with the central circle.

Finally and most importantly for us, the Hopf fibration is a mapping from a 3-sphere (the total space |$\mathcal {T}^3$|⁠; the exponent indicates the dimension of the space) to a 2-sphere (the base space |$\mathcal {B}^2$|⁠) such that each distinct circle (fiber) on the 3-sphere |$\mathcal {T}^3$| corresponds to a distinct point on the 2-sphere |$\mathcal {B}^2$|⁠. In the other direction, the total space |$\mathcal {T}^3$| is formed by circular fibers corresponding to points in the base space (the 2-sphere |$\mathcal {B}^2$|⁠). Furthermore, inverse images of points on the 2-sphere correspond to nonintersecting linked Villarceau circles on a torus in the 3-sphere when stereographically projected (Fig. 2).

An analytical definition of the Hopf fibration in coordinate geometry (expression 5) is given after we have made some preliminary remarks. For an elementary introduction to the Hopf fibration with visualizations, see Lyons (2003), Treisman (2009), and Ozols (2007), and for further details and proofs in modern topological language, see the textbook by Hatcher (2002) (chapter 4).

Let {X_i}_{i ∈ I} be a collection of sets indexed by a set I. The set of functions |$f: I\rightarrow \bigcup _{i\in I}X_i$| such that f(i) ∈ X_i for each i ∈ I is called the Cartesian (or direct) product of the family of sets {X_i}_{i ∈ I}. The Cartesian product is denoted by ∏_{i ∈ I}X_i or X₁ × X₂ × ….

For the purposes of visualization, we follow the direct construction of the Hopf fibration in real 4D space, |$\mathbb {R}^4$|⁠, given for example in Treisman (2009); we also give an alternative construction in complex space useful for simple calculations. As a valuable by-product, we thus obtain visualizations of objects embedded not only in |$\mathbb {R}^4$| but also in |$\mathbb {C}^2$|⁠, where |$\mathbb {C}^2=\mathbb {C}\times \mathbb {C}$| is the Cartesian product of two complex coordinate systems.

Analytical definition of the Hopf fibration. At first, we construct a mapping that projects a point on the 3-sphere |$\mathcal {T}^3$| to a point on the 2-sphere |$\mathcal {B}^2$|⁠. The unit 2-sphere |$\mathcal {B}^2$| in |$\mathbb {R}^3$| is the set of points given by the equation

$$\begin{eqnarray} x^2+y^2+z^2=1. \end{eqnarray}$$

(1)

The unit 3-sphere |$\mathcal {T}^3$| in |$\mathbb {R}^4$| is given by the analogous equation

$$\begin{eqnarray} x^2+y^2+z^2+w^2=1. \end{eqnarray}$$

(2)

Given complex numbers z₁ = x₀ + iy₀ and z₂ = z₀ + iw₀, under the identification of |$\mathbb {C}$| with |$\mathbb {R}^2$| we have z₁ = [x₀, y₀] and z₂ = [z₀, w₀] (the labeling x₀, y₀, z₀, w₀ refers to the location of a point with respect to the coordinate axes x, y, z, w, which will be graphically interpreted later). This way, the unit 3-sphere |$\mathcal {T}^3$| embedded in |$\mathbb {C}^2$| has the equation

$$\begin{eqnarray} |z_1|^2+|z_2|^2=1. \end{eqnarray}$$

(3)

Likewise, setting ζ = x + iy, we can reformulate equation (1) as an embedding of |$\mathcal {B}^2$| in |$\mathbb {C}\times \mathbb {R}$|⁠:

$$\begin{eqnarray} |\zeta |^2+z^2=1. \end{eqnarray}$$

(4)

The mapping

$$\begin{eqnarray} \begin{split} \displaystyle f:\mathbb {C}^2\rightarrow \mathbb {C}\times \mathbb {R}\text{ such that }\\ f(z_1,z_2)=(2z_1\overline{z_2},|z_1|^2-|z_2|^2) \end{split} \end{eqnarray}$$

(5)

is called the Hopf fibration. [At this point, the Hopf fibration is given by its existential definition. Other equivalent definitions of the Hopf fibration as a ratio of complex numbers (see Hatcher, 2002) or via quaternions (see Hanson, 2006) would require a much broader theoretical discourse and would not fit our perspective any better.]

Proposition 1.

The preimage [z₁, z₂] of the point [ζ, z] on |$\mathcal {B}^2\subset \mathbb {C}\times \mathbb {R}$| under the Hopf fibration (5) is on |$\mathcal {T}^3\subset \mathbb {C}^2$|⁠.

Substituting |$\zeta =2z_1\overline{z_2}$| and z = |z₁|² − |z₂|² into equation (4) yields

$$\begin{eqnarray} |2z_1\overline{z_2}|^2+(|z_1|^2-|z_2|^2)^2 &=& 1 \\ 4((z_1\overline{z_2})(\overline{z_1}z_2))+|z_1|^4-2(z_1\overline{z_1})(z_2\overline{z_2})+|z_2|^4 &=& 1 \\ (|z_1|^2+|z_2|^2)^2 &=& 1 \\ \text{and since } |z_1|^2+|z_2|^2\ge 0 \text{, we have }|z_1|^2+|z_2|^2 &=& 1. \end{eqnarray}$$

(6)

According to equation (3), the point [z₁, z₂] lies on |$\mathcal {T}^3$|⁠.

Next, we describe how to create a circle from a point on |$\mathcal {T}^3$| such that its image is a fixed point on |$\mathcal {B}^2$|⁠.

Proposition 2.

The Hopf fibration (5) maps all points of a circular fiber on |$\mathcal {T}^3\subset \mathbb {C}^2$| to one point on |$\mathcal {B}^2\subset \mathbb {C}\times \mathbb {R}$|⁠.

Let [z₁, z₂] be a point on the 3-sphere |$\mathcal {T}^3\subset \mathbb {C}^2$|⁠. Let |$\lambda =l_1+il_2\in \mathbb {C}$| be such that |λ|² = 1; i.e. λ represents a point on a unit circle embedded in |$\mathbb {C}$|⁠. From equation (6), for the point |$[\lambda z_1,\lambda z_2]\in \mathbb {C}^2$| it holds that

$$\begin{eqnarray} |\lambda z_1|^2+|\lambda z_2|^2=|\lambda |^2(|z_1|^2+|z_2|^2)=|z_1|^2+|z_2|^2=1, \text{for each }\lambda . \end{eqnarray}$$

(7)

Hence, the point [λz₁, λz₂] lies on |$\mathcal {T}^3\subset \mathbb {C}^2$|⁠.

Rewriting the point |$[\lambda z_1,\lambda z_2]\in \mathbb {C}^2$| in its parametric representation in |$\mathbb {R}^4$| with λ = l₁ + il₂, z₁ = x₀ + iy₀, and z₂ = z₀ + iw₀, we have

$$\begin{eqnarray} \begin{split} {\begin{pmatrix}\operatorname{Re}(\lambda z_1)\\ \operatorname{Im}(\lambda z_1)\\ \operatorname{Re}(\lambda z_2)\\ \operatorname{Im}(\lambda z_2)\\ \end{pmatrix}}= {\begin{pmatrix}l_1x_0-l_2y_0\\ l_1y_0+l_2x_0\\ l_1z_0-l_2w_0\\ l_1w_0+l_2z_0\\ \end{pmatrix}}. \end{split} \end{eqnarray}$$

(8)

For each |$[l_1,l_2]\in \mathbb {R}^2$|⁠, the last expression defines a set of points |$l_1\overrightarrow{u}+l_2\overrightarrow{v}$| in a plane in |$\mathbb {R}^4$| through the origin O = [0, 0, 0, 0] with the directional vectors |$\overrightarrow{u}=(x_0,y_0,z_0,w_0)$| and |$\overrightarrow{v}=(-y_0,x_0,-w_0,z_0)$|⁠. By equation (7), the set of all points [λz₁, λz₂] for each λ is the intersection of |$\mathcal {T}^3$| and a plane through its center. Hence, the set of all points [λz₁, λz₂] for |$\lambda \in \mathbb {C}$| is a unit circle on |$\mathcal {T}^3$|⁠. By equation (5) defining the Hopf fibration f,

$$\begin{eqnarray} f(\lambda z_1,\lambda z_2) &=& (2\lambda z_1\overline{\lambda z_2},|\lambda z_1|^2-|\lambda z_2|^2)=(2|\lambda |^2 z_1\overline{z_2},|\lambda |^2(|z_1|^2-|z_2|^2)) \\ &=& (2z_1\overline{z_2},|z_1|^2-|z_2|^2)=f(z_1, z_2). \end{eqnarray}$$

(9)

so that f maps [λz₁, λz₂] for |λ|² = 1 to the same point on the unit 2-sphere |$\mathcal {B}^2$| as [z₁, z₂].

Hopf coordinates. In our synthetic reconstruction, we construct the inverse mapping – for a point on |$\mathcal {B}^2$|⁠, we find the circle on |$\mathcal {T}^3$|⁠. The point on |$\mathcal {B}^2$| will be constructed in terms of its polar and azimuthal angles in the representation of |$\mathcal {B}^2$| in spherical coordinates. We thus seek a relation between the spherical coordinates of a point on |$\mathcal {B}^2$| and a point [z₁, z₂] on |$\mathcal {T}^3$| in its trigonometric representation z₁ = r_A(cos α + isin α) and z₂ = r_B(cos β + isin β) for r_A, r_B ≥ 0 and |$\alpha ,\beta \in \mathbb {R}$|⁠. From equation (6), for a point [z₁, z₂] on |$\mathcal {T}^3$| it holds that |$|z_1|^2+|z_2|^2=r_A^2+r_B^2=1$|⁠, and so there exists a unique |$\gamma \in \langle 0,\frac{\pi }{2}\rangle$| such that r_A = cos γ and r_B = sin γ. Then a point on |$\mathcal {T}^3$| has the following coordinates in |$\mathbb {R}^4$|⁠:

$$\begin{eqnarray} {\begin{pmatrix}\cos \gamma \cos \alpha \\ \cos \gamma \sin \alpha \\ \sin \gamma \cos \beta \\ \sin \gamma \sin \beta \end{pmatrix}}, \alpha , \beta \in \langle 0,2\pi ), \gamma \in \Big \langle 0,\frac{\pi }{2}\Big \rangle . \end{eqnarray}$$

(10)

With the use of this representation, the image f(z₁, z₂) of the point [z₁, z₂] under the Hopf fibration (5) has first coordinate

$$\begin{eqnarray} \begin{split} 2z_1\overline{z_2}=2\cos \gamma (\cos \alpha +i\sin \alpha )\sin \gamma (\cos \beta -i\sin \beta )=\\ 2\cos \gamma \sin \gamma (\cos \alpha \cos \beta +\sin \alpha \sin \beta +\\ i(\sin \alpha \cos \beta -\cos \alpha \sin \beta ))= \sin (2\gamma )(\cos (\alpha -\beta )+i\sin (\alpha -\beta )), \end{split} \end{eqnarray}$$

(11)

and second coordinate

$$\begin{eqnarray} \begin{split} |z_1|^2-|z_2|^2=r_A^2-r_B^2=\cos ^2\gamma -\sin ^2\gamma =\cos (2\gamma ). \end{split} \end{eqnarray}$$

(12)

Let 2γ = ψ ∈ 〈0, π〉 and α − β = φ ∈ 〈0, 2π) (taken modulo 2π, and likewise for all further operations with angles). Considering the real and imaginary parts in (11) and the third coordinate in (12), the coordinates of the image f(z₁, z₂) in |$\mathbb {R}^3$| are

$$\begin{eqnarray} \begin{split} {\begin{pmatrix}\sin (2\gamma )\cos (\alpha -\beta )\\ \sin (2\gamma )\sin (\alpha -\beta )\\ \cos (2\gamma ) \end{pmatrix}}= {\begin{pmatrix}\sin \psi \cos \varphi \\ \sin \psi \sin \varphi \\ \cos \psi \end{pmatrix}},\\ \psi \in \langle 0,\pi \rangle ,\varphi \in \langle 0,2\pi ). \end{split} \end{eqnarray}$$

(13)

We thus obtain the spherical coordinates of the image of a point on |$\mathcal {T}^3$|⁠, which lies on the unit 2-sphere |$\mathcal {B}^2$|⁠, and we have also reduced the number of parameters (from three α, β, γ to two φ, ψ). Finally, to construct the preimage of a point on |$\mathcal {B}^2$|⁠, we substitute φ and ψ into equation (10) parametrizing |$\mathcal {T}^3$| in |$\mathbb {R}^4$|⁠, thereby obtaining the Hopf coordinates of a 3-sphere:

$$\begin{eqnarray} {\begin{pmatrix}\cos \frac{\psi }{2}\cos (\varphi +\beta )\\ \cos \frac{\psi }{2}\sin (\varphi +\beta )\\ \sin \frac{\psi }{2}\cos \beta \\ \sin \frac{\psi }{2}\sin \beta \end{pmatrix}}, \beta ,\varphi \in \langle 0,2\pi ),\psi \in \langle 0,\pi \rangle . \end{eqnarray}$$

(14)

Represented in |$\mathbb {C}^2$|⁠, we have

$$\begin{eqnarray} {\begin{pmatrix}z_1\\ z_2 \end{pmatrix}}= {\begin{pmatrix}\cos \frac{\psi }{2}(\cos (\varphi +\beta )+i\sin (\varphi +\beta ))\\ \sin \frac{\psi }{2}(\cos \beta +i\sin \beta ) \end{pmatrix}}. \end{eqnarray}$$

(15)

Let us show that two fibers are disjoint (see also Treisman, 2009).

Proposition 3.

Hopf fibers are disjoint circles.

Let |$\mathcal {T}^3$| be a 3-sphere given by equation (2). Its intersection with the 3-space w = 0 is the (equatorial) 2-sphere |$\mathcal {B}^2$| with equation (1).

Consider a point A on |$\mathcal {T}^3$| with coordinates A[1, 0, 0, 0] in |$\mathbb {R}^4$|⁠. The set of points c_A = λ_AA for |$|\lambda _A|^2=1,\lambda _A\in \mathbb {C}$| is the unit circle defined by the rotation of A by λ_A about the origin in the plane (x, y). Now let B[x_B, y_B, z_B, w_B] be another point on |$\mathcal {T}^3$| and not on the circle c_A through A so that (z_B, w_B) ≠ (0, 0). The set of points c_B = λ_BB for |$|\lambda _B|^2=1,\lambda _B\in \mathbb {C}$| is by equation (7) a unit circle on |$\mathcal {T}^3$| with parametric representation

$$\begin{eqnarray} {\begin{pmatrix}\operatorname{Re}(\lambda _B \sqrt{x_B^2+y_B^2} (x_B+iy_B))\\ \operatorname{Im}(\lambda _B \sqrt{x_B^2+y_B^2}(x_B+iy_B))\\ \operatorname{Re}(\lambda _B \sqrt{z_B^2+w_B^2}(z_B+iw_B))\\ \operatorname{Im}(\lambda _B \sqrt{z_B^2+w_B^2}(z_B+iw_B))\\ \end{pmatrix}}. \end{eqnarray}$$

(16)

The unit circle c_B intersects the equatorial unit 2-sphere |$\mathcal {B}^2$| only if some point on c_B is in the 3-space w = 0. Let us again use the trigonometric representation to show that there are only two intersections of c_B with |$\mathcal {B}^2$| (just as for great circles on a 2-sphere intersecting its equator). Then, |$z_B + iw_B=\sqrt{z_B^2+w_B^2}(\cos \delta +i\sin \delta )$| for δ ∈ 〈0, 2π), and λ_B = cos ϰ + isin ϰ for ϰ ∈ 〈0, 2π). For the point on c_B in the 3-space w = 0, the equation

$$\begin{eqnarray} & \operatorname{Im}(\lambda _B \sqrt{z_B^2+w_B^2}(z_B+iw_B)))= \\ & \operatorname{Im}((\cos \varkappa +i\sin \varkappa )(\sqrt{z_B^2+w_B^2}(\cos \delta +i\sin \delta )))\\ & =\sqrt{z_B^2+w_B^2}\sin (\varkappa +\delta )=0 \end{eqnarray}$$

(17)

has, for (z_B, w_B) ≠ (0, 0), two solutions for λ_B: |$\varkappa =-\delta \mod {2\pi }$| or |$\varkappa =\pi -\delta \mod {2\pi }$| for each δ ∈ 〈0, 2π) . Therefore, c_B has only two antipodal points K₁ and K₂ in common with |$\mathcal {B}^2$|⁠.

Since for z-coordinates of c_B we have

$$\begin{eqnarray} && \operatorname{Re}((\cos \varkappa +i\sin \varkappa )(\sqrt{z_B^2+w_B^2}(\cos \delta +i\sin \delta )))= \\ && \sqrt{z_B^2+w_B^2}\cos (\varkappa +\delta )\ne 0\\ \text{ for }\varkappa &=& -\delta \text{ or }\varkappa =\pi -\delta , \end{eqnarray}$$

(18)

the points K₁ and K₂ on the circle c_B are not in the plane (x, y). Hence, the circles c_A and c_B, which are circular fibers in the Hopf fibration, are disjoint.

As we can rotate any circular fiber of the unit 3-sphere |$\mathcal {T}^3$| to the position of c_A, it follows that all circular fibers on |$\mathcal {T}^3$| are disjoint.

Stereographic projection. A reasonable choice to create a map of an ordinary 2-sphere (e.g. a map of a reference sphere of the Earth) is to take its stereographic projection from the North Pole onto a tangent plane at the South Pole. Despite the distortion of lengths, angles are preserved (the projection is a conformal mapping). As a result, circles on the 2-sphere not passing through the North Pole are projected to circles, and circles through the North Pole become straight lines. In our analogical 4D case, each point of |$\mathcal {T}^3$| is projected via projecting rays through a fixed point of |$\mathcal {T}^3$| – the center of projection – onto a 3-space touching |$\mathcal {T}^3$| at the point antipodal to the center of projection. As this stereographic projection of |$\mathcal {T}^3$| to a 3-space is conformal, all the circular Hopf fibers are projected to circles apart from the fiber through the center of projection, which projects to a line. We have already shown that the inverse images of points on |$\mathcal {B}^2$| in the Hopf fibration are disjoint circles. Moreover, under the stereographic projection, these disjoint fibers are projected onto linked circles (or a line; see Lyons, 2003). Given a circle on |$\mathcal {B}^2$|⁠, its inverse image under the Hopf fibration projects to a family of disjoint circles on |$\mathcal {T}^3$|⁠, and these are projected in the stereographic projection to a torus. Consequently, the stereographic images of all the fibers create nested tori; see Tsai (2006) for Lun-Yi Tsai’s exquisite artistic geometric illustrations.

3 Preliminary Constructions

3.1 Double orthogonal projection

The double orthogonal projection of 4-space onto two mutually perpendicular 3-spaces is a generalization of Monge’s projection of an object onto two mutually perpendicular planes (see Zamboj, 2018a). We briefly describe the orthogonal projection of a 2-sphere onto a plane in order to aid understanding of the double orthogonal projection of a 3-sphere onto two mutually perpendicular 3-spaces described subsequently.

In an orthogonal projection, the contour generator of a 2-sphere is a great circle – the intersection of the polar plane of the infinite viewpoint with respect to the 2-sphere (i.e. the plane perpendicular to the direction of the projection through the center of the 2-sphere) and the 2-sphere itself. The apparent contour of the 2-sphere is also a circle – the orthogonal projection of the contour generator onto the plane of projection (Fig. 4). Therefore, in this 3D case of Monge’s projection, we project a 2-sphere γ to disks γ₁ and |$\gamma ^{\prime }_2$| in two perpendicular planes (x, z) and (x, y′). Then, the plane (x, y′) is rotated about the x-axis into the plane (x, z) (the drawing plane) such that the y′-axis is rotated to form a y-axis that coincides with the z-axis but with opposite orientation.

$Double orthogonal projection of a 2-sphere γ onto the disk γ1 in the 2-space ν(x, z) and onto the disk $\gamma ^{\prime }_2$ in the 2-space π′(x, y′) rotated to γ2 in π(x, y) about the x-axis.$

Figure 4:

Double orthogonal projection of a 2-sphere γ onto the disk γ₁ in the 2-space ν(x, z) and onto the disk |$\gamma ^{\prime }_2$| in the 2-space π′(x, y′) rotated to γ₂ in π(x, y) about the x-axis.

Any point P in the 3-space (x, y, z) is projected orthogonally via its projecting rays to the conjugated image points P₁ (front view) and P₂ (top view after the rotation). The images of projecting rays of P coincide in a single line through P₁ and P₂, called the ordinal line (or line of recall) of P, and it is perpendicular to the x-axis in the drawing plane.

Analogously, we project a 3-sphere in 4-space (Fig. 5) orthogonally onto a 3-space. The contour generator of the 3-sphere is a 2-sphere – the intersection of the polar space of the infinite viewpoint with respect to the 3-sphere (i.e. the 3-space perpendicular to the direction of the projection through the center of the 3-sphere) and the 3-sphere itself. The apparent contour of the 3-sphere is a 2-sphere – the orthogonal projection of the contour generator (2-sphere) onto the 3-space of projection. Thus, in the double orthogonal projection both apparent contours in the 3-spaces Ξ(x, y, z) and Ω′(x, z, w′) of the 3-sphere Γ are 2-spheres Γ₁ and |$\Gamma ^{\prime }_2$|⁠, respectively. Then, we rotate Ω′(x, z, w′) about the plane π(x, z) to the 3-space Ξ(x, y, z) (the modeling 3-space) such that the w′-axis is rotated to form a w-axis that coincides with the y-axis but with opposite orientation. After the rotation, we say that Γ has two conjugated images in the modeling 3-space – the Ξ-image Γ₁ and the Ω-image Γ₂ (rotated |$\Gamma ^{\prime }_2$|⁠).

$Apparent contours of a 3-sphere Γ in the double orthogonal projection. The 2-sphere Γ1 in the 3-space Ξ(x, y, z) and the 2-sphere $\Gamma ^{\prime }_2$ in the 3-space Ω′(x, z, w′) rotated to Γ2 in Ω(x, z, w) about the plane π(x, z).$

Figure 5:

Apparent contours of a 3-sphere Γ in the double orthogonal projection. The 2-sphere Γ₁ in the 3-space Ξ(x, y, z) and the 2-sphere |$\Gamma ^{\prime }_2$| in the 3-space Ω′(x, z, w′) rotated to Γ₂ in Ω(x, z, w) about the plane π(x, z).

3.2 Visualization of a point in |$\mathbb {R}^4$| and |$\mathbb {C}^2$|

A point in 4-space is given by its two conjugated images (Ξ and Ω-image) lying on their common ordinal line, i.e. the line of coinciding rays of projection after the rotation in the modeling 3-space perpendicular to π(x, z). In the other direction, let P be a point with coordinates [x_P, y_P, z_P, w_P] in |$\mathbb {R}^4$|⁠. The Ξ-image P₁[x_P, y_P, z_P] and Ω-image P₂[x_P, z_P, w_P] are synthetically constructed in Fig. 6 using true lengths on the coordinate axes (concretely, in the modeling 3-space with the orthogonal coordinate system (x, y, z) given in Fig. 6, we should say that the images of P have coordinates P₁[x_P, y_P, z_P] and P₂[x_P, −w_P, z_P]. In an implementation, such coordinates would naturally be used if we wanted to construct the images from the parametric representation). The ordinal line of the point P is perpendicular to π(x, z) through P₁ and P₂. Moreover, if we interpret the point P to be in |$\mathbb {C}^2$| with coordinates P[a_P, b_P], then a_P = [x_P, y_P] is on the complex line generated by the real axes x, y, and b_P = [z_P, w_P] is on another complex line generated by z, w. Using the trigonometric representation, we have a_P = [r_Acos α, r_Asin α] and b_P = [r_Bcos β, r_Bsin β] (cf. equation 10).

Conjugated images P1, P2 of a point P(xP, yP, zP, wP) in the double orthogonal projection, and visualization of the complex coordinates of P.

Figure 6:

Conjugated images P₁, P₂ of a point P(x_P, y_P, z_P, w_P) in the double orthogonal projection, and visualization of the complex coordinates of P.

3.3 Visualization of a point on a 3-sphere

In an orthogonal projection of a 2-sphere onto a plane, points on the 2-sphere are projected onto a disk. To locate the position of a point in the image, we can draw its circle of latitude (in a plane perpendicular to the direction of the projection). Analogously, when we orthogonally project a 3-sphere onto a 3-space, we can locate a point on the 3-sphere by drawing its “2-sphere of latitude” (in a 3-space perpendicular to the direction of the projection). In Monge’s projection, planes parallel to the plane (x, z) intersect the plane (x, y) in lines parallel to x. The sections of a 2-sphere with planes parallel to (x, z) in Monge’s projection are shown in Fig. 7a; they create line segments in the top view and circles at their true size in the front view. In other words, if we imagine a 2-sphere passing orthogonally through a plane, their common intersection will be the tangent point extending continuously to the great circle and shrinking back to a point. Analogously, the 3-spaces parallel to the 3-space Ξ(x, y, z) intersect the 3-space Ω(x, z, w) in planes parallel to π(x, z). If a 3-sphere passes orthogonally through a 3-space, their common intersection is the tangent point, extending to the great 2-sphere and shrinking back to a point. Therefore, the intersections of the 3-sphere with a bundle of 3-spaces parallel to Ξ(x, y, z) are 2-spheres (Fig. 7b). Their Ξ-images are circles and Ω-images are 2-spheres at their true size. This construction should give us some insight into the visualization of points on tori inside a 3-sphere that will be carried out later.

$(a) Circular sections c1, …, c7 of a 2-sphere with planes γ1, …, γ7 parallel to (x, z) in Monge’s projection. The planes are given by their intersections pγ with the plane (x, y). The apparent contours of the sections are circles $c_1^1,\dots ,c_1^7$ in the front view and segments $c_2^1,\dots ,c_2^7$ in the top view. (b) Sections σ1, …, σ7 of a 3-sphere with 3-spaces Σ1, …, Σ7 parallel to the 3-space Ξ(x, y, z) are 2-spheres. The 3-spaces are given by their intersections $\omega ^\Sigma$ with the 3-space Ω. The Ξ-images $\sigma _1^1,\dots ,\sigma _1^7$ of the spherical sections are 2-spheres in their true shape and Ω-images $\sigma _2^1,\dots ,\sigma _2^7$ are disks.$

Figure 7:

(a) Circular sections c¹, …, c⁷ of a 2-sphere with planes γ¹, …, γ⁷ parallel to (x, z) in Monge’s projection. The planes are given by their intersections p^γ with the plane (x, y). The apparent contours of the sections are circles |$c_1^1,\dots ,c_1^7$| in the front view and segments |$c_2^1,\dots ,c_2^7$| in the top view. (b) Sections σ¹, …, σ⁷ of a 3-sphere with 3-spaces Σ¹, …, Σ⁷ parallel to the 3-space Ξ(x, y, z) are 2-spheres. The 3-spaces are given by their intersections |$\omega ^\Sigma$| with the 3-space Ω. The Ξ-images |$\sigma _1^1,\dots ,\sigma _1^7$| of the spherical sections are 2-spheres in their true shape and Ω-images |$\sigma _2^1,\dots ,\sigma _2^7$| are disks.

Let P be a point on a 3-sphere Γ (Fig. 8, Suppl. File 1). The point P lies on some 2-sphere σ in a 3-space Σ parallel to Ξ(x, y, z). Since Σ has the same w-coordinate as P, its Ω-image appears as the plane |$\omega ^\Sigma _2$| through P₂ parallel to π(x, z). The intersection of |$\omega ^\Sigma _2$| and Γ₂ is a circle σ₂ that is the boundary of the Ω-image of σ. The Ξ-image σ₁ through P₁ is a 2-sphere concentric with Γ₁ and has radius equal to the radius of σ₂.

$Conjugated images P1 and P2 of a point P on the spherical section σ of a 3-sphere Γ with a 3-space Σ parallel to Ξ(x, y, z). The Ξ-image P1 is on σ1, which is in its true shape. The Ω-image σ2 of the 2-sphere σ is a disk with the same radius as σ1. The Ω-image P2 lies on the disk σ2. In the interactive 3D model https://www.geogebra.org/m/yt27evc8 (or Suppl. File 1), the user can manipulate the position of P1 on σ1 and $\omega _2^\Sigma$ with the “Moving point” on the w-axis and observe the conjugated images of positions of the point P on the 3-sphere.$

Figure 8:

Conjugated images P₁ and P₂ of a point P on the spherical section σ of a 3-sphere Γ with a 3-space Σ parallel to Ξ(x, y, z). The Ξ-image P₁ is on σ₁, which is in its true shape. The Ω-image σ₂ of the 2-sphere σ is a disk with the same radius as σ₁. The Ω-image P₂ lies on the disk σ₂. In the interactive 3D model https://www.geogebra.org/m/yt27evc8 (or Suppl. File 1), the user can manipulate the position of P₁ on σ₁ and |$\omega _2^\Sigma$| with the “Moving point” on the w-axis and observe the conjugated images of positions of the point P on the 3-sphere.

3.4 Stereographic projection of a point on a 3-sphere onto a 3-space

Let us have a unit 3-sphere Γ with the center S = [0, 1, 0, 1] (cf. equation 2):

$$\begin{eqnarray} x^2+(y-1)^2+z^2+(w-1)^2=1. \end{eqnarray}$$

(19)

We project points of the 3-sphere Γ from its point N = [0, 2, 0, 1] onto the 3-space Ω(x, z, w): y = 0 that touches Γ at the antipodal point M = [0, 0, 0, 1]. Let P be a point on Γ with images P₁ and P₂ (Fig. 9, Suppl. File 2). The stereographic image P_S of the point P in Ω(x, z, w) is the intersection of the line NP and Ω(x, z, w). The line N₁P₁ intersects π(x, z) in the point |$P_{S_1}$|⁠. The intersection of the ordinal line through |$P_{S_1}$| and the Ω-image N₂P₂ is the desired point |$P_{S_2}$| that coincides with P_S in 4-space.

$The stereographic image PS of a point P on a 3-sphere Γ projected from the center N to the tangent 3-space Ω(x, z, w). The stereographic image PS lies in the 3-space Ω, so it is also the Ω-image $P_{S_2}$ constructed as the intersection of NP and Ω. Additionally, the spherical section σ from Fig. 8 is stereographically projected to the 2-sphere σS. For the analytic coordinates of PS, see equation (A3). In the interactive 3D model https://www.geogebra.org/m/xth6uszb (or Suppl. File 2), the user can, again, manipulate P1 on σ1 and $\omega _2^\Sigma$ by “Moving point” on w-axis and observe the stereographic images.$

Figure 9:

The stereographic image P_S of a point P on a 3-sphere Γ projected from the center N to the tangent 3-space Ω(x, z, w). The stereographic image P_S lies in the 3-space Ω, so it is also the Ω-image |$P_{S_2}$| constructed as the intersection of NP and Ω. Additionally, the spherical section σ from Fig. 8 is stereographically projected to the 2-sphere σ_S. For the analytic coordinates of P_S, see equation (A3). In the interactive 3D model https://www.geogebra.org/m/xth6uszb (or Suppl. File 2), the user can, again, manipulate P₁ on σ₁ and |$\omega _2^\Sigma$| by “Moving point” on w-axis and observe the stereographic images.

Since stereographic projection is a conformal mapping, we can also conveniently project the 2-sphere σ through the point P described in the previous section. The stereographic image σ_S of the 2-sphere σ is a 2-sphere σ_S with center G_S on the ordinal line through the center S₁ of σ₁. Its equatorial circle g_S is the image of the tangent circle g₁ on the sphere Γ₁ of the cone with the vertex N₁, i.e. the intersection of the polar plane of the pole N₁ with respect to σ₁ and σ₁ itself. Therefore, with the use of a point on g₁, we construct the image circle g_S of the circle g, and the center G_S of g_S is the center of the 2-sphere σ_S – the stereographic image of the 2-sphere σ.

4 Synthetic Construction of a Hopf Fiber

In this section, we illustrate geometrically the mathematical properties of the Hopf fibration given in Section 2 in images under the double orthogonal projection. This way, we synthetically construct a circular fiber on a 3-sphere from a point on a 2-sphere by elementary geometric constructions, thereby liberating it from its analytical description. For the purposes of computation, we used the equations of a 3-sphere |$\mathcal {T}^3$| with the center at the origin, but for the sake of visualization (to differentiate the Ξ and Ω-images) it proves more suitable to use a 3-sphere |$\mathcal {T}^3$| with center [0, 1, 0, 1]. Then, the 2-sphere |$\mathcal {B}^2$| has center with coordinates [0, 1, 0] in Ξ(x, y, z). Such a translation does not influence the properties of the Hopf fibration. This applies for Figs 10–12 and 14–18; the parametric equations corresponding to the visualizations are in the appendix.

$Construction of the Hopf fiber c on $\mathcal {T}^3$ corresponding to a point Q on $\mathcal {B}^2$. With the step-by-step construction https://www.geogebra.org/m/w2kugajz (or Suppl. File 3), the reader can follow the steps of the construction from the text in the Construction Protocol window. Use the arrows to move between the steps. From step 1 onward, the user can move the point Q on $\mathcal {B}^2$ and dynamically change all the dependent elements of the model and the resulting fiber. From step 3 onward, the user can also manipulate the angle β and observe positions of the conjugated images of P and P′ on the fiber. Steps 10 and 11 give the construction of the stereographic projection in Fig. 14.$

Figure 10:

Construction of the Hopf fiber c on |$\mathcal {T}^3$| corresponding to a point Q on |$\mathcal {B}^2$|⁠. With the step-by-step construction https://www.geogebra.org/m/w2kugajz (or Suppl. File 3), the reader can follow the steps of the construction from the text in the Construction Protocol window. Use the arrows to move between the steps. From step 1 onward, the user can move the point Q on |$\mathcal {B}^2$| and dynamically change all the dependent elements of the model and the resulting fiber. From step 3 onward, the user can also manipulate the angle β and observe positions of the conjugated images of P and P′ on the fiber. Steps 10 and 11 give the construction of the stereographic projection in Fig. 14.

Let Q ∈ Ξ(x, y, z) be an arbitrary point on |$\mathcal {B}^2$| (Fig. 10, Suppl. File 3). We will find its Hopf fiber – a circle c on |$\mathcal {T}^3$|⁠. Equations (13), (A1) and (14), (A2) give a relation between the spherical coordinates of the point Q (with parameters φ and ψ) and the Hopf coordinates of the fiber c (with parameters φ, ψ, and β). The construction proceeds by the following steps [see the step-by-step construction https://www.geogebra.org/m/w2kugajz (or Suppl. File 3)]:

Construct any point Q on |$\mathcal {B}^2$|⁠.
Find the angle φ (from equation 13): Construct the plane parallel to (x, y) through Q that cuts |$\mathcal {B}^2$| in a circle. The oriented angle between the radius parallel to the x-axis and the radius terminating in the point Q is the angle φ.
Construct an arbitrary angle β such that we can graphically add it to φ: First, translate φ to φ′ in the plane (x, y) with its vertex in the center of |$\mathcal {B}^2$|⁠, and the initial side in the direction of the non-negative x-axis. Now choose β with the same vertex and initial side in the terminal side of φ′. For implementation, it is enough to choose β on the top semicircle, as the points P and P′ constructed in step 7 dependent on β will be antipodal. Additionally, construct α such that α = φ′ + β (cf. equation 13). The angles α and β are arguments of the complex points a_P = [r_Acos α, r_Asin α] ∈ (x, y) and b_P = [r_Bcos β, r_Bsin β] ∈ (z, w) (see the derivation of z₁ and z₂ above equation 10).
Find the angle ψ (from equation 13): Construct the great circle of |$\mathcal {B}^2$| through Q with a diameter parallel to the x-axis. Choose a radius in the (y, z) plane to be the initial side of the angle ψ with the terminal side being the radius through Q.
Construct the moduli of the points a_P and b_P (cf. equation 10): Let γ be the half-angle of ψ and find its cosine by dropping a perpendicular onto the initial side of ψ. The length |$\cos \frac{\psi }{2}=\cos \gamma$| is the modulus r_A of the point a_P. Similarly, find the length |$\sin \frac{\psi }{2}=\sin \gamma$| on the radius in the direction of the x-axis, which is the modulus r_B of b_P.
Construct points a_P and b_P: Using the moduli and arguments of a_P and b_P, construct them according to Fig. 6.
Construct the Ξ and Ω-images of the point P: Having a_P and b_P, we finalize the images P₁ and P₂ on the parallels to the reference axes (as in Fig. 6).
Construct the antipodal point P′ on |$\mathcal {B}^2$|⁠: Parallel projection preserves central symmetry, and the images |$P^{\prime }_1$| and |$P^{\prime }_2$| are the reflections of P₁ and P₂ about the centers S₁ and S₂ of the images of |$\mathcal {T}^3$|⁠.
Construct the Hopf fiber corresponding to the point Q: The Hopf fiber is a circle c consisting of the locus of points P dependent on β. We have a point construction of P, which can be repeated for different choices of β even though, at this point, we comfortably use the GeoGebra tool to draw the locus. The orthogonal projections of c will appear as ellipses (or circles, or line segments) c₁ and c₂.

Varying β in the interactive applet, P moves on its fiber c. Moving with |$Q\in \mathcal {B}^2$|⁠, the whole fiber c moves on |$\mathcal {T}^3$|⁠.

To illustrate the construction, we consider the images of fibers corresponding to certain points on the base sphere |$\mathcal {B}^2$| on diameters parallel to the x-, y-, and z-axes (Fig. 11).

(With respect to the translated coordinates of the centers of |$\mathcal {B}^2$| and |$\mathcal {T}^3$|⁠)
(a), (b) The fibers of points [0, 1, ±1] (i.e. φ = 0 and ψ = 0, π) are in the planes (x, w) and (y, z), respectively. Thus, their conjugated images are a line segment and a great circle. In particular, for ψ = π the point on the base sphere lies on its fiber.
(c), (d) The fibers of points [± 1, 1, 0] (i.e. φ = 0, π and |$\psi =\frac{\pi }{2}$|⁠) lie in the plane of symmetry of the x- and z-axes and their conjugated images are congruent.
(e), (f) The fibers of points [0, 1 + ±1, 0] (i.e. |$\varphi =\frac{\pi }{2}, \frac{3\pi }{2}$| and |$\psi =\frac{\pi }{2}$|⁠) have their Ξ-images in the plane of symmetry of the y- and z-axes and Ω-images in the plane of symmetry of the x- and w-axes.

Figure 11:

Conjugated images of the fibers corresponding to the point Q in certain positions: (a) Q = [0, 1, −1], (b) Q = [0, 1, 1], (c) Q = [1, 1, 0], (d) Q = [−1, 1, 0], (e) Q= [0, 2, 0], (f) Q = [0, 0, 0].

We have already mentioned that fibers corresponding to two distinct points on the base 2-sphere |$\mathcal {B}^2$| create linked circles. Let Q and Q′ be two points on |$\mathcal {B}^2$| and c and c′ their fibers, respectively. We can easily observe from the conjugated images of c and c′ that the fibers are disjoint, for if c and c′ had a point of intersection R, their conjugated images |$c_1, c^{\prime }_1$| and |$c_2,c^{\prime }_2$| would intersect in the conjugated images R₁, R₂ of the point R. In the case that the conjugated images of a fiber are in a plane perpendicular to π (e.g. Fig. 12a), we must not swap the conjugated images of its points. Furthermore, we should note that the circles are linked on the 3-sphere |$\mathcal {T}^3$|⁠, but this property cannot be validated in the embedding 4-space. Analogously, imagine a point in a circular region on a 2-sphere embedded in 3-space. On the 2-sphere, the point cannot escape the bounding circle. However, if we remove the 2-sphere and leave only the 3-space, the point is not bounded at all. Hence, to establish the fibers’ interlinkedness, we need to understand the topology of the underlying 3-sphere |$\mathcal {T}^3$|⁠. Let us construct a great circle on |$\mathcal {B}^2$| through Q and Q′ and observe the motion of the fiber of a point Q_m on the circle moving from Q to Q′ (Fig. 12b). During this motion, the moving fiber twists along a surface. In fact, the generating fibers are always one of a pair of Villarceau circles around a torus (cf. Fig. 13 with a 3D parallel projection onto a plane of a torus generated by Villarceau circles). The toroidal structure of a 3-sphere will be seen more clearly in Section 5 with the use of stereographic projection.

$(a) Special position of the points Q and Q′ such that the conjugated images of disjoint fibers c and c′ intersect. In the Ξ-image, c1 and $c^{\prime }_1$ intersect in a point $R_1\equiv R^{\prime }_1$, but their Ω-images R2 ∈ c1 and $R^{\prime }_2\in c^{\prime }_2$ are distinct. Similarly, the point T′ on c′ has its Ω image $T^{\prime }_2\equiv R_2$, but their Ξ-images are distinct. (b) Motion of the point Qm between Q and Q′. The pink fibers correspond to positions of Qm on the shorter arc between Q and Q′, and the green fibers correspond to the motion of Qm on the longer arc. In the interactive model https://www.geogebra.org/m/ebjkx8pj (or Suppl. File 4), the user can with the scroll bars dynamically change the positions of Q and Q′ dependent (via equation A1) on parameters (φ, ψ) and (φ′, ψ′), respectively. The point Qm can move freely along the great circle through Q and Q′. The conjugated images of the fibers corresponding to Qm draw their traces in the modeling 3-space.$

Figure 12:

(a) Special position of the points Q and Q′ such that the conjugated images of disjoint fibers c and c′ intersect. In the Ξ-image, c₁ and |$c^{\prime }_1$| intersect in a point |$R_1\equiv R^{\prime }_1$|⁠, but their Ω-images R₂ ∈ c₁ and |$R^{\prime }_2\in c^{\prime }_2$| are distinct. Similarly, the point T′ on c′ has its Ω image |$T^{\prime }_2\equiv R_2$|⁠, but their Ξ-images are distinct. (b) Motion of the point Q_m between Q and Q′. The pink fibers correspond to positions of Q_m on the shorter arc between Q and Q′, and the green fibers correspond to the motion of Q_m on the longer arc. In the interactive model https://www.geogebra.org/m/ebjkx8pj (or Suppl. File 4), the user can with the scroll bars dynamically change the positions of Q and Q′ dependent (via equation A1) on parameters (φ, ψ) and (φ′, ψ′), respectively. The point Q_m can move freely along the great circle through Q and Q′. The conjugated images of the fibers corresponding to Q_m draw their traces in the modeling 3-space.

Figure 13:

A torus in a 3D space generated by revolution of Villarceau circles. All the generating circles are interlinked.

4.1 Construction of the stereographic image of a Hopf fiber

To see the circular structure of the Hopf fibration, we construct the fibers in stereographic projection (Fig. 14). We use the same center of projection and antipodal tangent space Ω(x, z, w) as in Section 3.4. Continuing from the previous construction:

10. Construct a stereographic image of the point P: Let N = [0, 2, 0, 1] be on |$\mathcal {T}^3$| and let the 3-space Ω(x, z, w) be tangent to |$\mathcal {T}^3$| at the point [0, 0, 0, 1]. The intersection of N₁P₁ with π(x, z) is |$P_{S_1}$|⁠. Dropping a perpendicular from |$P_{S_1}$| to the line N₂P₂ gives us the point |$P_{S_2}$| that is also the true stereographic image P_S.
11. Construct a stereographic image of the fiber c: We use the locus tool from GeoGebra to construct a locus of points P_S dependent on the angle β. The stereographic image c_S of c is a circle or a line segment (if c passes through N).

Again, by varying β the point P_S moves on c_S, and by varying the position of |$Q\in \mathcal {B}^2$| the circle c_s changes in such a way that it may cover the whole modeling 3-space. In the following section, we show how to move Q so as to obtain the Hopf tori.

$Construction of the stereographic image cS of the Hopf fiber c (cf. Fig. 9). The fiber cS is constructed as the locus of points PS and $P^{\prime }_S$ dependent on the angle β. Stereographic images are constructed in steps 10 and 11 in the step-by-step construction https://www.geogebra.org/m/w2kugajz (or Suppl. File 3) described in Fig. 10.$

Figure 14:

Construction of the stereographic image c_S of the Hopf fiber c (cf. Fig. 9). The fiber c_S is constructed as the locus of points P_S and |$P^{\prime }_S$| dependent on the angle β. Stereographic images are constructed in steps 10 and 11 in the step-by-step construction https://www.geogebra.org/m/w2kugajz (or Suppl. File 3) described in Fig. 10.

5 Hopf Tori Corresponding to Circles on |$\mathcal {B}^2$|

The geometric nature of the Hopf fibration becomes fully apparent when we visualize the tori of Hopf fibers corresponding to circles on the base 2-sphere |$\mathcal {B}^2$|⁠. With respect to the chosen stereographic projection, we divide the following constructions into two cases. First, we construct the tori on |$\mathcal {T}^3$| in |$\mathbb {R}^4(x,y,z,w)$| corresponding to circles on |$\mathcal {B}^2$| parallel with the plane (x, y) in the 3-space Ξ(x, y, z), and then the tori on |$\mathcal {T}^3$| corresponding to circles on |$\mathcal {B}^2$| with diameter parallel to z. Instead of point-by-point constructions, in the following interactive demonstrations (Suppl. Files 5 and 6) the objects are defined by their parametric representations [see equations (A4) of the circle k on |$\mathcal {B}^2$|⁠, (A5) of the corresponding torus κ, and (A6) of the stereographic image of the torus κ in the appendix], and the user can manipulate the angles φ and ψ using the sliders.

5.1 Hopf torus of a circle parallel to (x, y)

Let Q be a point on |$\mathcal {B}^2$| in Ξ(x, y, z) and k a circle parallel to (x, y) through the point Q (Fig. 15a, Suppl. File 5). From equation (13), we have the parametric coordinates of points on the circle k given by the angle φ′ for a fixed ψ:

$$\begin{eqnarray} k(\varphi ^{\prime })={\begin{pmatrix}\sin \psi \cos \varphi ^{\prime }\\ \sin \psi \sin \varphi ^{\prime }\\ \cos \psi \end{pmatrix}}, \varphi ^{\prime }\in \langle 0,2\pi ). \end{eqnarray}$$

(20)

Varying the angle β′ (positions of P on c), from equation (14) and the angle φ′ (positions of Q on k corresponding to distinct fibers c) we obtain the parametrization of a torus κ covered by the fibers on |$\mathcal {T}^3$|⁠:

$$\begin{eqnarray} \kappa (\beta ^{\prime },\varphi ^{\prime })={\begin{pmatrix}\cos \frac{\psi }{2}\cos (\varphi ^{\prime }+\beta ^{\prime })\\ \cos \frac{\psi }{2}\sin (\varphi ^{\prime }+\beta ^{\prime })\\ \sin \frac{\psi }{2}\cos (\beta ^{\prime })\\ \sin \frac{\psi }{2}\sin (\beta ^{\prime }) \end{pmatrix}}, \beta ^{\prime },\varphi ^{\prime } \in \langle 0,2\pi ). \end{eqnarray}$$

(21)

Figure 16a shows the double orthogonal projection of the torus κ and its generating circles corresponding to points on the circle k with their stereographic images.

$(a) Torus κ on $\mathcal {T}^3$ corresponding to a circle k on $\mathcal {B}^2$ parallel to the (x, y) plane. The torus is generated by fibers c above points Q along the circle k. (b) Torus μ on $\mathcal {T}^3$ corresponding to a circle m on $\mathcal {B}^2$ with a diameter parallel to the z-axis. Again, the torus is generated by fibers c above points Q on m.$

Figure 15:

(a) Torus κ on |$\mathcal {T}^3$| corresponding to a circle k on |$\mathcal {B}^2$| parallel to the (x, y) plane. The torus is generated by fibers c above points Q along the circle k. (b) Torus μ on |$\mathcal {T}^3$| corresponding to a circle m on |$\mathcal {B}^2$| with a diameter parallel to the z-axis. Again, the torus is generated by fibers c above points Q on m.

$(a) Blue and orange points on a circle on $\mathcal {B}^2$ parallel to the (x, y) plane and the corresponding family of circular fibers on $\mathcal {T}^3$ generating a torus. The stereographic projection of this torus onto Ω(x, z, w) is a torus of revolution. (b) Blue and orange points on a circle on $\mathcal {B}^2$ with a diameter parallel to the z-axis and their fibers. One of the fibers passes through the center of stereographic projection, so its image is a line. In the interactive models (a) https://www.geogebra.org/m/n4xg3sw6 (or Suppl. File 5) and (b) https://www.geogebra.org/m/vrasywpt (or Suppl. File 6), the user can vary the spherical coordinates of Q in equation (A1) and interactively change its circle on $\mathcal {B}^2$ and the corresponding torus. The user can also turn off the visibility of the objects on $\mathcal {B}^2$, the conjugated images in the double orthogonal projection, or the stereographic images.$

Figure 16:

(a) Blue and orange points on a circle on |$\mathcal {B}^2$| parallel to the (x, y) plane and the corresponding family of circular fibers on |$\mathcal {T}^3$| generating a torus. The stereographic projection of this torus onto Ω(x, z, w) is a torus of revolution. (b) Blue and orange points on a circle on |$\mathcal {B}^2$| with a diameter parallel to the z-axis and their fibers. One of the fibers passes through the center of stereographic projection, so its image is a line. In the interactive models (a) https://www.geogebra.org/m/n4xg3sw6 (or Suppl. File 5) and (b) https://www.geogebra.org/m/vrasywpt (or Suppl. File 6), the user can vary the spherical coordinates of Q in equation (A1) and interactively change its circle on |$\mathcal {B}^2$| and the corresponding torus. The user can also turn off the visibility of the objects on |$\mathcal {B}^2$|⁠, the conjugated images in the double orthogonal projection, or the stereographic images.

The conjugated images κ₁ and κ₂ of this torus are parts of cylindrical surfaces of revolution in |$\mathcal {T}_1^3$| and |$\mathcal {T}_2^3$|⁠. This is a straightforward consequence of the relationship between the point Q and the angle φ′. From equation (21) with a fixed ψ, we obtain the Ξ-image κ₁ as a part of a cylindrical surface of revolution about the axis parallel to z in the 3-space Ξ(x, y, z). Similarly, the Ω-image κ₂ is a part of a cylindrical surface of revolution about the axis parallel to x in the 3-space Ω(x, z, w).

5.2 Hopf torus of a circle with diameter parallel to z

Let Q be a point on |$\mathcal {B}^2$| and m a circle with a diameter parallel to z through the point Q (Fig. 15b, Suppl. File 6). The circle m has a parametric representation for a fixed angle φ and variable ψ′ (from equation 13):

$$\begin{eqnarray} m(\psi ^{\prime })={\begin{pmatrix}\sin \psi ^{\prime }\cos \varphi \\ \sin \psi ^{\prime }\sin \varphi \\ \cos \psi ^{\prime } \end{pmatrix}}, \psi ^{\prime }\in \langle 0,\pi \rangle . \end{eqnarray}$$

(22)

Varying the angle ψ′ (positions of Q on m) changes the moduli |$r^{\prime }_A$| and |$r^{\prime }_B$| of P on the corresponding fiber c (from equation 14). More precisely, |$r^{\prime }_A=\cos \gamma ^{\prime }=\cos \frac{\psi ^{\prime }}{2}$| and |$r^{\prime }_B=\sin \gamma ^{\prime }=\sin \frac{\psi ^{\prime }}{2}$| with the fixed angle φ induce a family of non-intersecting circular fibers c(β′) generating a torus:

$$\begin{eqnarray} \mu (\beta ^{\prime },\psi ^{\prime })={\begin{pmatrix}\cos \frac{\psi ^{\prime }}{2}\cos (\varphi +\beta ^{\prime })\\ \cos \frac{\psi ^{\prime }}{2}\sin (\varphi +\beta ^{\prime })\\ \sin \frac{\psi ^{\prime }}{2}\cos (\beta ^{\prime })\\ \sin \frac{\psi ^{\prime }}{2}\sin (\beta ^{\prime }) \end{pmatrix}}, \beta ^{\prime }\in \langle 0,2\pi ),\psi ^{\prime }\in \langle 0,\pi \rangle . \end{eqnarray}$$

(23)

In contrast to the previous case, in which the torus κ was generated by the parameter φ′, on which the first two coordinates depend, the torus μ is generated by the parameter ψ′, on which all four coordinates depend. Therefore, the conjugated images of the torus μ are more twisted. For the choice ψ′ = 0 and |$\beta ^{\prime }=\frac{\pi }{2}-\varphi$|⁠, we always obtain the point [0, 1, 0, 0] lying on the torus μ. This is the center N of the stereographic projection before applying the translation for visualization purposes, so the torus μ always contains a point that is stereographically projected to infinity. Moreover, it contains a fiber through this point, too. The stereographic image of this fiber is a line. See Fig. 16b for the full illustration of the double orthogonal projection of the torus covered by its circles and their stereographic images. Let μ and ν be tori generated by great circles m and n with a diameter parallel to the z-axis (Fig. 17). The circles m and n intersect in antipodal points U and W, which correspond to fibers u and v, respectively. These fibers were depicted earlier in Fig. 11a and b. Therefore, the tori μ and ν have two common fibers such that one conjugated image is always a segment and the second image is a great circle. By the above-mentioned property of the fiber through the center of stereographic projection (u in Fig. 17), the stereographic images μ_S and ν_S of the tori have a common line u_S.

$(a) Conjugated images of the tori μ and ν corresponding to the circles m and n, respectively. The circles m and n pass through the points U and V on the diameter of $\mathcal {B}^2$ parallel to the z-axis. The fibers u and v above points U and V, respectively, lie on both tori. (b) Stereographic images μS and νS of the tori. The stereographic images uS and vS of the fibers u and v are the intersecting line and circle of μS and νS. In the interactive model https://www.geogebra.org/m/k94dvfpx (or Supp. File 7), the user can manipulate the circles m and n by varying the parameter φ in equation (A4). The projections of tori μ and ν vary dependently on m and n. The visibility of the objects on $\mathcal {B}^2$, the conjugated images in the double orthogonal projection, or the stereographic image can be turned off.$

Figure 17:

(a) Conjugated images of the tori μ and ν corresponding to the circles m and n, respectively. The circles m and n pass through the points U and V on the diameter of |$\mathcal {B}^2$| parallel to the z-axis. The fibers u and v above points U and V, respectively, lie on both tori. (b) Stereographic images μ_S and ν_S of the tori. The stereographic images u_S and v_S of the fibers u and v are the intersecting line and circle of μ_S and ν_S. In the interactive model https://www.geogebra.org/m/k94dvfpx (or Supp. File 7), the user can manipulate the circles m and n by varying the parameter φ in equation (A4). The projections of tori μ and ν vary dependently on m and n. The visibility of the objects on |$\mathcal {B}^2$|⁠, the conjugated images in the double orthogonal projection, or the stereographic image can be turned off.

5.3 Nested Hopf tori corresponding to families of circles on |$\mathcal {B}^2$|

We summarize the toroidal structure of a 3-sphere in the following visualizations. For each circle k parallel to the (x, y) plane on the 2-sphere |$\mathcal {B}^2$|⁠, we obtain a torus κ. Figure 18a gives a model (see Suppl. File 8 for a video animation) of nested tori κ on |$\mathcal {T}^3$| corresponding to circles k on |$\mathcal {B}^2$|⁠. This family of disjoint tori contains only one fiber through the center of the stereographic projection, and hence the tori appear in the stereographic projection as nested tori of revolution including one line, which is their axis.

$(a) A family of circles on $\mathcal {B}^2$ parallel to (x, y) and with a diameter parallel to z; (b) the corresponding nested tori on $\mathcal {T}^3$, and their stereographic projection onto Ω(x, z, w). Colors (shades) refer to mutually related objects; in (b) the images of the torus highlighted in red correspond to the white great circle on $\mathcal {B}^2$. The visualizations are based on the parametrization in equation (A7).$

Figure 18:

(a) A family of circles on |$\mathcal {B}^2$| parallel to (x, y) and with a diameter parallel to z; (b) the corresponding nested tori on |$\mathcal {T}^3$|⁠, and their stereographic projection onto Ω(x, z, w). Colors (shades) refer to mutually related objects; in (b) the images of the torus highlighted in red correspond to the white great circle on |$\mathcal {B}^2$|⁠. The visualizations are based on the parametrization in equation (A7).

In the second case (Fig. 18b, Suppl. File 9), the family of circles m on |$\mathcal {B}^2$| with a diameter parallel to the z-axis forms nested tori μ on |$\mathcal {T}^3$|⁠. Each of these tori contains two common fibers in the special positions shown in Fig. 17. The stereographic image of one of the fibers is a line, which is the common line for all the stereographic images of the tori.

These families of tori cover the 3-sphere |$\mathcal {T}^3$| reparametrized by variables β′, φ′, and ψ′ as

$$\begin{eqnarray} \begin{split} \mathcal {T}^3(\beta ^{\prime },\varphi ^{\prime },\psi ^{\prime })={\begin{pmatrix}\cos \frac{\psi ^{\prime }}{2}\cos (\varphi ^{\prime }+\beta ^{\prime })\\ \cos \frac{\psi ^{\prime }}{2}\sin (\varphi ^{\prime }+\beta ^{\prime })\\ \sin \frac{\psi ^{\prime }}{2}\cos (\beta ^{\prime })\\ \sin \frac{\psi ^{\prime }}{2}\sin (\beta ^{\prime }) \end{pmatrix}}, \\ \beta ^{\prime },\varphi ^{\prime } \in \langle 0,2\pi ), \psi ^{\prime }\in \langle 0,\pi \rangle . \end{split} \end{eqnarray}$$

(24)

6 Further Applications

In the last section, we provide a few brief references to geometrically challenging applications of the Hopf fibration.

6.1 Cyclic surfaces

Using the above-mentioned method of visualization and construction of the Hopf fibration, we can study related properties of geometric surfaces and design shapes that are formed by disjoint circles. If we consider a point moving along an arbitrary curve on the 2-sphere |$\mathcal {B}^2$|⁠, the motion of the corresponding Hopf fiber creates a cyclic surface consisting of disjoint circles of variable radius on the 3-sphere |$\mathcal {T}^3$| embedded in |$\mathbb {R}^4$|⁠. Consequently, two curves intersecting in one point on a 2-sphere create two cyclic surfaces with only one common circle (the Hopf fiber of the point of intersection; Fig. 19a). Stereographic projection preserves circles (up to a circle through the center of projection) and so the stereographic images are cyclic surfaces intersecting in a circle in |$\mathbb {R}^3$|⁠, too (Fig. 19b).

$The intersection point of two curves on a 2-sphere $\mathcal {B}^2$. The corresponding fibers in the Hopf fibration form two cyclic surfaces on the 3-sphere $\mathcal {T}^3$ intersecting in one circle. The situation is visualized in the double orthogonal projection (a) and stereographic projection (b).$

Figure 19:

The intersection point of two curves on a 2-sphere |$\mathcal {B}^2$|⁠. The corresponding fibers in the Hopf fibration form two cyclic surfaces on the 3-sphere |$\mathcal {T}^3$| intersecting in one circle. The situation is visualized in the double orthogonal projection (a) and stereographic projection (b).

Finally, we can construct orthogonal and stereographic images of shapes consisting of cyclic surfaces or their parts connected by common circles. The case in Fig. 20 shows a union of three circular arcs (the vertices are tangent points of the corresponding circles) stereographically projected from the (x, z) plane onto a 2-sphere |$\mathcal {B}^2$|⁠. Then, we apply the inverse Hopf projection and construct conjugated images of the corresponding surfaces in |$\mathcal {T}^3$| in the double orthogonal projection. After the stereographic projection from |$\mathcal {T}^3$| to the 3-space Ω(x, z, w), we obtain a 3D model of the shape as the union of parts of cyclic surfaces. The common points of each pair of circular arcs become the common circles of each pair of parts of the cyclic surfaces.

$A planar shape created by three circular arcs in the (x, z) plane is stereographically projected onto the 2-sphere $\mathcal {B}^2\in \mathbb {R}^3$. The corresponding fibers in the Hopf fibration into $\mathcal {T}^3\in \mathbb {R}^4$ form three connected parts of cyclic surfaces, each two connected by a common circle. A part of the shape is depicted in (a) the double orthogonal projection and (b) the stereographic projection. The whole shape is visualized in (c) the double orthogonal projection and (d) the stereographic projection.$

Figure 20:

A planar shape created by three circular arcs in the (x, z) plane is stereographically projected onto the 2-sphere |$\mathcal {B}^2\in \mathbb {R}^3$|⁠. The corresponding fibers in the Hopf fibration into |$\mathcal {T}^3\in \mathbb {R}^4$| form three connected parts of cyclic surfaces, each two connected by a common circle. A part of the shape is depicted in (a) the double orthogonal projection and (b) the stereographic projection. The whole shape is visualized in (c) the double orthogonal projection and (d) the stereographic projection.

6.2 4D modulations

The Hopf fibration was used as a constructive tool in classical optical communications to design 4D modulations in Rodrigues et al. (2018). From the geometric point of view, nPolSK-mPSK modulations are constructed by n vertices of a polyhedron inscribed in the base 2-sphere generating n fibers in the 3-sphere, and each fiber contains m points. The authors demonstrated their main results on 14PolSK-8PSK modulation generated by tetrakis hexahedron, which is a union of a hexahedron and an octahedron with 14 vertices. The visualization of this arrangement supplemented with its double orthogonal projection is in Fig. 21.

Figure 21:

Visualization of the 14PolSK-8PSK modulation in the double orthogonal projection and stereographic projection given by a tetrakis hexahedron with 14 vertices.

6.3 Twisted filaments

Our final application is inspired by twisted toroidal structures appearing in biological and synthetic materials. In this context, Atkinson et al. (2019), Grason (2015), and Kléman (1985) studied the problem of twisted filament packings. Geometric models are constructed with the use of the Hopf fibration, in which fibers play the role of filament backbones. To construct equally spaced filaments in the 3-sphere |$\mathcal {T}^3$|⁠, we consider equally spaced disks on the base 2-sphere |$\mathcal {B}^2$|⁠. Visualizations of such arrangements based on the vertices of Platonic solids (and triangular case) projected to the 2-sphere |$\mathcal {B}^2$| in the double orthogonal projection and also stereographic projection are in Figs 22 and 23. Tangent points of the circles on |$\mathcal {B}^2$| correspond to tangent circles along the filaments in |$\mathcal {T}^3$|⁠. Consequently, the number of neighboring filaments in |$\mathcal {T}^3$| is the same as the number of neighboring disks in |$\mathcal {B}^2$|⁠.

Figure 22:

Triangular, tetrahedral, and hexahedral arrangements of vertices on the base 2-sphere and their corresponding twisted filaments with backbones visualized in the double orthogonal projection and in the stereographic projection.

Figure 23:

Octahedral, icosahedral, and dodecahedral arrangements of vertices on the base 2-sphere and their corresponding twisted filaments with backbones visualized in the double orthogonal projection and in the stereographic projection.

As an example of more complex structure of filaments packing, we chose buckminsterfullerene with 60 vertices, 12 pentagonal, and 20 hexagonal faces (Fig. 24).

Figure 24:

Vertices of a buckminsterfullerene projected to the base 2-sphere, their corresponding filament backbones in the double orthogonal projection, and a close-up to stereographic images of the corresponding twisted filaments.

https://iopscience.iop.org/article/10.1088/1367-2630/ab1c2d.

7 Conclusion

By the use of elementary constructive tools in the double orthogonal projection of 4-space onto two mutually perpendicular 3-spaces rotated into one 3D modeling space, we have described a synthetic step-by-step construction of a Hopf fiber on a 3-sphere embedded in 4-space that corresponds to a point on a 2-sphere. The virtual modeling space is accessible in supplementary interactive models created in the interactive 3D geometric software GeoGebra 5, in which the reader can intuitively manipulate fundamental objects and achieve a sense of the fourth dimension through two interlinked 3D models. The choice of the method of visualization plays a significant role in several aspects. First, two conjugated 3D images of a 4D object carry all the necessary information to determine this object uniquely. For example, if we had only one image of parallel sections of a 3-sphere in Fig. 7b, we would miss important details for reconstruction. The interpretation of two 3D images as one object indeed assumes some training and experience. However, in the case of projection onto a plane, we would need at least three images for a visual representation. Another advantage of the double orthogonal projection is that synthetic constructions generalize constructions in Monge’s projection. Thus, for example, the localization of a point on a 3-sphere or constructions of stereographic images are elementary. Furthermore, with the use of this technique, we have visualized a torus formed by the Hopf fibers corresponding to a circle on the 2-sphere and shown it in two different positions of the circles on the 2-sphere. The Hopf fibration is usually visualized in the stereographic projection based on an analytical representation. Since intuition in 4D visualization, including the double orthogonal projection, is often misleading, stereographic images, in this case, support our reasoning. Therefore, the tori were projected to synthetically constructed stereographic images in the modeling 3-space, revealing the true nature of the Hopf fibration. The final visualizations show how the points of the 2-sphere cover the 3-sphere by their corresponding fibers on the nested tori, and the whole modeling 3-space when stereographically projected. In this way, we have built a mathematical visualization of the Hopf fibration in which we have presented a constructive connection between the base space (a 2-sphere), fibers covering the total space (a 3-sphere), and the stereographic images, and through which we can study and explain properties of the Hopf fibration in a way that does not depend on its analytical description. However, for the purposes of verification and implementation, the objects are supported by their parametric representations used in the classical analytical approach.

We finished by giving a short application of the Hopf fibration for constructing shapes on the 3-sphere and in stereographic projection in 3-space. The method provided promises future applications in visualizing other curves on a 2-sphere and their corresponding surfaces generated by their Hopf fibers on a 3-sphere in 4-space, along with their corresponding stereographic images in the modeling 3-space. The double orthogonal projection method is also likely to be useful for visualizing and analysing the properties of further 3-manifolds embedded in 4-space.

Acknowledgments

I wish to thank Lada Peksová (Charles University) for her help with the topological aspects and Filip Beran (Charles University) for revision and valuable suggestions. I also thank the anonymous referees for their valuable suggestions, which helped to improve the exposition. The work was supported by the grant Horizon 2020 MaTeK Enhancement of research excellence in Mathematics Teacher Knowledge (No 951822).

Conflict of interest statement

None declared.

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