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arxiv: 2003.09236 · v3 · submitted 2020-02-29 · 🧮 math.HO · math.DG

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

Michal Zamboj This is my paper

Pith reviewed 2026-05-24 15:03 UTC · model grok-4.3

classification 🧮 math.HO math.DG
keywords Hopf fibrationdouble orthogonal projectionsynthetic geometry4-dimensional spacestereographic projectionnested torigreat circles3-sphere
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The pith

Double orthogonal projection constructs the fibers of the Hopf fibration synthetically in 4-space.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how to use double orthogonal projection of 4-space onto two perpendicular 3-spaces to build the fibers of the Hopf fibration directly from points on a 2-sphere. This synthetic approach avoids reliance on coordinate-based computer graphics and instead uses geometric constructions. A reader would care because it makes the four-dimensional mapping of circles to points accessible through step-by-step drawing and dynamic models. The result visualizes the fibers as great circles forming nested tori under stereographic projection.

Core claim

The method of double orthogonal projection is used for a direct synthetic construction of the fibers of a 3-sphere from the corresponding points on a 2-sphere. The fibers of great circles on the 2-sphere create nested tori visualized in a stereographic projection onto the modeling 3-space. Each step of the synthetic construction is supported by its analytical representation to highlight connections between the two interpretations.

What carries the argument

Double orthogonal projection of 4-space onto two mutually perpendicular 3-spaces, preserving topological incidence relations to construct Hopf fibers as great circles.

If this is right

  • The constructed fibers remain great circles on the 3-sphere.
  • Nested tori appear in the stereographic projection to 3-space.
  • Dynamic three-dimensional models show the 3-sphere, 2-sphere, and fiber interrelations simultaneously.
  • Analytical representations support each synthetic step to connect the two views.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same projection technique could be tested on other circle bundles or fibrations in 4-space.
  • It offers a route to check topological properties via incidence geometry rather than coordinates alone.
  • Dynamic models built this way may serve as concrete aids for exploring 4D relations that lack 3D analogs.

Load-bearing premise

The double orthogonal projection preserves the topological incidence relations of the Hopf fibration so that the constructed fibers remain great circles without additional embedding assumptions.

What would settle it

A specific constructed fiber that is not a great circle on the 3-sphere or that violates the incidence relation to its base point on the 2-sphere under the projection.

Figures

Figures reproduced from arXiv: 2003.09236 by Michal Zamboj.

Figure 1
Figure 1. Figure 1: Illustration of the Hopf fibration. Two distinct points [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The stereographic images of circular fibers corresponding t [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: M¨obius band with central circle as the base space and segme [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Double orthogonal projection of a 2-sphere [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Apparent contours of a 3-sphere Γ in the double orthogona [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: 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. 3.2 Visualization of a point in R4 and C2 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, … view at source ↗
Figure 7
Figure 7. Figure 7: (a) Circular sections c 1 , . . . , c7 of a 2-sphere with planes γ 1 , . . . , 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 , . . . , c7 1 in the front view and segments c 1 2 , . . . , c7 2 in the top view. (b) Sections σ 1 , . . . , σ7 of a 3-sphere with 3- spaces Σ1 , . . . , Σ … view at source ↗
Figure 8
Figure 8. 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 … view at source ↗
Figure 9
Figure 9. Figure 9: 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 PS2 constructed as the intersection of NP and Ω. Additionally, the spherical section σ from [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Construction of the Hopf fiber c on T 3 corresponding to a point Q on 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 Construc￾tion Protocol window. Use the arrows to move between the steps. From step 1 onward, the user can move the point Q on B 2 and dynamically change all the depende… view at source ↗
Figure 11
Figure 11. Figure 11: Conjugated images of the fibers corresponding to the poin [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. 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 ′ 1 intersect in a point R1 ≡ R′ 1 , but their Ω-images R2 ∈ c1 and R′ 2 ∈ c ′ 2 are distinct. Similarly the point T ′ on c ′ has its Ω image T ′ 2 ≡ R2, but their Ξ-images are distinct. (b) Motion of the point Qm between Q and Q′ . The pink fibers correspond to posi… view at source ↗
Figure 13
Figure 13. Figure 13: A torus in a three-dimensional space generated by revolut [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Construction of the stereographic image cS of the Hopf fiber c (cf [PITH_FULL_IMAGE:figures/full_fig_p022_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: (a) Torus κ on T 3 corresponding to a circle k on 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 T 3 corresponding to a circle m on B 2 with a diameter parallel to z-axis. Again, the torus is generated by fibers c above points Q on m. 5 Hopf tori corresponding to circles on B 2 The geometric nature of the Hopf fibration becomes fully … view at source ↗
Figure 16
Figure 16. Figure 16: (a) Blue and orange points on a circle on [PITH_FULL_IMAGE:figures/full_fig_p024_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: (a) Conjugated images of the tori µ and ν corresponding to the circles m and n, respectively. The circles m, n pass through the points U, V on the diameter of 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… view at source ↗
Figure 18
Figure 18. Figure 18: (a) A family of circles on B 2 parallel to (x, y) and with a diameter parallel to z; (b) the corresponding nested tori on 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 corresponds to the white great circle on B 2 . The visualizations are based on the parametrization in equation (31). 27 [PITH_… view at source ↗
Figure 19
Figure 19. Figure 19: The intersection point of two curves on a 2-sphere [PITH_FULL_IMAGE:figures/full_fig_p028_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: A planar shape created by three circular arcs in the ( [PITH_FULL_IMAGE:figures/full_fig_p029_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Visualization of the 14PolSK-8PSK modulation in the double ort [PITH_FULL_IMAGE:figures/full_fig_p030_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Triangular, tetrahedral, and hexahedral arrangements [PITH_FULL_IMAGE:figures/full_fig_p031_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Octahedral, icosahedral, and dodecahedral arrangemen [PITH_FULL_IMAGE:figures/full_fig_p032_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Vertices of a buckminsterfullerene projected to the base [PITH_FULL_IMAGE:figures/full_fig_p032_24.png] view at source ↗
read the original abstract

The Hopf fibration mapping circles on a 3-sphere to points on a 2-sphere is well known to topologists. While the 2-sphere is embedded in 3-space, four-dimensional Euclidean space is needed to visualize the 3-sphere. Visualizing objects in 4-space using computer graphics based on their analytical representations has become popular in recent decades. For purely synthetic constructions, we apply the recently introduced method of visualization of 4-space by its double orthogonal projection onto two mutually perpendicular 3-spaces to investigate the Hopf fibration as a four-dimensional relation without analogy in lower dimensions. In this paper, the method of double orthogonal projection is used for a direct synthetic construction of the fibers of a 3-sphere from the corresponding points on a 2-sphere. The fibers of great circles on the 2-sphere create nested tori visualized in a stereographic projection onto the modeling 3-space. The step-by-step construction is supplemented by dynamic three-dimensional models showing simultaneously the 3-sphere, 2-sphere, and stereographic images of the fibers and mutual interrelations. Each step of the synthetic construction is supported by its analytical representation to highlight connections between the two interpretations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper claims to deliver a direct synthetic construction of the Hopf fibration's fibers (great circles on S^3) from points on the base S^2 by applying the double orthogonal projection of 4-space onto two mutually perpendicular 3-spaces; the resulting fibers are visualized as nested tori via stereographic projection into modeling 3-space, with each synthetic step accompanied by its analytic representation and supported by dynamic 3D models.

Significance. If the construction is shown to be independent of coordinate verification for the preservation of incidence and great-circle character, the work supplies a new projection-based visualization technique for 4-dimensional topological relations that have no direct lower-dimensional analogue, complementing existing analytic and computer-graphics approaches to the Hopf fibration.

major comments (2)
  1. [Abstract and construction steps] Abstract (paragraph on the method) and the central construction: the claim that the double orthogonal projection 'preserves the topological incidence relations of the Hopf fibration so that the constructed fibers remain great circles' is load-bearing, yet the manuscript appears to verify this property via the supplied analytic representations rather than deriving it solely from incidence axioms internal to the projection; this risks making the construction circular with respect to the target fibration.
  2. [Method description] The choice of the two mutually perpendicular 3-spaces and the handling of potential degeneracies in the projection (e.g., when fibers align with projection directions) are not shown to be parameter-free or axiomatically forced; if these choices are fixed by the analytic model of the Hopf map rather than by synthetic rules alone, the generality of the method is limited.
minor comments (1)
  1. The dynamic models are described as showing 'simultaneous' views of S^3, S^2 and stereographic images, but the manuscript does not specify the software or file formats used to make the models reproducible by readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed review and valuable feedback on our manuscript. We have carefully considered the major comments and provide point-by-point responses below. We believe the construction can be clarified to address the concerns while maintaining the synthetic nature of the approach.

read point-by-point responses

  1. Referee: [Abstract and construction steps] Abstract (paragraph on the method) and the central construction: the claim that the double orthogonal projection 'preserves the topological incidence relations of the Hopf fibration so that the constructed fibers remain great circles' is load-bearing, yet the manuscript appears to verify this property via the supplied analytic representations rather than deriving it solely from incidence axioms internal to the projection; this risks making the construction circular with respect to the target fibration.

    Authors: We agree that the manuscript relies on analytic representations to confirm that the constructed fibers are great circles of the Hopf fibration. The synthetic construction uses the incidence properties of the double orthogonal projection to build the fibers from points on the base, but the identification with the standard Hopf fibration is verified analytically. This is not intended to be circular; the projection method is applied generally, and the analytic part serves to link it to the known fibration. To strengthen the presentation, we will revise the abstract and relevant sections to more clearly separate the synthetic steps from the verification, emphasizing that the preservation follows from the projection's properties as established in the referenced prior work on the method. revision: partial

  2. Referee: [Method description] The choice of the two mutually perpendicular 3-spaces and the handling of potential degeneracies in the projection (e.g., when fibers align with projection directions) are not shown to be parameter-free or axiomatically forced; if these choices are fixed by the analytic model of the Hopf map rather than by synthetic rules alone, the generality of the method is limited.

    Authors: The double orthogonal projection onto two mutually perpendicular 3-spaces is a fixed feature of the visualization method we employ, as introduced in our prior publications on the topic. The specific orientation is chosen to allow clear visualization of the nested tori without excessive overlap, but we acknowledge that the manuscript does not explicitly demonstrate that this choice is the only possible one or fully axiomatically derived without reference to the analytic model. We will add a discussion in the method section explaining the rationale for the choice based on synthetic geometric considerations and address potential degeneracies by noting how the dynamic models avoid or handle them. This will clarify the generality. revision: yes

Circularity Check

0 steps flagged

No significant circularity; synthetic construction remains independent of target result

full rationale

The paper presents a direct synthetic construction of Hopf fibers via double orthogonal projection of 4-space, with each geometric step supported by (but not reduced to) its analytical representation. No equations or claims equate the output fibers or incidence relations to fitted parameters, self-defined quantities, or load-bearing self-citations. The projection method is applied as an external tool to produce the fibration visualization, and the derivation chain does not collapse the claimed result to its inputs by construction. This is the expected non-finding for a geometric construction paper whose central steps are incidence-based rather than algebraic fits.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction rests on standard properties of Euclidean 4-space, orthogonal projections, and the known incidence structure of the Hopf fibration. No free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Orthogonal projections from 4-space onto two mutually perpendicular 3-spaces preserve incidence and distance relations needed for the fibration.
    Invoked when the abstract states that the projection method enables direct synthetic construction of the fibers.
  • standard math The 3-sphere and 2-sphere are embedded in 4-space and 3-space respectively in the standard way.
    Background fact used to define the starting 2-sphere and target 3-sphere.

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