Demonstration model of the Higgs boson SU(2) -> SU(2)³ U(1) / U(1)² Symmetry breaking using an origami one-stone structure

Abstract

Demonstration model for the didactic illustration of group-theoretical symmetry structures, in particular of SU(2) modulations, comprising an asymmetric, aperiodic one-stone tile (monotile) according to Smith (2024), characterized by: • an interior angle structure according to 90° − 120° − ( 90° − 120° − 180° − 240° ) − 120° − 90° − 120° − 270° − 120° − 120° − 90° − 240° − 90° − 240° , • an edge length sequence according to 3 − 3 − 1 ( 1 − 1 − 2 ) − 1 − 3 − 3 − 1 − 2 − 1 − 3 − 3 − 1 − 1, where the sections in parentheses complement the previously known aperiodic monotile and allow the representation of three separate SU(2) modulation ranges.

Description

[0001] The Standard Model of particle physics is based on the gauge symmetry SU(3) C × SU(2) L × U(1) Y which describes the fundamental interactions (with the exception of gravity). Starting from this explicit symmetry, didactic reductions can be developed that illustrate essential principles in low-dimensional models. In particular, the breaking of SU(2) symmetry is associated with the Higgs boson. This SU(2) symmetry can be intuitively compared to the topology of a Möbius strip. However, it is extremely difficult for students and learners to imagine such symmetries and their breaking. Without a concrete geometric or physical illustration, the underlying principles of modern particle physics remain abstract and difficult to access.

[0002] The task, therefore, is to provide a means by which symmetries and their refraction can be experienced visually and haptically.

[0003] The invention relates to a mechanical demonstration model that physically illustrates the spontaneous symmetry breaking of a reduced SU(2)-based group architecture. State of the art

[0004] As early as 1865, Möbius, in his investigation into determining the contents of polyhedra, described the possibility of closing a strip of paper not only into a two-sided cylinder, but also, with a half turn, into a one-sided zone. This construction represents the first known description of what was later called a "Möbius strip".

[0005] Furthermore, Lemaitre (1927), Jackiw (1985), and Teitelboim (1983) have shown that complex structures can be described by low-dimensional geometries. Lemaitre illustrated a two-dimensional spacetime on the surface of a sphere in his cosmological model, while Jackiw and Teitelboim developed consistent models of gravity in (1 + 1) dimensions.

[0006] Furthermore, Lemaître (1931) introduced the concept of the "primeval atom," which was considered the origin of the expanding universe. This "primeval atom" represents an independent cosmogonic hypothesis that complements the historical line of physical illustrations and at the same time suggests that future developments in particle physics would offer new insights into the question of origins.

[0007] Dirac (1958, Chapter 11, pp. 262 ff.) demonstrated in this sense, for more modern particle physics, that spinor states change sign under a 2π rotation and only reappear after 4π. The so-called "plate trick" demonstration, as described by Feynman et al. (1965, Vol. III, Chapter 11, §11-4, pp. 11-6 ff.), illustrates this fact. A more modern presentation can be found in Penrose (2004, Chapter 11, pp. 220 ff.), where the belt trick is explicitly explained as an illustration of the double superposition SU(2) → SO(3).

[0008] In addition, the tile system for aperiodic surface covering patented by Penrose (1979) shows that fivefold rotational symmetry can be physically realized even without translational symmetry. The Penrose tiles thus demonstrate the possibility of geometrically encoding local symmetry principles without requiring global periodicity. This forms a conceptual bridge to the representation of nontrivial modulation structures within closed SU(2) cells.

[0009] A recent example of the physical realization of the so-called "belt trick" can be found in one-dimensional spin models. Rahul and Murugesh (2019) show that in a Heisenberg ferromagnet chain, a helicoidal spin structure can be continuously transformed into a configuration twisted by 2π, with identity only being reached after 4π. This mechanism illustrates the simple connected structure of SU(2) and leads to the splitting of the configuration space into two topological sectors with different energy gaps.

[0010] Furthermore, Wochnowski (2019) derived an inverse harmonic potential by modifying the one-dimensional Kronig-Penney model, which formally corresponds to the "Mexican hat" potential of the Higgs mechanism. The one-dimensionality is essential here, as it allows for a clear didactic reduction and simultaneously illustrates how continuous symmetry is maintained in the Lagrangian form but spontaneously broken in the ground state.

[0011] Kalmbach (2021) has provided didactic illustrations of symmetry breaking models S4 → D3 and CPT → Z2 × Z2 in connection with quark and Higgs geometry, and others using the so-called MINT-WIGRIS model (among others, Kalmbach, 2019). In these models, the symmetry product U(1) × SU(2) × SU(3) of the Standard Model is embedded in a projective geometry that includes Euclidean space as a special case, but goes beyond it. The haptic-optical models serve to illustrate projective, group-theoretical relations. Using the so-called MINT-WIGRIS experiment kit, a haptic-optical model is developed that employs plug-in elements, light chains, and projections to vividly demonstrate the interactions in the deuteron. The models consist of rotors, octahedrons and membranes, which can be assembled by learners themselves and are supplemented by videos.This provides an optically and mechanically perceptible understanding of nuclear forces and symmetry processes.

[0012] These works demonstrate that both topological sectorizations and spontaneous symmetry breaking can be visualized in simple models. However, a mechanical demonstration model that physically links these ideas to SU(2) symmetry breaking has not yet been disclosed. invention

[0013] The model according to the invention builds on this tradition by unfolding a two-dimensionally Möbius-like folded pentagonal structure in three-dimensional space, thus making the breaking of an SU(2) symmetry physically perceptible. The starting point is a pure SU(2) symmetry, which—unlike in the standard model—is not located in a flat Minkowski space between SU(3) C and U(1) Y is embedded. Instead, it exists independently in the (1 + 1) JT-metric, as proposed by Kiesewetter-Köbinger (2025a) in his shell of Higgs-Bose-Einstein condensate with a pure non-abelian SU(2) metric. In contrast to classical Kaluza-Klein compactifications [cf. Kaluza (1921); Klein (1926); Appelquist et al. (1987)] and holographic boundary projections in the sense of AdS / CFT [cf. Maldacena (1999); Gubser et al. (1998); Witten (1998)], an independent, non-abelian SU(2) metric is developed here, comparable to the nonlinear HEFT structure according to Krause (2016). Only this independence allows essential properties of highly complex particle physics to be represented with a technically simple symmetry field. This eliminates the need for complex technical projection structures, in which the metric tensor living in the SU(2) topology would have to be mapped to a tensor defined in the parallel conceived Minkowski spacetime.This also simplifies the didactic concept, because learners only have to concentrate on one metric.

[0014] The body according to the invention is formed from a 13-sided, flat, and asymmetrical shape developed by Smith et al. (2024), which is also referred to in the literature as the "Einstein tile". This Einstein tile can be divided into 16 superimposed right-angled triangles with angles of 30°, 60°, and 90° and side ratios by just two incisions. (1:3:2) Fold it together. The resulting structure initially remains completely flat, allowing for an origami-like pre-shaping without yet establishing a topological closure or Möbius connection. This flat initial form serves as a didactically accessible introduction to the subsequent symmetry breaking.

[0015] In contrast, a structurally more symmetrical approach, better suited to the SU(2) topology, involves developing from an equilateral triangle that is divided into two mirror-symmetric halves and expanded along its outer edges to form a mirror-symmetric pentagon. This more symmetrical approach will be described below.

[0016] Paper, such as that used for origami, is recommended as a material for this form; however, other thin, foldable and tear-resistant materials are also suitable, provided they can be joined to form a very tight Möbius strip.

[0017] By unfolding this triangle twice according to Fig. This results in a mirror-symmetric pentagon with the interior angles (120°, 90°, 120°, 120°, 90°) and the aspect ratios (3:1:2:1:3). Drawings Fig. The diagram in the upper left shows the elementary triangle as it becomes a basic pentagon when bisected and the two halves unfolded. This basic pentagon serves as the foundation for the SU(2) fold, which is shown on the right. Using the indicated guide lines and a little skill with the tape, this simple SU(2) shape can be folded fairly securely. Fig. Figure 1 shows the demonstration model according to the invention in its fully unfolded form: an asymmetrical one-stone tile analogous to Smith et al. (2024) – an aperiodic monotile – with all three unfolding arrows of the base pentagon. The outline corresponds to the 13-sided, flat, and asymmetrical shape developed by Smith et al. (2024), which is referred to in the literature as the “Einstein tile.” An additional angled incision and three fold lines make the four congruent base pentagons and their asymmetrical arrangement immediately visible. The unfolding arrows show how the asymmetrical demonstration model emerges from the mirror-symmetrical base pentagon and serves to represent a SU(2)→SU(2)3×U(1)U(1)2−symmetry breaking These arrows serve as symbols. In the Standard Model of particle physics, these arrows correspond to the generators W. 1 , W 2 ', W 3the electroweak interaction and mark three independent modulation directions within the structure. Fig. shows the basic pentagon, equipped with half a real part sine wave and half an imaginary part cosine wave, which according to Fig. The right-hand side is to be connected to form a Möbius strip. The phase shift Θ imposed by the geometry illustrates the complex 4π-wave function of the ground state of the closed SU(2) cell: ϕ(τ,ρ,Θ)=Aei(ωτ−kρ+Θ).

[0018] If no transparent film is used, the reverse side must also be printed with the mirrored half of the imaginary part of the cosine wave to enable a phase-correct Möbius connection. This conveys to the students the special nature of SU(2) symmetry: that the second 2π orbit through a mirror world leads back to the identity.

[0019] With repeated rotations, the phase shift accumulates according to the abelian addition rule until, at the limit 2π, a spontaneous symmetry breaking occurs due to a sign change. This dialectic between abelian phase addition and the non-abelian group structure of SU(2) makes the mechanism of symmetry breaking visible in group-theoretical terms. Didactically, one can draw a comparison with the electromagnetic near field of a dipole antenna, which exhibits a phase shift of π at the dipole but no phase shift in the far field.

[0020] The phase shift Θ can be interpreted both as a local modulation phase of the wavefunction and as a projection of a scalar coupling field [ϕ]. In this interpretation, Θ corresponds to a geometrically induced mixing phase – analogous to the Weinberg angle θ. W- while [ϕ] modulates the scalar coupling of the system, comparable to a dilaton field or an effective cosmological constant Λ. In the closed state, the trace effect of [ϕ] remains confined within the cell; only with the topological transition to the open structure - for example through folding rupture or phase decoupling - can the unconfined modulation component escape along an open edge into free space and become effective there as a trace of A.

[0021] The spatial structure of the closed SU(2) pentagon is intrinsic and not externally embedded. The geometric effect of the modulation structure can be described within the framework of Einstein's field equations. Rμν−12Rgμν+Λgμν=κTμν It is best formulated in a fully symmetrical representation: κTμν+(12R−Λ)gμν−Rμν=0μν

[0022] Here, R denotes µv the Ricci tensor, R the scalar curvature, g µvthe metric, A the cosmological constant, κ the coupling factor and T µv the energy-momentum tensor with tensorial matter and tensorial curvature in scalar equilibrium with the tensorial geometry as a zero-sum game of the de Sitter vacuum.

[0023] Depending on the interpretation of the diagonal modulation, two complementary pathways of effect can be discussed didactically using the demonstration model: • With a trace-free phase shift ±Θ, T remains µv unchanged, and the relaxation occurs exclusively via the geometric parts of the equation - in particular via the scalar-coupled part 1R. • If, however, the magnitude [ϕ] is treated as a scalar modulation, a contribution to T is generated. µv , which amplifies the amplitude A or the frequency ω of the wave function. This corresponds to an energetic coupling to the Higgs boson structure in the sense of a coherent Bose-Einstein condensate.

[0024] Equation (2) thus serves as a formal framework for distinguishing between purely geometric phase modulation and energetic coupling within the closed SU(2) system.

[0025] Both pathways of action can be represented geometrically in the model and allow a differentiated discussion of symmetry breaking in the closed SU(2) system.

[0026] This Möbius-like folded basic pentagon can not only demonstrate the scalar Higgs field, but also give an idea of ​​the primeval atom postulated by Lemaitre (1931) in the context of modern quantum physics, with an inflationary spatial expansion caused by the off-diagonal elements R τρ , R ρτ driven by the Ricci tensor (Kiesewetter-Köbinger, 2025b) and fractally similar to a Pythagoras tree with 1τ13ρ(1τ)+1τ43ρ(3ρ(1τ)+1τ133ρ(3ρ(3ρ(1τ)+1τ)+1τ403ρ(3ρ(3ρ(3 ρ(1τ)+1τ)+1τ)+1τ1213ρ(3ρ(3ρ(3ρ(3ρ(1τ)+1τ)+1τ)+1τ364⋮3n+1−12 grows. geometry

[0027] Starting from the bottom corner, the demonstration model has the following interior angles: Interior angle: 90°−120°−(90°−120°−180°−240°)−120°−90°−120°−270°−120°−120°−90°−240°−90°−240°

[0028] The corresponding edge lengths (in the same direction of rotation) are: Edge lengths: 3−3−(1−1−1−2)−1−3−3−1−2−1−3−3−1−1

[0029] The bracketed angle and length segments distinguish the demonstration model from the monotile described by Smith et al. (2024) and enable the subsequent splitting into three separate SU(2) regions. This structural extension allows for a didactically tangible representation of generator separation within a closed modulation system and forms the basis for the hands-on teaching of symmetry breaking and group unfolding.

[0030] The short edges and fold lines are preferably joined to form Möbius-like loops using transparent adhesive tape (such as Scotch tape) on paper or film materials. Overhead transparencies are particularly suitable, as their tear resistance, dimensional stability, and light transmission visually reinforce the modulation structure. The reversible adhesive bonds allow for controlled folding and unfolding of the demonstration model without compromising its geometric integrity.

[0031] Ideally, the Möbius folding process begins at the upper left pentagon, which has the longest connecting line to the single-stone monotile and requires particular skill, as the middle pentagon must also be folded. Next, the upper right pentagon is connected – here, too, the narrow-side connection to the middle pentagon and the immediate attachment of the left folded section demand considerable precision. The third outer pentagon can then be folded into a Möbius solid as easily as a standalone base pentagon.

[0032] While the basic pentagon and the one-stone monotile can still be described as beautiful, the completely asymmetrically twisted basic model with its three Möbius connections appears almost ugly - a truly aesthetic break in symmetry. didactics

[0033] Fig. This shows a possible labeling of the demonstration model for didactic purposes and allows it to be produced and offered as a commercial craft kit for schools. This exemplary assignment illustrates the development of the structure. SU(2)3×[U(1)U(1)2] To be understood didactically.

[0034] The three 2 × 2 matrix blocks along the Cartesian axes (x, y, z) represent local symmetry sectors in which color transitions and modulation processes are encoded by specific operators. The off-diagonal elements represent exchange processes between color states, while the diagonal elements capture the corresponding modulation within the respective channel. The central modulation point [ϕ] forms the color-neutral singlet and carries the scalar coupling across the entire structure. The depicted folding allows for a haptically and visually perceptible realization of the configuration geometry and serves as a basis for the didactic teaching of the underlying group symmetry.

[0035] It becomes apparent that the central [U(1)U(1)2] The pentagon within the depicted folding structure cannot, in principle, be completely closed. The associated electromagnetic modulation (EM) component, represented by the photon as the corresponding gauge boson, can escape into free space via the remaining open edge, as can the free neutrino of the weak interaction (according to Nikitin (2020), this also applies to the dilaton as a monopole wave of dark matter and can be easily explained in this demonstration model via [ϕ]). This illustrates the special structural position of the electromagnetic and electroweak fields within the depicted geometry and explains its unconstrained effect in macroscopic space.

[0036] Compared with the grouping of gauge bosons in the conventional GUT model according to Georgi and Glashow (1974), in which the fermions of a generation are organized in common SU(5) representations, the matrix structure of the demonstration model offers a visually and haptically memorable representation of the underlying symmetry structure - especially with regard to the confining properties of the color charges and the special position of the electromagnetic and electroweak fields.

[0037] The following schematic representation is not to be understood as an algebraic reduction of the SU(5) structure according to Georgi and Glashow (1974), but as a functional grouping of the mediator bosons according to their action domain: G3 G1 + iG2 G4 + iG5 X X G1 - iG2 -G3 G6 + iG7 X X G4 - iG5 G6 - iG7 [φ] −23G8 X X X X X Z 0 W + X X X W - γ

[0038] The following are included: • G1 to G8 for the eight gluons of the strong interaction (SU(3)), • Z 0 , W ±, γ for the physically observable bosons of the electroweak interaction (SU(2) × U(1)), • X for the hypothetical leptoquark bosons (X, Y) which enable the coupling between quark and lepton sectors and are responsible for proton decay within the framework of the SU(5) model.

[0039] The inventive modeling and the exemplary labeling with physically known particle names serve as didactic orientation and enable a memorable communication of the underlying symmetry structure - especially with regard to the definition of the color charges and the free action of the electromagnetic field.

[0040] The technical effect of the demonstration model consists in the haptically and visually perceptible representation of group-theoretical modulation processes within a closed SU(2) 3-System with a central U(1) modulation component. Through targeted folding and matrix structure, confinement processes, color transitions, and the special status of electromagnetic and electroweak fields are made didactically comprehensible. The open edge of the central U(1) pentagon allows the physically motivated representation of non-confined fields (photon, neutrino).

[0041] The students will be able to connect two of the three pentagonal components to form Möbius strips using the paper demonstration model, provided they have some skill. However, the third component—the one with the long fold line to the middle pentagon—reveals that the entire structure must be bent, which requires considerable dexterity. This resistance provides a vivid illustration of the close coupling of the three SU(2) 3 -groups in the strong QCD interaction.

[0042] As a craft kit for teaching modern symmetry structures in school lessons, it is suitable for commercial use. literature

[0043] Thomas Appelquist, Alan Chodos, and Peter George Oliver Freund. Modern Kaluza-Klein Theories. Addison-Wesley Publishing Company, 1987. ISBN 978-0-201-09829-7.

[0044] Paul Adrien Maurice Dirac. The principles of quantum mechanics. Clarendon Press, 4th edition, 1958.

[0045] Richard P. Feynman, Robert B. Leighton, Matthew Sands, and EM Hafner. The Feynman Lectures on Physics; Vol. I. American Journal of Physics, 33(9):750-752, September 1965. doi: 10.1119 / 1.1972241.

[0046] Howard Georgi and SL Glashow. Unity of All Elementary-Particle Forces. Physical Review Letters, 32(8):438-441, 1974. doi: 10.1103 / PhysRevLett.32.438. ;

[0047] SS Gubser, IR Klebanov, and AM Polyakov. Gauge theory correlators from non-critical string theory. Physics Letters B, 428(1-2): 105-114, 1998. doi: 10.1016 / 80370-2693(98)00377-3.

[0048] R. Jackiw. Lower Dimensional Gravity. Nucl. Phys. B, 252:343-356, 1985. doi: 10.1016 / 0550-3213(85)90448-1.

[0049] Gudrun HE Kalmbach. DE 10 2018 005 425 A1 Mint-wirgis, June 2019.

[0050] Gudrun HE Kalmbach. 4-dimensional lattice models for the quantum range. World Journal of Advanced Research and Reviews, 12(1): 175-181, October 2021. doi: 10.30574 / wjarr.2021.12.1.0003.

[0051] Theodor Kaluza. On the problem of unity in physics. Proceedings of the Royal Prussian Academy of Sciences, pages 966-972, January 1921.

[0052] Swen Kiesewetter-Köbinger. Higgs shell collapsar with 1+1 JT SU(2) junction horizon, September 2025a. URL https: / / www.researchgate.net / publication / 396406325_Higgs_shell_collapsar_with_11_JT_SU2_junction_horizon.

[0053] Swen Kiesewetter-Köbinger. Lemaitres primeval atom as an SU(2) Planck cell, 2025b. in preparation.

[0054] Oskar Klein. Quantum theory and five-dimensional relativity theory. Zeitschrift für Physik, 37(12):895-906, 1926. doi: 10.1007 / BF01397481.

[0055] Claudius Krause. Higgs effective field theories. Text. PhD Thesis, Ludwig-Maximilians-Universität München, September 2016. URL https: / / edoc.ub.uni-muenchen.de / 19873 / .

[0056] Abbé Georges Lemaître. A homogeneous universe of constant mass and rayon croissant renders compte de la vitesse radiale des extra-galactic nebulae. Annales de la Société Scientifique de Bruxelles, 47:49-59, January 1927.

[0057] Abbé Georges Lemaitre. The Beginning of the World from the Point of View of Quantum Theory. Nature, 127(3210):706-706, May 1931. doi: 10.1038 / 127706b0.

[0058] Juan Maldacena. The Large-N Limit of Superconformal Field Theories and Supergravity. International Journal of Theoretical Physics, 38 (4):1113-1133, 1999. doi: 10.4310 / ATMP.1998.v2.n2.a1.

[0059] August Ferdinand Möbius. Über die Bestimmung des Inhalts eines Polyeders. 1865.

[0060] Igor Nikitin. On dark stars, Planck cores and the nature of dark matter. 21:221-246, 2020. doi: 10.48550 / arXiv.2102.07769.

[0061] Roger Penrose. US 4,133,152 Pattern for Surface Covering, 1979.

[0062] Roger Penrose. The road to reality: a complete guide to the laws of the universe. Jonathan Cape, London, 2004.

[0063] O. R. Rahul and S. Murugesh. Rogue breather modes: Topological sectors, and the ‚belt-trick‘, in a one-dimensional ferromagnetic spin chain. Chaos, Solitons & Fractals, 122:262-269, May 2019. doi: 10.1016 / j.chaos.2019.02.012.

[0064] David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss. An aperiodic monotile. Combinatorial Theory, 4(1), July 2024. doi: 10.5070 / C64163843.

[0065] Claudio Teitelboim. Gravitation and hamiltonian structure in two spacetime dimensions. Physics Letters B, 126:41-45, June 1983. doi: 10.1016 / 0370-2693(83)90012-6.

[0066] Edward Witten. Anti de Sitter space and holography. Advances in Theoretical and Mathematical Physics, 2(2):253-291, 1998. doi: 10. 4310 / ATMP.1998.v2.n2.a2.

[0067] Carsten Wochnowski. A Simple Explanation for the Higgs Mechanism Given by a Modified Kronig-Penney Model. Journal of High Energy Physics, Gravitation and Cosmology, 5(4):1112-1122, October 2019. doi: 10.4236 / jhepge.2019.54064. QUOTES INCLUDED IN THE DESCRIPTION

[0000] This list of documents cited by the applicant was automatically generated and is included solely for the reader's convenience. The list is not part of the German patent or utility model application. The DPMA accepts no liability for any errors or omissions. Cited patent literature

[0000] DE 10 2018 005 425 A1

[0049] US 4,133,152

[0061] Cited non-patent literature

[0000] Dirac (1958, Chapter 11, p. 262 ff

[0007] Plate Trick demonstration, as described by Feynman et al. (1965, Vol. III, Chapter 11, §11-4, pp. 11-6 ff

[0007] Penrose (2004, Kap. 11, S. 220 ff

[0007] Smith et al. (2024 [0014, 0029] Georgi and Glashow (1974 [0036, 0037] Thomas Appelquist, Alan Chodos, and Peter George Oliver Freund. Modern Kaluza-Klein Theories. Addison-Wesley Publishing Company, 1987. ISBN 978-0-201-09829-7

[0043] Paul Adrien Maurice Dirac. The principles of quantum mechanics. Clarendon Press, 4 edition, 1958

[0044] Richard P. Feynman, Robert B. Leighton, Matthew Sands, and E. M. Hafner. The Feynman Lectures on Physics; Vol. I. American Journal of Physics, 33(9):750-752, September 1965. doi: 10.1119 / 1.1972241

[0045] Howard Georgi and S. L. Glashow. Unity of All Elementary-Particle Forces. Physical Review Letters, 32(8):438-441, 1974. doi: 10.1103 / PhysRevLett.32.438.

[0046] SS Gubser, IR Klebanov, and AM Polyakov. Gauge theory correlators from non-critical string theory. Physics Letters B, 428(1-2): 105-114, 1998. doi: 10.1016 / 80370-2693(98)00377-3

[0047] R. Jackiw. Lower Dimensional Gravity. Nucl. Phys. B, 252:343-356, 1985. doi: 10.1016 / 0550-3213(85)90448-1

[0048] Gudrun HE Kalmbach. 4-dimensional lattice models for the quantum range. World Journal of Advanced Research and Reviews, 12(1): 175-181, October 2021. doi: 10.30574 / wjarr.2021.12.1.0003

[0050] Theodor Kaluza. On the Unity Problem of Physics. Proceedings of the Royal Prussian Academy of Sciences, pages 966-972, January 1921

[0051] Swen Kiesewetter-Köbinger. Higgs shell collapsar with 1+1 JT SU(2) junction horizon, September 2025a. URL https: / / www.researchgate.net / publication / 396406325_Higgs_ shell_collapsar_with_11_JT_SU2_junction_horizon

[0052] Swen Kiesewetter-Köbinger. Lemaitres primeval atom as an SU(2) Planck cell, 2025b

[0053] Oskar Klein. Quantum theory and five-dimensional relativity. Zeitschrift für Physik, 37(12):895-906, 1926. doi: 10.1007 / BF01397481

[0054] Claudius Krause. Higgs effective field theories. Text. PhD Thesis, Ludwig-Maximilians-Universität München, September 2016. URL https: / / edoc.ub.uni-muenchen.de / 19873 /

[0055] Annales de la Société Scientifique de Bruxelles, 47:49-59, January 1927

[0056] Abbé Georges Lemaitre. The Beginning of the World from the Point of View of Quantum Theory. Nature, 127(3210):706-706, May 1931. doi: 10.1038 / 127706b0

[0057] Juan Maldacena. The Large-N Limit of Superconformal Field Theories and Supergravity. International Journal of Theoretical Physics, 38 (4):1113-1133, 1999. doi: 10.4310 / ATMP.1998.v2.n2.a1

[0058] Igor Nikitin. On dark stars, Planck cores and the nature of dark matter. 21:221-246, 2020. doi: 10.48550 / arXiv.2102.07769

[0060] Roger Penrose. The road to reality: a complete guide to the laws of the universe. Jonathan Cape, London, 2004

[0062] O. R. Rahul and S. Murugesh. Rogue breather modes: Topological sectors, and the ‚belt-trick‘, in a one-dimensional ferromagnetic spin chain. Chaos, Solitons & Fractals, 122:262-269, May 2019. doi: 10.1016 / j.chaos.2019.02.012

[0063] David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss. An aperiodic monotile. Combinatorial Theory, 4(1), July 2024. doi: 10.5070 / C64163843

[0064] Claudio Teitelboim. Gravitation and hamiltonian structure in two spacetime dimensions. Physics Letters B, 126:41-45, June 1983. doi: 10.1016 / 0370-2693(83)90012-6

[0065] Edward Witten. Anti de Sitter space and holography. Advances in Theoretical and Mathematical Physics, 2(2):253-291, 1998. doi: 10. 4310 / ATMP.1998

[0066] Carsten Wochnowski. A Simple Explanation for the Higgs-Mechanism Given by a Modified Kronig-Penney-Model. Journal of High Energy Physics, Gravitation and Cosmology, 5(4):1112-1122, October 2019. doi: 10.4236 / jhepge.2019.54064

[0067]

Claims

[1] Demonstration model for the didactic illustration of group-theoretical symmetry structures, in particular of SU(2) modulations, comprising an asymmetric, aperiodic one-stone tile (monotile) according to Smith (2024), characterized by : • an interior angle structure according to 90°−120°−(90°−120°−180°−240°)−120°−90°−120°−270°−120°−120°−90°−240°−90°−240°, • an edge length sequence according to 3−3−1(1−1−1−2)−1−3−3−1−2−1−3−3−1−1, where the sections in parentheses complement the previously known aperiodic monotile and allow the representation of three separate SU(2) modulation ranges. [2] Demonstration model according to claim 1, characterized by that there is between • the fourth and fifteenth, • the fifth and ninth and • the ninth and thirteenth edges to form a mirror-symmetric pentagon consisting of four congruent pentagons • with an interior angle structure according to 120°−90°−120°−120°−90°, • and an edge length sequence according to 3−1−2−1−3 It can be folded up. [3] Demonstration model according to claim 2, characterized by , that the two short edges of the folded pentagon of length 1 are joined, in particular with adhesive tape, to form a Möbius strip. [4] Demonstration model according to claim 1, characterized by , that each of the three outer unfolded symmetrical pentagons is joined at their two short edges and fold lines of length 1, in particular with adhesive tape, to form a Möbius strip.

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