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Draft:Triangle Theory by Jordan Miller

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Triangle Theory

Introduction Physics seeks to describe the fundamental structure of the universe. Traditional models like String Theory propose vibrating one-dimensional strings, but no experimental evidence has yet confirmed these structures. Triangle Theory offers an alternative: reality emerges from rigid, interconnected triangles forming a universal mesh. By analyzing the behaviors of this mesh, we can explain matter, energy, and fundamental forces. Principles of Triangle Theory 1. Rigidity: Triangles are perfectly rigid; they do not stretch, shrink, or deform individually. Permanent Connection: Triangles are always bonded to at least one other triangle, forming a continuous structure. 2. Extreme Stability: Triangles are extraordinarily stable and do not break apart under any normal physical conditions. 3. Flexible Mesh: Although individual triangles are rigid, the mesh flexes through angle changes at the edges and vertices. 4. Bonding Mechanism: Connections are maintained by topological linking and flexible bonding forces, allowing slight bending but no disconnection. Triangles are also slightly attracted to each other. 5. Triangle size never exceeds 1Ut Mesh Structures and States of Matter Solids In solids, triangles are tightly packed, predominantly connected edge-to-edge. The mesh is rigid, allowing minimal flexibility and resulting in the hardness and strength observed in solid materials. Liquids 18 In liquids, triangles are connected more loosely, often vertex-to-vertex. This looser arrangement allows localized flexibility, enabling liquids to flow while maintaining overall cohesion. Gases In gases, triangles are connected and interact naturally because like it says in the core rules, triangles are naturally attracted to each other. Emergent Forces Gravity Massive regions cause large-scale bending of the triangle mesh. Objects follow the curves in the mesh, manifesting as gravitational attraction. Gravity is therefore a geometric effect of the mesh’s bending. Magnetism Magnetism arises from organized twisting within regions of the mesh. These twisted flows create fields that influence the motion of other nearby matter and charges. Fundamental Particles as Mesh Patterns Particles such as electrons and quarks are interpreted as stable, localized twists, knots, or concentrated patterns within the triangle mesh. Different twisting patterns result in different particle properties, including charge, mass, and spin. Wave Propagation (Sound) Sound waves are small oscillations propagating through denser regions of the mesh. These oscillations represent collective flexing of triangles at connection points. In regions where the mesh is too loose (vacuum), sound cannot propagate, matching experimental observations. Three Fundamental Forces of Triangles 1. Weak Attraction Force This force models a natural, background-level attraction between all triangles, regardless of their location. It behaves similarly to gravity on very small scales — long-range and always attractive — and helps explain why even loosely connected regions of the mesh (like gases or empty space) tend to remain part of the same structure over time. It does not cause bonding or collision, but maintains large-scale cohesion in the universe-wide mesh. ● Acts over long distances). 19 ● Becomes negligible at extremely large scales but dominates in low-density regions. ● Conceptually similar to a gravitational-like potential, but acts between fundamental units. 2. Strong Repulsion Force This force prevents triangles from overlapping or collapsing into each other when they get too close. It behaves similarly to quantum exclusion or electrostatic repulsion: once two triangles get within a very short distance, this force rapidly increases and pushes them apart. This maintains the structural geometry of the mesh by keeping triangle cores from occupying the same space. ● Dominates at very short distances. ● Prevents singularities or mesh collapse. Works like a hard-core potential in particle physics. 3. Bonding Force This is a short-range force that activates only between neighboring triangles. It binds them into stable, flexible connections — edge-to-edge or vertex-to-vertex. This is what forms the actual “mesh.” The bonding force is analogous to the strong nuclear force: very powerful but only effective over extremely short distances. Once formed, these bonds maintain their strength and prevent the mesh from falling apart, even under stress. ● Extremely strong, but only acts at bond-forming distances. ● Not only attractive — also elastic, meaning it can stretch slightly and return to its original shape. ● Explains why the mesh bends and flexes, but never rips or unlinks. Future Work Triangle Theory is in its early stages. Many important steps remain to grow and test the theory. In the future, scientists could: ● Build full equations for how triangle mesh bending causes gravity, fields, or motion. ● Create computer simulations of triangle mesh regions in a 3D space. ● Study how triangle vibrations could produce sound, heat, or wave-like effects. ● Compare triangle-particle behaviors to known particles like electrons, photons, or neutrinos. ● Look for possible predictions the theory makes that could be tested in the lab or with telescopes. 20 Triangle Theory offers a new geometric way to understand matter, energy, and space. There is still much to discover. Experimental Predictions and Next Steps Triangle Theory is ready for experimental exploration. Possible tests include: ● Simulating triangle meshes at small scales to measure bending-based curvature and energy flow. ● Modeling sound or wave propagation through triangle structures to compare against known physics. ● Comparing particle energy levels from triangle knots to observed masses of real particles (like electrons or photons). ● Looking for predicted behaviors that differ slightly from current physics (such as localized gravity effects or energy transfer dynamics). If any of these predictions match real-world data — especially in ways that existing theories do not — that would be strong evidence supporting Triangle Theory as a true model of physical reality. Future experiments, simulations, or mathematical proofs will determine if Triangle Theory describes the actual structure of space, matter, and forces. Conclusion Triangle Theory offers a unified model where the structure of space and matter emerges naturally from a simple geometric framework. By focusing on rigid, interconnected triangles, it explains the behavior of solids, liquids, gases, gravity, magnetism, and fundamental particles. Future work will involve developing detailed mathematical models to describe the flexing, twisting, and knotting behaviors quantitatively. Fundamental Properties Units 1�� = 10 : Unit Triangle : Average measurement for triangle −21 ������ 1�� = 1. 248 · 10 : Tri : Mass of a equilateral triangle with height 1Ut −33 ����� 1�� = 1. 248 · 10 : TriForce : Unit of force −51 ������� Constants �� = 4. 1 �� = 53. 2 �� = 71. 5 21 Equations (Force(F) measured in Tf(Triangle force), Mass(m) measured in Tr, and distance(d) measured in Ut) � = : Equation for Weak attraction force : m1 and m2 are masses of both triangles �� �1 �2 (0.4�) � = : Equation for Strong repulsion force : m1 and m2 are masses of both triangles �� �1 �2 (0.4�) 4 � = : Equation for Bond attraction force : m1 and m2 are masses of both triangles �� �1 �2 (0.4�) 25 � = 10( �1 � )�(� 1 , � 2 ) + ( 0.5�1 � ) �ℎ��� �(� 1 , � 2 ) = { (1+�2 15) −1 �1 ≥25 1+�2 15 �1 <25 Equation for Charge particle attraction/repulsion : E1 is the energy of particle and E2 is the energy of the triangle � : Equation for how much energy can be stored in an electron : m is mass of � = 40 3 π� 3 + 10� electron without energy and r is radius of electron. Each triangle is defined by three points in space: A = (x₁, y₁, z₁), B = (x₂, y₂, z₂), C = (x₃, y₃, z₃) These are the triangle’s corners. The edges of the triangle are: AB = B - A BC = C - B CA = A - C These represent the straight sides between the corners. The normal vector of the triangle is: n = AB × AC This vector shows the direction the triangle is facing. The bending angle between two triangles is: θbend = cos⁻¹( (n₁ ⋅ n₂) / (||n₁|| ||n₂||) ) This measures how much two triangles are bent relative to each other. (Optional) The bending energy stored in the bend is: Ebend = k ⋅ (θbend)² where k is a stiffness constant. This represents how much energy is stored in the bend. Triangle Particle Model (Electron Example) 22 A particle in Triangle Theory is a small, stable twist in the triangle mesh. We model the simplest particle using 4 triangles forming a small pyramid (a tetrahedron). Each triangle shares edges and vertices with its neighbors, creating bending between them. The bending angle at each shared edge is: θbend ≈ 70° = 1.22 radians The bending energy per connection is: Ebend= k × (θ_bend)² where k is the mesh stiffness constant. Assuming k = 10 Tf: Ebend per connection = 10 × (1.22)² = 14.9 Tf With 6 bending connections in the tetrahedron: Etotal = 6 × 14.9 = 89.4 Tf Thus, the particle traps about 89.4 ,Tf of energy inside its structure, giving it mass and stability. After some time, using this equation i was able to get ChatGPT to generate me an accurate image following all the laws and math of Triangle theory, and I generated a tetrahedron: A tetrahedron is a three-dimensional shape with four triangular faces, six edges, and four vertices. It's also known as a triangular pyramid and is the simplest type of polyhedron. (not an electron just a model using these calculations) Triangle Twist Bending angle (in radians): θbend = 20° = 0.349 radians 23 Bending energy formula (per connection): Ebend= k × θbend² Assuming stiffness constant: k = 10 Tf Then: Ebend= 10 × (0.349)² = 10 × 0.122 = 1.22 Tf Number of bending connections: 5 Total bending energy for twist: Etotal = 5 × 1.22 = 6.1 Tf Triangle theory 2D engine This is an engine me and Louis created ourselves, using c# https://drive.google.com/file/d/1vUy8MRkj_N4KjQ9A--9GvHyojaAie_yL/view?usp=sharing (controls are space key,arrow keys, -,+,Z,X,C,Q,E,T,P) by the way, you can drag the triangles) Types of triangles (I will call each triangle by there color in the simulation for the sake of simplicity) Normal triangle (white triangle in simulation): They try to stay away from each other, but when there are many of them, the pull of them pulls them together. Antimatter triangle(Red triangle in simulation): when 2 of them, they repel and come back together, they try not to touch, when many of them they form a clump, when one red one white the white one chases the red one. When there are many white and one or more red, white is repealed from red. If white and red touch they annihilate each other and release energy “Negative” matter triangle(blue triangle in simulation) same rules as white triangle, but When blue and white get too close they try to stay together without touching each other. “Negative” antimatter(green in simulation): same rules and do the same thing as anti matter but negative anti matter attract each other. Negative matter and normal matter don't annihilate each other, white matter is attracted to green matter. When Red and green touch they start to attract each other and spin fast. NOTE:(The only triangles we know exist are the normal ones and energy triangles) Energy transfer between triangles Energy triangles get energy from charged triangles, they spread their energy to other uncharged triangles until there is equal energy through all of them. Eventually these charged triangles will stop spreading energy because the energy is equally spread out amongst the triangles. Quantum Hamiltonian – Triangle Theory 24 To describe quantum behavior in Triangle Theory, we define a Hamiltonian (H) that governs the energy and evolution of triangle configurations through space and time. The total Hamiltonian is the sum of bending energy, bond energy, triangle interaction energy, and optional external potentials: H = H_bend + H_bond + H_interaction + H_potential → This expresses the total energy of a triangle configuration as a combination of internal and external effects. 1. Bending Energy Term H_bend = Σ⟨i,j⟩ k * (θ_ij)^2 → For each pair of connected triangles i and j, bending energy depends on the square of the angle between them, scaled by a stiffness constant k. 2. Bonding Energy Term H_bond = Σ⟨i,j⟩ (1/2) * k_b * (d_ij - d₀)^2 → This models the "spring-like" elastic energy of triangle bonds, where d₀ is the preferred (rest) distance between triangle centers. 3. Interaction Energy Term H_interaction = Σ(i ≠ j) [ -W_a / d_ij + S_r / d_ij^4 ] → Every pair of triangles experiences long-range attraction and short-range repulsion, controlled by constants W_a (weak attraction) and S_r (strong repulsion). 4. External Potential Energy (Optional) H_potential = Σ_i V(r_i) → You can add an optional external energy term for each triangle, based on its position in space or curvature of the surrounding mesh. Time Evolution of Quantum States i * ħ * d/dt |ψ(t)⟩ = H * |ψ(t)⟩ → This is the Schrödinger equation, showing how a quantum triangle state evolves over time based on the total Hamiltonian. State Representation |ψ⟩ = |positions, angles, bonds, twists⟩ → Each quantum state can be described by the arrangement, bending, and connection pattern of triangles. Formal Topology – Triangle Theory 25 To describe the underlying structure of the triangle mesh, we treat it as a topological space using the language of simplicial complexes. This allows us to define connections, boundaries, and shapes rigorously. 1. Triangle Mesh as a Simplicial Complex Definition: A simplicial complex is a collection of simplices (points, lines, triangles, tetrahedra, etc.) glued together along shared faces. → Each triangle in your mesh is a 2-simplex, and the entire mesh is a union of connected 2-simplices. 2. Vertex Set Let V = {v₁, v₂, v₃, ..., v} be the set of all triangle vertices in space. → This is the complete list of all points (x, y, z) that make up the triangle corners. 3. Triangle Set (2-Simplices) Each triangle Tᵢ = [v_a, v_b, v_c] is an ordered triple of vertices (a 2-simplex). → A triangle is defined by connecting three vertices into a flat, oriented face. 4. Triangle Mesh Space (K) Let K = {T₁, T₂, ..., T} be the set of all triangles in the mesh. → The full mesh is the union of all triangles, forming a 2-dimensional topological space embedded in 3D. 5. Edge and Vertex Adjacency Two triangles are said to be edge-adjacent if they share a common edge. Two triangles are vertex-adjacent if they share a single vertex. → This defines how local connectivity works in the mesh. 6. Boundary The boundary ∂K is the set of edges that belong to only one triangle. → The boundary defines the "edges" of the mesh where it stops or is open. 7. Topological Properties ● Connectedness: K is connected if there's a path between any two triangles via shared edges. → This ensures the mesh behaves as a continuous space. ● Compactness (optional): If the mesh is finite and closed, it's compact. → This matters if you're modeling a closed universe or particle. ● Orientability: If every triangle can be consistently oriented, the mesh is orientable. → Required for defining spin and flow direction. 8. Homology (for Defects and Particles) Use simplicial homology to detect holes, loops, or twisted regions in the mesh. ● H₀(K): Counts connected components. → Helps detect isolated mesh regions. ● H₁(K): Counts independent loops or holes. → Used to detect twist-based particles or vortex-like structures. ● H₂(K): Detects voids (3D holes if you extend the mesh). → Important if you use tetrahedra for 3D versions. 9. Curved Surfaces If triangle angles sum to less than 180° at a vertex, the space is positively curved. If they sum to more than 180°, it is negatively curved. → This lets you approximate mesh-based curvature, similar to smooth manifolds. 10. Mesh as a Discrete Manifold (Optional) A triangle mesh with well-defined neighborhoods and curvature can be treated as a piecewise-linear 2D manifold. → This allows the mesh to model curved surfaces and gravity geometrically. Regge-Style Curvature Model – Triangle Theory To simulate gravity in a discrete triangle mesh, we adopt ideas from Regge Calculus, a method of approximating general relativity using simplices (triangles and tetrahedra). 1. Concept of Discrete Curvature In smooth space, curvature is continuous and defined via derivatives. In Regge Calculus, curvature is concentrated at edges or vertices where triangles meet. → We approximate spacetime curvature by computing angle deficits around edges or points. 2. Triangle Mesh in 3D (Optional Step for Full Regge Calculus) Extend your mesh to include tetrahedra (3-simplices) instead of only flat triangles. → This allows 3D curvature modeling and mimics spacetime more accurately, but it's optional if you're staying in 2D. 3. Vertex Angle Sum (2D Regge Curvature) Let θ₁, θ₂, ..., θ be the angles of triangles meeting at a vertex v. Then the curvature at vertex v is: 2 K(v) = 2π - Σθᵢ → The difference between 2π and the total angle at the vertex gives the discrete Gaussian curvature. 4. Edge Deficit (3D Regge Curvature) Let θ₁, θ₂, ..., θ be the dihedral angles (3D angles) between faces around edge e. Then the curvature at edge e is: K(e) = 2π - Σθᵢ → If the angles around an edge in 3D don't add up to 2π, the space is curved at that edge. 5. Einstein-Hilbert Action in Regge Form The total gravitational action in discrete form is: S_gravity = Σ_e K(e) * L(e) Where: ● K(e) is the curvature at edge e, ● L(e) is the length of edge e (in triangle units). → This mimics the integral of curvature used in general relativity, but over discrete edges. 6. Equations of Motion (Discrete Einstein Equations) By varying the action S_gravity with respect to triangle edge lengths, you get: δS_gravity / δL(e) = 0 → This gives the discrete analog of Einstein's field equations — the triangle mesh "wants" to distribute curvature consistently. 7. Gravitational Bending in Triangle Theory Large-scale mesh bending = curvature = gravitational field → Triangle Theory naturally produces gravity if you interpret large-scale bending and angle deficits as curvature affecting motion. 8. Geodesics (Paths of Least Action) Particles follow paths along the mesh that minimize their action: Geodesic path = minimal cumulative angle bending → This replaces smooth-space geodesics with triangle-to-triangle sequences that trace natural curves through the mesh. 9. Curvature and Mass Massive objects = regions of concentrated bending energy = high angle deficits → Mass arises from the geometry itself, and bending curvature alters the motion of nearby particles, just like in general relativity. 2 10. Space-Time Interpretation (Optional Extension) If you allow time-like dimensions and moving triangle states: Curvature evolves over time → dynamic gravity → Triangle General Relativity → You can define a time-step evolution of triangle configurations that simulate gravity waves and spacetime deformation.

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