It is possible to determine whether many convex regions "have a common part as a whole" using only a small number of combinations. When you choose any three things, there is always a common element. A central point or shared space exists that satisfies all conditions. A theorem for finding the commonality of the whole from local overlaps. 複数の凸状ボリュームが重なり、中央の金色に光る共有アトリウムへ収束する The golden light represents the "common area" of each domain. Spaces with different uses exist independ

If you move a disc or sphere continuously without cutting or flying it, at least one point will always remain in its original position. There is at least one point where the "actual floor position" and the "position depicted on the map" perfectly coincide. Architecture where everything appears to be in motion, yet there is always one unmoving center. 黒い曲面屋根：連続的に変形する空間 A thin, golden light at the center of the stage is the "fixed point." Even though the entire surrounding spac

"A four-dimensional complex surface with many holes, deep symmetry, and beautifully balanced curvature." K3 surfaces are smooth surfaces in complex 2D and real 4D, with a representative example being a quartic surface in projective space. 1. A type of Calabi-Yau K3 surfaces can be viewed as two-dimensional versions of Calabi-Yau manifolds; that is, highly balanced spaces where the curvatures cancel each other out as a whole. 2. The structure of the holes is rich. Unlike a sim

The sense of time, rhythm, and changes that occur within living organisms 1. Body clock 2. Time for Growth 3. Seasonal Time 4. Metabolism time Time that manifests as changes in life 成長し、変化し、劣化し、回復し、季節に反応する空間 Biological time is "the structure of time that living organisms possess within themselves." Mathematical Architecture space

"A theory that defines 'quantities' such as length, area, volume, and probability for very general spaces." Geometry → “shape” Topological space → "connection" Measure theory → "How much does it exist?" 1. Set XXX 2. σ-additive family (sigma algebra) 3. Measure μ\muμ Fourier analysis Quantum mechanics Chaos theory geometry information theory These connections lead to → probability theory → "designing density" → from "object" to "distribution field". 「空間内に何がどれくらい存在するかを見る」 Arch

"A three-dimensional object in which there is no distinction between inside and outside." 1️⃣ There is no front or back. 2️⃣ There are no boundaries. 3️⃣ In 4th dimension, they do not self-intersect. The sequence is: city → building → city → building. 「内外の連続」＋「有機的な膨張」 Landscape architecture Looking at the cross-sectional diagram makes the mechanism of a Klein bottle instantly clear. 建物に入った瞬間が存在しない 「街の中に“外”と“内”が存在しない都市」 　 "The city itself is inverting into space." The boundar

The shape changes depending on the viewpoint, and perspective, vanishing points, and depth themselves constitute the structure. Perspective / Vanishing Point / Infinite Horizon Architecture Projective Variety Architecture Parallel lines intersect at infinity. 射影多様体建築の原理 Mathematical Architecture space

"A movement that flows over time in the 'most natural way' within space." 「空間の形そのものが、運動の運命を決める現象」 “都市の流れの増幅器” Navier-Stokes equations Navier-Stokes equations [\rho\left(\frac{\partial \mathbf{u}}{\partial t}+\mathbf{u}\cdot\nabla\mathbf{u}\right) -\nabla p+\mu\nabla^2\mathbf{u}+\mathbf{f}] (\rho)：Fluid density (\mathbf{u}): Velocity field (p): pressure (μ): viscosity coefficient (\mathbf{f}): External force This equation is, water flow air vortex turbulence Thermal convection

(An energetic perspective) Energy (Hamiltonian) → Time evolution → Motion There is an "invisible topography (energy undulation)" in space, and particles flow through it. "Architecture that designs movement itself" 動いていないのに動いて見える "Dynamic Space Architecture" Mathematical Architecture space ■ Gumroad https://shomei.gumroad.com/l/apmao ■ Lemon Squeezy https://shomei.lemonsqueezy.com/ ■ Payhip https://payhip.com/MathematicalArchitectureS ■ BOOTH https://shomei-a.booth.pm

The world where form, sound, and vibration modes correspond. Visualization of spatial shape = eigenfunctions → "Architecture with overlapping frequency layers" 「音の干渉縞」 Humans become "particles in the sound field" The space itself is a mixer Mathematical Architecture space

Space itself is a "structure that possesses power." "Architecture that designs changes in space," rather than "architecture that creates form." The realm of designing "relationships themselves" → "Architecture that visualizes the distribution of invisible forces" 空間＝位置と運動のペア system Content Relationship with Symplectic Mandelbrot Shape generation ❌ Wrong Lehman Face multilayer space △ Related topology connection ◎ Core non-commutative interaction ◎ Deeply related Yang Mills pl

"Mathematics is the process of unifying disparate, localized pieces of information into a coherent, unified whole structure." ① Segmentation (local) ② Connection (restriction mapping) ③ Integration (gluing) "A collective building that retains differences in its components while perfectly aligning only its boundaries." ①形態 × ②構造 × ③運動 を統合 Core Geometry Mathematical Architecture space

"Geometry that deals with the relationships between motion and energy" Geometry that governs the motion of the universe Geometry of movement paths → Human movement × Space 流れが凍結した建築 If topology is about "preserving shape," then symplectic architecture is about "preserving motion." Mathematical Architecture :

"A structure in which distant places or spaces are directly connected within a building." The "instantaneous" connection of space → "architecture that eliminates distance" The internal space is topologically broken. Visual perception and movement patterns do not align. "Time" also becomes a design subject. Opening the sliding door of a Showa-era house reveals a futuristic city—this is a "disconnecting jump." "Architecture that breaks down and reconnects distance, continuity,

"The study of mathematically dealing with patterns of randomly (accidentally) placed points, lines, and surfaces." Mathematical modeling of random spatial structures 建築と都市の中間 "Cities are architecture that grow like living organisms." "Architecture that has come into existence" rather than "architecture that has been designed" Mathematical Architecture space

"An integer version of complex numbers with the symmetry of a triangle (60°)" Eisenstein integers → triangular lattice (60°) → hexagonal tile (honeycomb structure) "The most stable planar structure" honeycomb crystal A configuration in which force is evenly distributed. "The structure chosen by nature" "Unifying everything under the rule of 60 degrees." 「中心を持たない文化」 Space that grows diagonally ↑ "Architecture where crystals proliferate like living organisms, losing their bound

Physical laws do not change even when a certain "transformation of perception" is applied. The forces of the universe = gauge symmetry In architecture, mathematics is incredibly powerful. Yang-Mills theory Wilson Loop Holographic principle Chern–Simons theory Gauge/Gravity compatible When viewed from a distance A single massive mass of fluid As you get closer A collection of countless misaligned spaces As we got closer You can see the "flow (force) of light" running along the

Space + Time = Spacetime Black Hole ● ↓↓↓ ───∪∪∪─── Gμν + Λgμν = c48πGTμν Black Hole Architecture physics concepts architectural elements Spacetime distortion Nonlinear grid of exterior walls Gravity well Settlement pattern towards the center Event horizon ring-shaped facade The bending of light Light facade (reflection and convergence) physics concepts architectural elements Spacetime distortion Nonlinear grid of exterior walls Gravity well Settlement pattern towards the cen

A structure in which points are arranged in a regular and infinite pattern in space. A perfectly formed supercrystal → A gigantic symmetrical city → A cosmic-scale urban grid 無限に続く構造 静かな建築 structure = light 24 Dimensions of Light Mathematical Architecture space

A collection of exceptional symmetries that do not belong to any large pattern (an infinitely continuing family). Monster group Mathieu Group Conway Group Fisher group, etc. Sporadic group: "An unknown creature that suddenly appears (only one exists)." "A structure that deviates from theory, yet somehow works perfectly." 全体ルールが存在しないのに、崩壊しない構造 "Experiences are continuous, but space is not." "A cross-section where a different building emerges no matter where you cut it." Mathem