The Japanese architectural world possesses a unique culture where tradition and innovation merge. In particular, in the field of modern architecture, architects who incorporate mathematical concepts into spatial design, creating buildings with a sense of weightlessness that transcends gravity, are attracting attention. They translate abstract theories such as chaos theory, topology, and higher-dimensional geometry into visual and spatial forms, opening up new possibilities fo

A mathematical and scientific way of thinking that interprets the overall structure and flow from the relationship between "points" and "lines". Point = person, station, building, room, information, urban hub Line = connection, road, communication, relationship, movement, influence A structure in which people naturally gather, move, meet, and engage in a chain reaction of activities. 複数の広場や屋上テラスが、網目状の動線で接続された文化施設 A giant living room in the city An urban hub where people, acti

The field of spatial analysis requires handling complex data structures and diverse variables. Kenjiro Kimura's proposed spatial analysis method using multivariable polynomials is attracting attention as a technique that significantly improves accuracy and efficiency compared to conventional methods. This article clearly explains the characteristics and specific applications of Kimura's method, exploring the future of spatial analysis. 多変数多項式による空間解析ーあるホール What is spatial anal

Architecture can be viewed not merely as a physical "thing," but as a "continuous space" where boundaries blend together, surfaces transform, and space flows without interruption. This perspective transcends conventional architectural concepts and invites us into the world of mathematical architecture, incorporating mathematical structures and topology. This time, we will introduce a new perspective on architecture centered on continuity and topology, along with specific meth

"The phenomenon in which the boundary grows according to the gradient of the field." 1. Dendritic crystals (snow crystals) 2. Electric discharge (lightning) 3. The boundary between oil and water "The most unstable, yet the most natural" 「空間が自己生成する」 拡散律速凝集 CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=641439 サフマン・テイラー不安定性 By Claire Trease, Kingston University - Own work, CC BY-SA 4.0,https://commons.wikimedia.org/w/index.php?curid=45344174 Now let's consider sk

"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

As an ebook nears completion, anticipation grows, along with thoughts about its content and how it can be used. In particular, ebooks dealing with themes gaining attention in contemporary design fields, such as #Parametric Design, #Generative Art, and #Algorithmic Architecture, become valuable learning resources for students and creators. This article will introduce the appeal of an ebook nearing completion and offer practical ways to utilize it. What is the appeal of ebooks?

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

"A method for dividing the region closest to a certain point." "Space is distributed autonomously." plant leaf vein cell bark geology Cracked dry land living thing Giraffe feather pattern structure universe galaxy distribution Instead of "designing the form," we "arrange the relationships." “空間が自ら育ったような建築” Voronoi ↔ Delaunay: "Mathematics that generates spatial order solely through the relationships between points" Object Design　→　Field Design Mathematical Architecture space

"In a sufficiently large system, 'order' will inevitably emerge." “An order that appears without design.” 1. A pattern emerges despite the random arrangement. 2. Cities unilaterally acquire a "core" 3. "Necessity within coincidence" exterior Giant random grid city countless units Seemingly chaotic However, partial symmetry will always appear. The golden light reveals an "unavoidable structure". Interior Random passage labyrinthine circulation However, walking always converges

Architecture is becoming "biological." Architecture changes based on its "state" rather than its "purpose." Growing due to congestion Extinction due to depopulation Mutation through activity Evolves through connection “進化する建築” How do vacation homes multiply? How it multiplies internally 時間的可変性 Now, let's create a desert island using this math game. Autonomous Grid Architecture Mathematical Architecture space

"A phenomenon in which the interface grows due to the 'field gradient'." A phenomenon in which shapes self-organize and grow in accordance with an "invisible flow." The easiest image to understand When you drop ink into water, In some places, it spreads quickly. Some places are slow The tip is more likely to grow Then, branching Lightning arborescent coral-like It takes on a complex shape. "Space expands autonomously." スライムモールド都市 1. Lightning 2. Crystal growth 3. DLA ( Diffus

"A 'natural-looking randomness' where similar patterns are repeated even when the scale is changed." Natural shapes like mountains, clouds, smoke, and water surfaces 「設計しないのに、リアルになる空間」 Perlin noise generates smooth, pseudo-random patterns that are widely used in computer graphics and procedural generation. Simplex noise is a noise generation algorithm used in computer graphics and procedural generation. Mathematical Architecture space "If you like this architectural artwork,

"Line (knot)" → "Plane (impossible orient)" → "Bundle structure of space" 👆 "Three-dimensional space is made up of 'bundles of circles'." ① Base (foundation space) = the framework of the city ② Fiber = flow/circulation ③ Connection = Control Rules "Intertwining of lines" → "Structure of bundles" → "Space itself" 建築＝静止 → 建築＝流体 "Walking Architecture" It is an important fundamental concept in differential geometry, topology, and theoretical physics. Mathematical Architecture sp

As I've mentioned before, this is a framework of mathematical physics that attempts to describe physics using light rays and causal structures as its fundamental objects , rather than spacetime itself . Roger Penrose proposed a profound theory that connects quantum gravity, scattering amplitude, and complex geometry. 「時空は本当に基本的な存在なのか」 Let's consider the bridge between Tokyo and Yokohama. The paths of light propagating at the speed of light and conformal symmetries are natural

トポロジーによる不安感 "Architecture that first gives you the feeling of unease with topology." 建築は、私たちの空間認識を挑戦します。特に、トポロジーの概念は、私たちの理解を超えた新しい形を生み出します。以下の要素がこの建築の特徴です。 連続した表面 動線の逆転 内外の消失 境界の消失 価値の再発見 "Lost, discovered, and cyclical" becomes valuable. このような建築は、私たちに新しい視点を提供します。私たちの周囲の世界を再発見する手助けをします。建築は、私たちの感覚を刺激し、思考を促進します。 Mathematical Architecture 未来的なメビウスストリップ建築 この建築は、単一の連続した非向きの建物であり、ねじれたループを形成します。外部、内部、断面が一つのイメージに融合しています。カットアウェイの視点から見ると、構造がどのように機能するかが明らかになります。 構造の特徴 屋根、壁、床の統合: すべてが一つの

"The smallest unit of finite symmetry that cannot be further decomposed." "Symmetry like that of 'elementary particles' in a finite world" → Finite simple group = "Prime number" in the world of groups Finite simple group = symmetric atom The concept of a room disappears. There is no hallway Structure and traffic flow are integrated. ① You get lost (but it's not broken) → The symmetry is so strong that you lose your sense of direction. ② No center → Everything feels like the c

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 that goes as far as "the relationship of the relationship of the relationship..." "Relationships are more important than material things." dimension Content 0 subject 1 Relationship of the subject 2 Relationship 3 Relationship of relationships n Infinite Layers The most abstract theory → A universe where structures are stacked hierarchically. 多層ネットワーク空間建築 Symplectic → “The geometry of motion” Mathematical Architecture space