A space that expands infinitely within a circle, yet visually it appears to be contained within a finite circle— a rather不思議 (mysterious/strange) space. ① Straight lines curve ② Distorted sense of distance ③ Infinitely many parallel lines A visualization of hyperbolic geometry 有限の円に見えるけど、内部は無限に広がる曲がった空間 "The center is stable, while the outer perimeter can extend infinitely." Endless corridor Space with gravity at its center A city that stretches outwards Mathematical Archit

"A space created by gathering all the 'shapes (structures)' that satisfy certain conditions, and then classifying and organizing them." "A space in which geometric and algebraic objects are classified by 'grouping similar objects together as one.'" ● Circle Moduli ● Moduli of a donut (torus) Not "space itself," but "space of space." "A space that uses differences in shape as coordinates." Mathematical Architecture space

"A polygon with an infinite number of sides" "Apeirogon" comes from the Greek words apeiros (infinity) + gonia (horn). "A way of thinking that views a circle as a 'polygon that is infinitely divided'" 終わらない回廊都市 ● Architecture with an endless corridor ● Deconstruction of a circular hall ● Architecture where only the boundary exists Mathematical Architecture space

"A phenomenon where a state changes suddenly, almost as if it's flying, after having been changing slowly." The form "cusp" Space is gradually compressed. In a sense, "the view suddenly opens up." The flow of movement suddenly branches or reverses. Architecture that creates a continuous space, yet the experience is discontinuous. 連続変化 → 突然の空間ジャンプ The ball is in a valley. The terrain gradually changes. At a certain moment, it jumps into another valley. This "jumping" becomes

The outer space becomes incredibly complex. Strange topology 絡まり地獄 Hilbert Hotel Architecture A crazy space Mathematical Architecture space

It's continuous (not interrupted), yet it can't be differentiated anywhere (the slope cannot be defined). A space where details continue endlessly f(x)=n=0∑∞​ancos(bnπx) Curved surfaces vibrate infinitely; slabs become fractalized ; light creates abnormal shadows. Mathematical Architecture Mathematical Architecture space

"Cities may appear random, but in reality, they are made up of self-similarity (fractals) and nonlinearity (chaos)." ① Hierarchical design of scale 👉 This creates a "comfortable atmosphere" ② Network optimization 👉 Resilient to disasters and congestion ③ Blurring of boundaries 👉 Spaces with chaotic "fluctuations" "Designing growth rules" 「建築というより小都市」 "A cultural hall without a central point"

"A geometric space for treating time and space as a single entity." 4th dimension (3-dimensional space + time) → The very stage of the theory of relativity → "A collection of trajectories extended in the time direction" "A coordinate system of the universe that includes time" Mathematical Architecture 's "Riemannian Manifold Architecture"

A spherical surface that looks like a "sphere" but behaves in an "abnormally messy" way in space. The surface is infinitely intricate. A seemingly endless series of tiny thorns and ribs. The distinction between "inside" and "outside" is intuitively broken down. The internal structure becomes incredibly complex. The sense of spatial continuity is broken. The concept of "boundaries" goes haywire. "Architecture that is closed yet open" アレクサンダーのホーンド球面 Infinite Horn Corridor

Architecture is not merely the construction of space; it is deeply intertwined with mathematical principles. In particular, translating abstract mathematical concepts such as chaos theory, topology, and higher-dimensional geometry into spatial ideas and visual forms makes it possible to create architectural designs that defy gravity and possess a sense of weightlessness. This article explores the connection between architecture and mathematics , providing concrete insights u

"A single line, yet a mysterious curve that completely fills the entire plane." space-filling curve 「回廊＝建築全体」 A structure that defies intuition: "lines become surfaces" space Mathematical Architecture

It's not something that can be cut with a knife. "Something that goes against intuition" The trick of the spatial structure itself → "Manifold and topological architecture" The discrepancy between "space," "infinity," and "rules" バナッハ＝タルスキー "Destruction of spatial preservation" Mathematical Architecture space

"The space takes on a different shape each time." for example: Waves on the water's surface → constantly changing shape Clouds → Boundaries are random Cracks → Spreading irregularly Random walk (how a drunk person walks) Fractals (self-similar random shapes) Random graph (network) 迷宮型文化施設 (Incredibly beautiful) (Cosmic) (The city is shaking) Conflict between different programs

A world of "not smooth spaces" Thinking in terms of distance and measure Viewing in terms of extremes and scales Approach it probabilistically Treat as a discrete structure "A space where you can't find peace anywhere."   「世界は本当に滑らかなのか？」 Differentiable = Ideal/Smooth Not differentiable = realistic, rough, complex 👉 Non-differentiable geometry is **"geometry that is close to reality"**

A situation where rules exist, but unpredictable actions occur. for example weather Water flow Movement of crowds of people Moving endlessly within complex shapes → somewhere between order and disorder "Architecture where getting lost itself becomes an experience." 秩序のある迷宮

A collection of points floating randomly in space A space that "exists, yet lacks coherence." "A city that exists but is disconnected" → "A space where continuity has been completely destroyed" “存在とは何か”を問う建築 A fantastical landscape of a disconnected city and futuristic skyscrapers.

Beautiful vs. Wild tame = "gentle" vs wild = "wild" ① The “extreme behavior” of space ② "Geometry where intuition doesn't work" Wild topology = "A continuous yet uncontrollably complex space." 未来的な有機的建築複合体

🌀 Main line flow Swirling curves: The spiral flow that dominates the entire screen creates a cosmic sense of movement. Fluid Wavy Lines: Red, pink, blue, and orange colors merge in a wave-like pattern, representing the flow of energy. Convergence towards the center: A composition in which all lines guide the viewer's gaze towards the central light (sun). 💧 The effect of vertical lines The dripping vertical lines visible at the edges create a surrealistic "melting" effect. T

Defining architecture itself as geometry and determining how waves behave. ① Definition of the space (hall) itself The interior of the hall is treated as a three-dimensional space. Ω⊂R3\Omega \subset \mathbb{R}^3Ω⊂R3 Boundaries (walls, ceilings, floors, columns, openings) ∂Ω\partial \Omega∂Ω Let's assume that. The coordinates of each point are x=(x,y,z)x = (x,y,z)x=(x,y,z) ② The basic equation for waves propagating in space (time-dependent) The fundamental form of sound, ligh

Slow zoom-in motion A slightly enhanced glowing effect. 24fps / Full HD Landscape prompt Vertical video, chaotic dense urban structures, cyberpunk atmosphere, dark tones, slow zoom in, cinematic lighting, ultra realistic マッピング picture Night view + building silhouettes