Self-intersecting surfaces that appear when a projective plane is pushed into three-dimensional space.one "A surface that has no distinction between front and back, but is forced to appear as such in three-dimensional space." exterior: A cultural hall where three large curved shells overlap in the center, making it impossible to tell which is the front and which is the back . Introspection: The walls, ceiling, and floor continuously reverse, creating a space where the sense o

Algebraic spaces that operate using multiplication It's not a "point," but a space created by ratios, magnifications, and scale transformations. Algebraic, lattice-based, controlled deformations The same structure, but on different scales, chains together in a multiplicative manner. Algebraic Torus Architecture　 Multiplicative Space / Scaling Geometry A cultural landscape where scales proliferate in a chain reaction. Mathematical Architecture space

"A network that spreads information to almost the highest level with very few connections." "Despite the small number of connections, the entire system connects surprisingly well." "It's practically impossible to create a more efficient network than this." ① Minimal connections yet a huge cycle ② No central point ③ "Local" and "whole" coincide Actual applications Communication network semiconductor quantum computing data center Cryptography AI Network Distributed computing 最適

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

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?

"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

"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

"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

"What kind of movement would make this world 'most natural'?" "The force that wants to move" - "The force that wants to stay still" "Architecture that arises spontaneously from the principles of motion," rather than "architecture that has been designed." fluid architecture Continuous surface City optimized for traffic flow Field-generated architecture Crowd flow analysis architecture “空間そのものが法則になる建築” More natural cities: underground cities and underwater cities ~ like ruins M

"A hotel that's fully booked but can still accommodate guests." 無限を説明するための思考実験 部屋は 1号室、2号室、3号室…と無限にある。すでに全部満室 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

A concept of urban design that applies random movement to urban forms, probabilistically generated curved cities normal city + + + + + + + + + + + + + + + Brown City ~ ~ ~~ ~ ~~ ~ ~~ ~~~ Cities are "generated like terrain." ground Gentle, random undulations road No straight lines It continues to curve slightly. building Embedding in a curved surface In other words City = Mega-landscape parametric city 「ブラウン多様体都市」 The space itself is a performance device; the cross-section is