259.509 Talk
This course is in all assigned curricula part of the STEOP.
This course is in at least 1 assigned curriculum part of the STEOP.

2022W, VO, 2.0h, 2.0EC


  • Semester hours: 2.0
  • Credits: 2.0
  • Type: VO Lecture
  • Format: Hybrid

Learning outcomes

After successful completion of the course, students are able to 'think' as architects, in the design process. with greater mathematical imagination. They have acquired a basic familiarity with the kinds of spaces theoretical mathematics is capable of opening up.

Subject of course

The lecture course “TALK” investigates philosophical theories on relevant discourses in architecture and culture, including among others topics from economy, ecology, politics, ethics, art, pedagogy, the natural, the engineering and the juridical sciences.

This semesters VO “TALK” will focus on mathematical thinking. It is the invariant and explicit part with regard to mathematical concepts that an architect self-evidently meets daily in her work – such as ‘form,’ ‘rule,’ ‘structure,’ ‘number,’ ‘field,’ ‘group,’ ‘set,’ ‘system’, ‘type,’ ‘proportion’, ‘volume’, ‘area,’ or ‘power’. Many people believe that in mathematics, one follows only rules, mechanically and with no deliberation. In many ways this is true, of course. But at the same time mathematical concepts have a history and are part of sociocultural contexts. We will put special attention also to this philosophical and inventive part that belongs to mathematics too.Having an awareness of this sociocultural scope of deliberation that mathematical thinking provides is a basic literacy highly relevant with regard to the capacities of thinking abstractly; perhaps it is as relevant for the kind of abstract thinking in which every architect needs to be proficient as being able to master basic grammar or basic arithmetics is with regard to dialectical thinking in general.

This course will be co-taught by Elias Zafiris, a theoretical physicist and professor in mathematics and logics, and Vera Bühlmann, professor for architecture theory and philosophy of technics. 

*** Kein spezielles Vorwissen oder Affinität mit Mathematik wird vorausgesetzt. Der Kurs knüpft an allgemeine Mathematik an (Matural Level) ***


The semester will be structured around recorded lectures on the ATTP YouTube Channel, with live discussion meetings every second week (on ZOOM). The course ends before Christmas break. The lectures will be the subject of the exam after Christmas.

Introduction // THURSDAY 13. October 2022 (live or on ZOOM (tbd), 16:00-18:00 Uhr)


Eleven Lectures for these Modules are available pre-recorded at the ATTP YouTube Channel

1.    Module – Cosmos, Earth and Firmament (Arithmetic) 

  • On the Notion of Number – Arithmos and The Method of Abstraction
  • Numbers as Multitudes, Magnitudes, and Powers in Space and Time
  • Arithmetic Systems: Integer – Rational – Irrational – Real – Imaginary
  • Congruence – Polynomials and Roots – Law of  Solvability
  • Transcedental Numbers – Exponentiation and Logarithmization
  • Atoms – Prime Numbers – Law of Unique Factorization

2.    Module – Observatory, Measure and Treasure (Geometry) 

  • On the Notion of Geometric Space – Measurement and Homothesis
  • Vectors and Linear Spaces – Linear Transformations
  • Metric Distance and Angle – Euclidean Spaces
  • Perspective, Parallelism, and Geodesic Curves
  • Atlases and Manifolds 
  • Intrinsic Curvature – The 3 types of Geometric Spaces

3.    Module – Mechanae, Art and Techné (Algebra) 

  • On the Notion of Algebraic Structure – Operations and Closure
  • Groups – Symmetries and Actions
  • Homomorphisms – Partition Kernels and Images  – Isomorphisms
  • Conjugation and Division – Ideals and Irreducibility – Varieties
  • Abstraction of Scalarity – Rings and Fields (Körper)
  • Dilation and Contraction of Scalarity – Adjunctions

4.    Module – Agriculture, Knots and Fields (Topology) 

  • On the Notion of Local-to-Global Relations – Form and Shape
  • Order and Continuity – Lattices and Inverse Images
  • Homeomorphisms – Coverings and Filters
  • Simple and Multiple Connectivity – Obstacles and Obstructions
  • Localized Identity, Amalgamation, and Sheaves of Germs
  • Twisting – Winding – Encycling: Homology and Homotopy

5.    Module – Meteora, Cycles and Aion (Analysis and Synthesis)

  • On the Notion of Infinitesimals – Differential Forms 
  • Invariants and Differential Equations
  • Law of Epiphaneia – Cocycles and Coboundaries
  • Spectrum – Spectral Resolutions – Exact Sequences
  • Replication – Connections and Holonomy
  • Integration of Forms and Cohomology 

Teaching methods

Theory lectures with demonstrations in exercises; discussions. A text reader will backup the content from the lectures. 

Mode of examination




Examination modalities

A selection of 3 tasks will be offered for the students to chose one, and to write a short essay that shows the student's application of the learnt themes to an architectural artefact. There will be a deadline to submit the essays via TUWEL. Video documentation as well as a text reader will be provided for the students to solve their exam tasks in proficient manner.

The Final Exam paper is TBD

Course registration

Begin End Deregistration end
19.09.2022 08:00 16.11.2022 08:00 13.11.2022 08:00


Study CodeObligationSemesterPrecon.Info
066 443 Architecture Not specified


Elias Zafiris, Natural Communication (Basel: Birkhäuser, 2020). Excerpts.

Elias Zafiris, Mathematical Thinking. An Involution for Architects (unpublished manuscript).