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Shapes and Patterns (F14)

Instructors: Herbert Edelsbrunner, Carl-Philipp Heisenberg

Teaching Assistants: Vanessa Barone, Daniel Capek, Florian Pausinger


This is a semester-long course revolving around the general topics of Shapes and Patterns.  In biology, shape is intimately connected to function, and the quantification of shape and its variation is at the heart of biological organization.  In neuroscience, the dynamic pattern of brain activity is overlaid on an intricate arrangement of neuronal cells.  In physics, the sometimes deterministic, random, chaotic behavior of systems has a profound impact on our understanding of how the world works.  The course material follows mathematical categories, focusing on Symmetric,  Smooth, and Fractal Shapes at different times during the semester.  The translation between vocabularies in different disciplines will be emphasized.  Throughout the course, we will encourage interdisciplinary approaches through the formation of mixed teams of students.


The backbone of the course structure is mathematical, and the meat are topics from the sciences, balancing theory and applications within each topic.  To encourage interactions across the disciplines, we will organize the students in mixed groups of about four members.  Each group will work on a class project throughout the semester.  Possible projects include background search, computer
simulation, mathematical investigation, etc.  Here is a preliminary list of possible topics:

     1-  Diffusion in different environments.  What are relevant scenarios inbiology?  What is known about this question in mathematics and physics?  Do simulations under varying conditions.  Karla, Bor, Enikö, Kreso.

     2-  Cellular aggregates and shape changes.  What is known in mathematics and computer science about these local changes?  What are the biological scenarios where they are relevant?  Pepa, Julio, Cheta, Catarina.

     3-  Synchronization and relevance in somitogenesis.  What is the fundamental mechanism that leads to synchroniation?  What is the specific realization that plays out in the development of somites?  Andreas, Shayan, Shamsi, Jason.

     4-  Measuring cells.  Study conventional measures such as length, area, volume to cells.  How about different types of curvature?  What is the biological relevance?  Minji, Ran, Corinna, Simon.

     5-  3D print cells.  We have one and soon two 3D printers.  To print cells (eg neurons), we need to acquire their shape, represent it in the computer, send it to the printer.  Stephan, Ruslan, David, Zuzka.

     6-  Rotations in space.  In representing rotations in 3D space, quaternions have a distinct advantage over Euler angles.  Study the algebra of quaternions.  Use it to sample the space of rotions and or rigid motions.  Aglaja, Thomas, Johannes, Bernhard.


Exams and Grade

There will be homework questions asked during the lectures and solutions will be collected three times during the semester.  There will be a written final exam at the end of the course.  The grade assignment will depend on the

     -  class participation:   10%
     -  class project:            35%
     -  homework:                20%
     -  final exam:                35%


Schedule (subject to change)

Date Topic (Math) Topic (Nature) Notes Assignments
Tue Oct 07 Introduction and Organization      
Thu Oct 09   Evolution of symmetric forms Lecture 01  
Tue Oct 14 Symmetry groups   Lecture 02  
Thu Oct 16   Chirality Lecture 03  
Tue Oct 21 Lattices   Lecture 04  
Thu Oct 23   Cell sorting in development Lecture 05  
Tue Oct 28 Voronoi tessellations   Lecture 06  
Thu Oct 30 Knots and links   Lecture 07 Homework I
Tue Nov 04 Project proposals      
Tue Nov 11   Surface tension Lecture 08  
Thu Nov 13   Shape formation in plants Lecture 09  
Tue Nov 18 Curves and surfaces   Lecture 10  
Thu Nov 20 Morse theory   Lecture 11  
Tue Nov 25   Lateral inhibition Lecture 12  
Thu Nov 27 Singularities   Lecture 13 Homework II
Tue Dec 02 Project progress report     Homework IIb
Thu Dec 04 Project progress report      
Tue Dec 09 Geometric probability   Lecture 14  
Thu Dec 11 Intrinsic volume   Lecture 15  
Tue Dec 16   Single cell migration    
Thu Dec 18   Collective migration Lecture 17  
Thu Jan 08 Fractal dimension   Lecture 18 Homework III
Tue Jan 13   Gene networks    
Thu Jan 15   Patterns in the brain Lecture 20  
Fri Jan 23 Project final reports      
Tue Jan 27 Preparation for final exam      
Thu Jan 29 Final exam      


Hermann WeylSymmetry.  Princeton University Press, Princeton, New Jersey, 1952.  (For background on symmetry; mathematics, biology, and architeture.)

Herbert Edelsbrunner.  Geometry and Topology for Mesh Generation.  Cambridge University Press, Cambridge, England, 2001.  (For background on Voronoi diagrams and Delaunay triangulations; mathematics and algorithms.)

D'Arcy Wentworth Thompson On Growth and Form.  Cambridge University Press, Cambridge, England, 1961.  (Classic work on biological form.  Interesting ideas but slow reading.)

René Thom.  Structural Stability and Morphogenesis.  Benjamin, Reading Massachusetts, 1975.  (A broad approach to morphology based on smooth functions; mathematics and philosophy but rather controversial.)

Benoit B. Mandelbrot.  The Fractal Geometry of Nature.  Freeman and Company, New York, New York, 1983. (A standard text on fractals.)

Colin C. Adams.  The Knot Book.  Freeman and Company, New York, New York, 1994.  (A textbook on the mathematical theory of knots; very pleasant reading.)

Yukio Matsumoto.  An Introduction to Morse Theory.  American Mathematical Society, Providence, Rhode Island, 2000.  (An introductory text for Morse theory.)

J. W. Bruce and P. J. Giblin.  Curves and Singularities.  Cambridge University Press.  (An introduction to curves but also surfaces; pleasant reading.)


Left right a/symmetry

Blum, M., Weber, T., Beyer, T., and Vick, P. (2009). Evolution of leftward flow.Semin Cell Dev Biol 20, 464–471.

Lee, J.D., and Anderson, K.V. (2008). Morphogenesis of the node and notochord: the cellular basis for the establishment and maintenance of left-right asymmetry in the mouse. Dev Dyn 237, 3464–3476.


Cell sorting

Krens SF1, Heisenberg CP. Cell sorting in development. Curr Top Dev Biol. 2011;95:189-213. (Review on cell sorting theories and cell sorting events in development)

Lecuit T1, Lenne PF. Cell surface mechanics and the control of cell shape, tissue patterns and morphogenesis. Nat Rev Mol Cell Biol. 2007 Aug;8(8):633-44. (on the idea that the function of cell-cell adhesion is increasing contact size)

Krieg M1, Arboleda-Estudillo Y, Puech PH, Käfer J, Graner F, Müller DJ, Heisenberg CP. Tensile forces govern germ-layer organization in zebrafish. Nat Cell Biol. 2008 Apr;10(4):429-36. (cited in the lecture)

Maître JL1, Berthoumieux H, Krens SF, Salbreux G, Jülicher F, Paluch E, Heisenberg CP. Adhesion functions in cell sorting by mechanically coupling the cortices of adhering cells. Science. 2012 Oct 12;338(6104):253-6. (cited in the lecture)

Maître JL1, Heisenberg CP. Three functions of cadherins in cell adhesion. Curr Biol. 2013 Jul 22;23(14):R626-33. (cited in the lecture)


Gene Networks

Molecular Genetics of Bacteria - Jeremy W Dale, Simon F Park - 5th edition (can be borrowed from the library or from Calin Guet)


Recitation slides can be found here:

Additional references are mentioned during class or can be retrieved from the lectures pdfs published on-line.