Topology in cinema and mathematics

The exhibition about film architecture in Tchoban foundation engages the visitor with the topic of cinematic spaces, which are subset of real physical space, but sometimes tries to use the means of art to give the viewer an opportunity to experience spaces different from our day-to-day perception. Based on this introduction this article discussed the basics of the generalization of spaces in mathematics and shows a way to define a Hilbert space, as an equivalent to our surroundings, starting with a topological space, as a space with least restricitons.

What?

Exhibition “German Film Architecture from 1918 to 1933” Tchoban foundation – Museum for Architektural Drawing, Christinenstraße 18a, 10119 Berlin

 Prices?

5€ adult / 3€ reduced

 

 

 

 

 

Introduction

On 12th of June the Tchoban foundation held a vernissage on a new exhibition “German Film Architecture from 1918 to 1933”.

The building of the Tchoban foundation is immediately visible from the outside. It stands at the end of a row of the houses in Christianenstraße, visible from three sides. The facade of the four-storey building, which was designed by S. Kuznetsov and S. Tchoban himself, shows enlarged elements of archtectural drawings and signales hereby the pupose of the structure. The sculptural design on the outside continues inside of the building, which, despite being made from concrete, glass and steel, implements the architectural elements from the surrounding historic buildings. The ground floor host the library and the reception. The visitor has to climb up to the first and second floor, to access the exhibition halls.

Current exhibition covers the topic of the architecture in the German movies from 1918 to 1933. This period was essential for the development of modern art, which was expressed in all creative areas including cinema.This influence can be seen in the sketches and drawings, which were prepared for the films featured in the exhibition: “The Cabinet of Dr. Caligari”, “Die Nibelungen”, “Der Golem, wie er in die Welt kam”, “Metropolis”, “Der blaue Engel” usw. These movies are mostly expressionistic and the pieces were drawn by Emil Hasler, Robert Herlth, Otto Hunte, Erich Kettelhut, Hans Poelzig, Franz Schroedter and Hermann Warm.

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The spaces in the mostly silent movies of this epoche were of vital importance. The space in the movies was build, using materials and methods close to the architecture with similar intention. The audience should be able to subconciously transform the view in the movie into an impression. Wide and open spaces usually transport a feeling of freedom and easyness of mind, as closed and narrow rooms stand for danger and imprisonment. Therefore changes in spaces and proportions had significant meaning in the influence of the viewers impressions and feelings. Also light and the lines had significant value for the creation of the cinematic spaces. The light and darkness can mediate similar feelings as wide and closed spaces, so can do the lines. Straight lines have different emotional impact compared to the zigzag or broken lines.

The production of the movie sets gave the architects the possibility to implement stunning, out-of-scale projects, which would never get approved to be built in real life. The reaction of the audience to the new and daring archtectural solutions could be tested.

Discussion

In the introduction the projection of a two dimensional space (paper) to a three dimensional space (drawing of a seton paper and models built for the set) and back to a two dimensional space of a movie screen were described. How is this possible mathematically? Especially, if we take in account that our day-to-day life is happening in a three dimensional space, which sometimes has to be described as four dimensional (three spatial and one temporal dimensions, e.g. in relativity or quantum mechanics), and our universe might evolve into five dimensional state.

To explain this we have to introduce a general idea of a space, as it is used in mathematics. As we discussed in our article about scientific method, mathematical generalization is crucial to scientific approach of understanding. Most spaces used to model reality are Hilbert spaces, which are topological spaces with certain restriction, which we would like to explain below (Sunder, 2000).

Space in mathematical sense is just a set of objects, which follows certain rules called structure. Usually the objects in the space are called points, but e.g. a set of cars spans a car space. The highest layer in the space hierarchy is reserved for topological spaces, which have least restrictions. There are different possibilities to define this kind of space. One of them is to use the neighborhoods (Sunder, 2000).

Hier

Bildergebnis für topological space
Hierarchy of mathematical spaces (Wikipedia)
  1. First rule for the set of objects X is that it has to have a possibility of having zero elements in it (like during school holidays there are no cars left in Prenzlauer Berg).
  2. To have a proper topological space we have to assign a function N(x) to the points in the set X, which allows us to choose non-empty subsets in X (let’s say all cars in Friedrichshain or all cars in Prenzlauer Berg). The elements of N(x) are called neighbourhoods of x, where x is a point of set X (so Friedrichshain is geographical neighbourhood in Berlin, but for a car x all cars in Friedrichshain are neighbourhood. Basically, the cars in car space do not know or do not care about geography, just about cars).
  3. Set X with function N is called a topological space with N a neighborhood topology:
    • If N is a neighbourhood of x (i.e., NN(x)), then xN. In other words, each point belongs to every one of its neighbourhoods.
    • If N is a subset of X and includes a neighbourhood of x, then N is a neighbourhood of x. I.e., every superset of a neighbourhood of a point x in X is again a neighbourhood of x.
    • The intersection of two neighbourhoods of x is a neighbourhood of x.
    • Any neighbourhood N of x includes a neighbourhood M of x such that N is a neighbourhood of each point of M.

There are also linear spaces, which have predefined operators. These operators obey linearity. However, in linear spaces it is not possible to define a circle or a perpendicular line, so we will ignore them here (partly, as some spaces can be both – topological and linear) (Sunder, 2000).

Above we defined a structure of a topological space. Now we have to add a metric, to define a metric space, which is a subset of topological spaces. Metric basically enables us to compute a distance between two objects in a topological space. This distance has to uphold some additonal qualities (Sunder, 2000):

  • the distance between the same object and itself has to be zero
  • the distance between two objects has to be larger than zero
  • the distance from object x to object y has to be equal the distance from object y to object x
  • the sum of distances between objects x and z, and, y and z has to be large than the distance between x and y.
In taxicab metric three paths have the same length (red, blue and yellow), as in Euclidean metric on the plane the green path has length and is the unique shortest path. (Wikipedia)

A distance however is not enough, as a distance has to be measured, if we want to access a space close to what we are living in. This measurability is ensured through introduction of a norm. A norm is a function with a positive value, which can be used to measure distance (e.g. meter mile or a car length) (Sunder, 2000).

https://ece.uwaterloo.ca/~math211/Laboratories/02/images/norm1a.png
The one norm of the blue vector is the sum of the absolute values of the vector components shown in red  (University of Waterloo)

Additionally we have to define an inner product, which is our usual multiplication if we talk about scalar values (numbers, sometimes with a unit, like e.g. temperature), but for vectors it also results in a number. A vector has not only a value and an unit, but also a direction, like forces or velocity. The inner product has to be positive, linear and conjugate symmetric and allows to define an angle between two vectors, which seems ridiculous to our day-to-day life, but obviously is not that straight forward as we are used to think (Sunder, 2000).

Geometric interpretation of the angle between two vectors defined using an inner product (Wikipedia)

As you stayed with us so long, I will not try and explain the last criterium for definition of a Hilbert space and therefore of a space, which is scientific equivalent to our day-to-day space. This criterium is completeness, which is a complicated, but necessary mathematical criterium. If you are intereseted you can look it up by yourself as additional task (Sunder, 2000).

At this point we have finished out journey from a topological space or just a set of objects to the definition of a Hilbert space, which is very close to the space we are living in and is used to model e.g. thermodynamics, quantum physics, etc. We wanted to show this for three reasons:

  1. Generalization is important as it enables scientists to model and calculate the outcomes not only outcomes for a specific case, but also ideally for all cases.
  2. The upscaling generalization (going back from the “real” Hilbert space to topological space) allows to look into possible spaces using same tools, but based on the laws, which are confirmed at least for a part of a space (we just don’t know that universe is homogeneous and e.g. spedd of light is constant everywhere – we just observed it as far as we can look).
  3. Even such terms as “angle”, which we consider simple are sometimes complicated to describe, so it can be used in science or computer models.

Summary

In this article we spoke about the exhibition “German Film Architecture from 1918 to 1933”, where interplay of two and three dimensional spaces in architectural drawings, on set and on screen is shown. To explain the general idea of “space” we introduced the mathematical concept of space and showed how to define a “Hilbert” space, which is often used to solve real world physical problems (Academic Press, 1979; Foundations of Physics, 2012).

References

  1. V. S. Sunder, Functional Analysis: Spectral Theory. Institute of Mathematical Sciences (2000)
  2. A list of further literature is given here: https://mathoverflow.net/questions/72419/a-good-book-of-functional-analysis
  3. Any textbook on functional analysis should help you advance your understanding of spaces in mathematics
  4. M. Reeb and B. Simon, Methods of modern mathematical physics. III. Scattering theory. Academic Press (1979)
  5. B. Schumacher and M. D. Westmoreland, Modal quantum theory. Foundations of Physics, 42 (7): 918 – 925 (2012)

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