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MATHS MEETS ARTS
Minimal Surfaces: Artists' Views - Maths Meets Arts Tiger Team
Maths Meets Arts began as a research project by Dr. Katrin Leschke, exploring artists' approaches on mathematics research, specifically on minimal surface theory. This led to a broader desire for a space that allows a cross-disciplinary approach bringing together mathematicians and artists. Thus the Maths Meets Arts Tiger Team was created, an ongoing collaboration for both educational and artistic outcomes through the conversation of disciplines.
'f: Countless Deformations'
This interactive dance performance was created through the collaboration with Dr.Katrin Leschke and her research on minimal surface theory. f:Countless Deformations proposes an open dialogue between the languages of algebra, dance, sound and motion sensing while looking for physical manifestations of mathematical processes behind minimal surface creation. One dancer, one musician and one digital artist co-exist in a space where members of the audience are able to shape the choreography by choosing mathematical propositions which performers use as improvisation formulas. The choreography, sound and light are thus put together as pieces of a puzzle with audience members experiencing the process of composition while having the time to explore key mathematical ideas concepts. The work seeks to re- think & re-imagine the body as brain and bring to life the workbook of a mathematician.
Check out the f: Countless Deformations archives below to learn more about the maths behind the project.
Useful links:
m:iv info page
More about Minimal Surfaces Artists' Views
Check out our virtual festival
f: countless deformations
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the maths behind the performance
A Minimal Surface is a surface that locally minimises its area.
Minimal surfaces occur in nature, for example withdrawing a wire loop from soap water, the resulting soap film will minimise the area with the given boundary.
information presented sits within the context of research undertaken by Dr.Leschke at the University of Leicester through the international research programme m:iv.
*surface: In mathematics, a surface is a geometrical shape that resembles a deformed plane
Key:
*area: the space occupied by a flat shape or the surface of an object
*saddle point: where mean curvature is 0
*plane: a flat, two-dimensional surface that extends infinitely far
*catenoid: a type of surface, arising by rotating a catenary curve about an axis
THE FINITE TOPOLOGY CONJECTURE -----------------
An orientable surface M of finite topology with genus g and r ends, r ≠ 0, 2, occurs as a topological type of surface if and only if r ≤ g + 2
Topology is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing. Minimal surface* study falls within this area.
Therefore, we are looking at a point of the surface that is enclosed in a curved boundary and which occupies the least area possible.
A minimal surface has vanishing average curvature ("vanishing mean curvature"): taking the curvature of all curves through a point, the average of the largest/smallest curvatures should vanish: each point looks like a saddle*.
Mathematicians observe the properties of the 'zoomed in' points and come to a conclusion about the nature of the entire surface. If the constituent points of the surface minimise area then that is a minimal surface.
Minimal surfaces exists in lots of shapes and forms. To better understand these complex concepts, simplify and narrow down the area of study, mathematicians may choose to only focus on one type of minimal surface, here: the embedded surface.
For a surface to be consider embedded it has to fulfil two preconditions. If we extend the edges of the shape out to infinity these should never cross each other (no self intersections) and should never come into contact (no touching points).
We want our surfaces to have no boundary as this enforces the surface to go out to infinity. For the purpose of this research, the edges or 'ends' (r) have to behave nicely. That means that as they extend into infinity the either look like a plane* or a catenoid*. That way they will also never intersect or touch!
Surfaces may have holes.
The number of holes is called the genus (g) of the surface. A sphere for example has genus zero, a torus has genus 1.
To study existing and discover new minimal surfaces mathematicians use a variety of tools including complex maths and computer software. Their research involves deforming known minimal surfaces to discover new ones that might be able to maintain the properties mentioned above.
What mathematicians call a minimal surface is not exactly what occurs in nature. A surface in mathematics doesn't have "thickness" nor does a minimal surface minimize area*. However if we zoom in to a point of the surface we find that it locally does.
