University of Iowa Eight Annual Sonia Kovalevsky
April 6, 2013 Mathematics and Art
WS talk topics
Examples of mathematics explorations/projects High School Students and their
teachers/parents might work on. These are examples connecting mathematics with the
arts in some ways.
1. Create Lego sculptures of ‘interesting’ surfaces
2. The fourth dimension in visual, literary and media: examples and explorations of
mathematical representations of four dimensional objects in three dimensional space via
projections.
1. Mathematics with Legos
Lego sculptures of mathematical objects such as surfaces, minimal surfaces:
http://www.ams.org/mathimagery/
click on
Lipson's Lego Sculptures
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http://www.ams.org/mathimagery/thumbnails.php?album=29
We will focus on questions related to construction of Lego models of minimal surfaces
such as the Scherk, and Enneper surfaces shown in Lipson’s graphics. There are many
other interesting minimal surfaces not constructed by Lipson and we will talk about those
too. There are parameterizations of these surfaces arising in many ways such as the socalled Weierstrass representations which may be used as a set of ‘instructions’ for how to
take cube-shaped Lego blocks, thought of as the ‘points’ or ‘pixels’ and actually place
them in space to construct what are something like piecewise-linear or polyhedral
approximations to the actual surfaces. The mathematics involved in creating those
instructions includes interesting and non trivial mathematics which will be applicable in
study of higher math and STEM courses (e.g. engineering in understanding CAD design,
computer science in understanding how computers display 3D objects on a 2D computer
screen).
What are the mathematical questions arising in trying to construct such Lego
representations?
What is our starting ‘background’ 3D? What model, what axes, what scales on axes,
etc. How can we ‘best’ put coordinates on this model (e.g. rectangular but how about
cylindrical, spherical or other?).
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What kind of surfaces should we look at and why? Examples could be minimal
surfaces b/c they have so many interesting application and mathematical ideas
surrounding them: soap film/area minimization properties, as interfaces of chemicals, as
parts of cells, and mathematically as ‘extremal’ surfaces. Plenty of other examples are
possible too.
How is the surface described? Geometrically (e.g. take a curve in the x-y plane and
rotate it in space around a line, like the catenoid), with explicit or implicit single equation
in x-y-z (e.g. gyroid), parametrically (e.g. pretty much all surfaces)?
How can we take the given description of the surface and use it to place the Legos in
space to build the surface out of Lego ‘bricks’?
Are some parameterizations better than others for such constructions? Lipson told us he
would create a set of instructions for placement of the Lego bricks in space (email 4-113:
“I actually computer-generate lego plans for the mathematical sculptures, which specify
exactly which 'pixels' in space need to contain Lego. The catch is that these plans express
everything in terms of 1x1x1 bricks, so a model built this way would not hold together.
The process of building the shape out of the bricks I actually have available in such a way
that it holds together structurally I do by hand as I go along. Sometimes it can be quite
difficult!
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I sometimes play around with the parametrisation quite a lot before I find one I'm happy
with. I recall spending quite some time in Mathematica before I was satisfied with the
Klein bottle, for example.”
DPGraph examples:
DPGraph www.dpgraph.com is a 2 and 3D computer program that shows curves and
surfaces in space defined by equations. It allows you to interactive rotate the objects in all
directions with keyboard keys, to zoom in and zoom out the perspective, and to slice the
objects into sections in many different ways.
UI has a site license for DPGraph which makes it a free download for any UI student,
faculty, staff member and their families.
http://www.dpgraph.com/math-art-9.html
Shows a catenoid, Enneper, Sherk, gyroid
Params
Catenoid, by Carlos Cesar de Araujo
graph3d(x^2+y^2=a^2*(cosh(z/a))^2)
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Enneper's Surface by Brian Tenneson
graph3d(729* x^6 - 2187* x^4* y^2 + 2187* x^2* y^4 - 729* y^6 - 729* x^4* z +
1458* x^2* y^2* z - 729* y^4* z + 4374* x^4* z^2 - 4374* y^4* z^2 - 3888* x^2*
z^3 + 1215* x^4* z^3 - 3888* y^2* z^3 + 6318* x^2* y^2* z^3 + 1215* y^4* z^3 +
6480* x^2* z^4 - 6480* y^2* z^4 - 5184* z^5 + 3888* x^2* z^5 + 3888* y^2* z^5 +
432* x^2* z^6 - 432* y^2* z^6 + 1152* z^7 - 64* z^9=0 )
Parametric Repn of Enneper via Weierstrass repn:
http://en.wikipedia.org/wiki/Weierstrass%E2%80%93Enneper_parameterization
enneper[u_, v_] := {u - u^3/3 + u*v^2, -v + v^3/3 - v*u^2, u^2 - v^2}
=Real part of {z - z^3/3, 1/3 I z (3 + z^2), z^2}
= Weierstrass repn using
From Wikipedia, the free encyclopedia
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In mathematics, the Weierstrass–Enneper parameterization of minimal surfaces is a classical piece of differential geometry.
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Alfred Enneper and Karl Weierstrass studied minimal surfaces as far back as 1863.
Let ƒ and g be functions on either the entire complex plane or the unit disk, where g is meromorphic and ƒ is analytic, such that
wherever g has a pole of order m, f has a zero of order 2m (or equivalently, such that the product ƒg2 is holomorphic), and let c1, c2, c3
be constants. Then the surface with coordinates (x1,x2,x3) is minimal, where the xk are defined using the real part of a complex integral,
as follows:
The converse is also true: every nonplanar minimal surface defined over a simply connected domain can be given a parametrization of
this type.[1]
For example, Enneper's surface has ƒ(z) = 1, g(z) = z.
NB IN THE WS PARAM WE USED ƒ(z) = 2, g(z) = z. This still gives the Enneper surface.
Gyroid
graph3d(cos(x) * sin(y)+ cos(y) * sin(z)+ cos(z) * sin(x) = 0)
http://en.wikipedia.org/wiki/Gyroid
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“Applications
In nature, self assembled gyroid structures are found in certain surfactant or lipid
mesophases[5] and block copolymers. In the polymer phase diagram, the gyroid phase is
between the lamellar and cylindrical phases. Such self-assembled polymer structures
have found applications in experimental supercapacitors,[6] solar cells[7] and nanoporous
membranes.[8]
Gyroid membrane structures are occasionally found inside cells.[9]
Gyroid structures have photonic band gaps that make them potential photonic
crystals.[10] Gyroid structures have been observed in biological structural coloration
such as butterfly wing scales, inspiring work on biomimetic materials.[11][12] The
gyroid mitochondrial membranes in tree shrew cones might have an optical
function.[13]”
Helicoid
Scherk
; Scherk's Surface, by Carlos Cesar de Araujo
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graph3d(e^z*cos(x)-cos(y)=0)
graph3d(e^z*cos(x)-cos(y)=0)
Helicoid wiki
http://en.wikipedia.org/wiki/Helicoid
0. Mathematics of the Fourth Dimension in Literature and Art and Manipulatives
The fourth dimension and geometric realizations of it have motivated artists and writers
for more than a century.
http://pavlopoulos.wordpress.com/2011/03/19/painting-and-fourth-dimension-cubismand-futurism/
Examples in painting:
One can find examples of artwork depicting four-dimensional objects in the
“Cruxifixion” painting by Dali
notorious example is Salvador Dali’s “Crucifixion” or “Corpus Hypercubus” (1954), a painting where Jesus Christ is depicted crucified upon the
cross – like three dimensional net of a hypercube, the four dimensional analog of a cube.
Below is the url for a really nice view of the hypercube/tesseract also being ‘unwrapped’
into the 3D net, cross-like figure appearing in Dali’s Cruxificxion painting.
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http://www.youtube.com/watch?v=BVo2igbFSPE
Paintings by Picasso also showing multiple ‘sides’ or views of a person, considered as the
3D person changing in time give more examples (e.g. Nude Descending Staircase #2,
http://en.wikipedia.org/wiki/Nude_Descending_a_Staircase,_No._2
and
Left: Pablo Picasso, "Demoiselles D' Avignon" (1907). Right: Jean Metzinger, "Tea Time" (1911)
Examples in literature:
Flatland Edware Abbott Abbott 1884 (movies too)
Wrinkle In Time Madeline L’Engle 1962
Sphereland: A Fantasy About Curved Spaces and an Expanding Universe Dionys Burger
1965 [a sequel to Flatland, a novel by "A Square" (a pen name of Edwin Abbott Abbott).]
Boy Who Reversed Himself William Sleator 1986
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The Planiverse A. K. Dewdney, 1984
Flatterland Ian Stewart 2001 [non-Euclidean geometry, also a sequel to Flatland]
Examples in media
Little Girl Lost
(The Twilight Zone 1962) http://thelastdrivein.com/tag/little-girl-lost/
The Borderland
(The Outer Limits 1963)
http://www.hulu.com/watch/63094?utm_source=tvcom&utm_medium=aff&utm_campai
gn=tvcomaff&cmp=913
Behold Eck
(The Outer Limits1964) http://www.youtube.com/watch?v=3TUDv9-PghM
Flatland (multiple movies have been made, most recently
http://www.flatlandthemovie.com/)
New: Sphereland the movie.
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Lego/Zometool constructions of Hypercubes, tesseracts, etc. How EXACTLY do you
represent 4D objects in 3D and 2D? Mathematics of projects from 4D to 3D and 2D
such as stereographic projections, their analytic/algebraic representations, make for
excellent mathematics work for students and teachers.
Construction of the result objects with Legos, Zometools, Tinker Toys, etc. and computer
animations, make for good mathematics projects .
As in the case of the previous Lego constructions the mathematics involved in creating
those instructions includes interesting and non trivial mathematics which will be
applicable in study of higher math and STEM courses (e.g. engineering in understanding
CAD design, computer science in understanding how computers display 3D objects on a
2D computer screen).
Examples of visual images of 4D objects on the web:
Hypercubes:
http://www.math.uiowa.edu/~wseaman/DGImage5310022.htm
http://www.youtube.com/watch?v=t-WyreE9ZkI
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Examples of methods of creating visual images of 4D objects:
We can use the idea of stereographic projection to map 4D objects to 3D objects. By
analogy, here is what stereographic projection mapping 3D objects to 2D objects looks
like.
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2D to 1D
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3D to 2D
YouTube animation of stereographic project of images on the sphere to the plane
http://www.youtube.com/watch?v=6JgGKViQzbc
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(From our HyperCube rotation Mathematica file)
The image of the cube under stereographic projection from
( 0 , 0 , 0 , l ) onto the 3 - space w = - b is shown.
StProj[l_, b_][{x_, y_, z_, w_}] :=
{(b + l)/(l - w) x, (b + l)/(l - w) y, (b + l)/(l - w) z}
Stereographic projection from ( 0 , 0 , l ) in 3 - space onto the 2 - space z = - b is shown.
StProj[l_, b_][{x_, y_, z_}] :=
{(b + l)/(l - z) x, (b + l)/(l - z) y}
Stereographic projection from ( 0 , l ) in 2 - space onto the 1 - space y = - b is shown.
StProj[l_, b_][{x_,}] :=
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{(b + l)/(l - y) x}
Projects: derive these equations. Use them to program a computer program to
create the computer displays. How do you modify the equations to show rotating
hypercubes or surfaces etc.? Lots of work with analytic geometry, matrices for
rotations and maybe some other linear algebra.
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