These panels were constructed by David Jordan and his father while David was a senior math major at the University of Oregon (fall-winter 2005). They are installed in Hilbert Space in Deady Hall.
| Klein bottle and diagram indicating that area/circumference = radius/2. | Cardano's construction of roots of a cubic and diagram relating the nautilus shell to the golden ratio and the Fibonacci numbers. |
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| Closeups of both pairs of panels.
Click on any of the images to get large versions of the photographs! | |
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Topology: The red figure in the leftmost panel is a representation of the Klein bottle. It is a variant of the typical drawing of the 3-D representation which involves a self-intersection in order to embed the Klein bottle in 3-D.
Various features of this panel suggest the fundamental group of
the Klein bottle. In the small central aqua area, one can see the
semi-direct product symbol. The green suggests a
Geometry: In the second panel we see the division of a disk into a number of wedges, and the reassembly of those wedges into a figure which approximates a rectangle. The approximation to a rectangle becomes more and more precise as the wedges become narrower.
If we let the radius be r, and start with the circumference being 2*pi*r (even if we don't know the value of pi, we see that our "rectangle" will have area r*(pi*r) (the radius times half the circumference), so the original disk has that same area.
I believe this argument is attributed to the Egyptians.
Algebra: This is a representation of techniques published by Cardano (1501-1576) for finding the roots of a cubic (based on results also known to Tartaglia and del Ferro).
This method starts by simplifying to the case where the quadratic
term is 0. So we are solving an equation of the form x3 +
px + q = 0. One can easily check that if there are numbers u and v so
that
(1) u3 + v3 + q = 0
and
(2) uv + p/3 = 0
then x = u + v is a solution to the equation, and the other two
solutions are wu + w2v and w2u + wv where w is a
primitive complex cube root of 1.
We can solve for u by taking v = -p/3u from equation (2) and
substituting in equation (1). This gives the quadratic equation in
u3:
(u3)2 + q(u3) -(p3)/27 = 0
which can be solved for (u3) using the quadratic formula,
and then cube roots can be taken.
There will be 6 solutions for u (two square roots in the quadratic formula, 3 cube roots), but only three choices for u+v.
We can start by taking x to be a real root, in the case (I think) p real, and when the radicand that occurs in the quadratic formula is real and negative (these restrictions guarantee that all 3 roots will be real). In this case, u and v are complex conjugates, and the other roots are also real. In the panel, x is the point on the horizontal axis at the far right. u is (say) the point half-way along to x but above the horizonal axis where the white triangle in the first quadrant meets the blue triangle in the first quadrant. v is the point below u in the fourth quadrant at the bottom of the place where the two triangles meet.
In the second quadrant we can see wu, wv, and wx in a similar figure. In the third quadrant we can see w2u, w2v, w2x in another similar figure. The point furthest to the left on the horizontal axis is made by wu + w2v (constructed in the second quadrant) or symmetrically as w2v + wu (constructed in the third quadrant). This is a second root of the cubic.
We can also see a line from wv down to the horizontal axis in the second quadrant and a line up w2u to the horizontal axis in the third quadrant. This point on the horizontal axis where these lines meet (close to the origin) is the third root, wv + w2u = w2u + wv.
Superimposed over this picture is a similar picture where the radicand which occurs when solving for u is positive. This leads to the case of one real root, and a pair of conjugate complex roots (which are in the second and third quadrants).
Analysis: This is a picture that captures the golden ratio and the curve of the nautilus shell.
Let f be the golden ratio. The picture is made by starting with a small 1xf rectangle. Place a square next to it of side the longer of the two sides. We now have a 1x (1+f) rectangle. Place a square next to that of side the longer of the existing two dimensions (1+f). Continue this way. At each stage, the rectangle made of all the squares added so far is a golden rectangle. And a curve through the meeting of each new square with the previous rectangle which stays on the outside of the figure constucted so far is the curve of the nautilus shell!
An examination of a large version of the close up reveals that there are two colors to the background squares. Some are light purple, and some are more or less clear. This probably isn't a reference to Tetris, and it probably means something, but I don't know what!