At the MSRI in Berkeley, there is a marble sculpture by Helaman Ferguson showing Klein’s quartic surface.
This is a Riemann surface of genus 3 with 168 automorphisms. Our Euclidean brains have a hard time seeing all these. Let’s start with an automorphism of order 7, and a tiling of the plane by π/7 triangles:
Fourteen of them fit around a common vertex (at the center of our hyperbolic universe), and the black geodesic indicates how to identify edges of the green-yellow 14-gon (repeat the pattern by 2π/7 rotations). Euler will tell you that the identification space has genus 3. A little miracle is that these π/3 triangles fit nicely into a tiling by π/3 heptagons. This becomes evident like so:
The geodesic we used to indicate the 14-gon identification pattern becomes a geodesic in the heptagon tiling that passes through edge midpoints of eight consecutive heptagons, and all such geodesics will be closed on the identification space. This allows to define this surface also as an identification space of 24 heptagons (using the same geodesics). As this description is intrinsic to the heptagon tiling, it is invariant under all symmetries of that tiling, which include rotations of order 2 and order 3, in addition to the order 7 rotation.
Why is this surface called a quartic? Replacing the hyperbolic π/7 triangles with Euclidean (1,2,4)π/7 triangles in three different ways and keeping the identifications, we obtain three different translation structures on the Klein quartic, which define a basis of holomorphic 1-forms. Playing with their divisors show that these 1-forms satisfy the equation x³y+y³z+z³x, showing that the canonical curve of Klein’s surface is a quartic curve in the complex projective plane.
I have written before about the perspective vertically down, and complained that in Indiana, you only see mud or decaying leaves. So, let’s have a look.
What is this stuff? I have only seen it at the DePauw Nature Park, near water. It is likely organic, but never green. Is there a zombie-plant whose natural state of existence is that of decay?
But not everything is decaying. Roots are feeling their way, and algae cover everything in wondrous patterns.
Tiniest plants remind us that we are little, too.
Hence let us rest…
In high school I usually loathed art class. But once we started an architecture project that got me excited: We were tasked to design our own house, with all bells and whistles. I decided on the rooms being regular hexagons, arranged in an annulus of six, with a center hexagon without windows.
This is the motivation behind today’s surface. Six hexagons, arranged as above, with the additional stipulation that when you exit a hexagon you re-enter another hexagon as indicated by the arrows (and extended by rotational symmetry). You can easily check that when you travers the rooms by always leaving through opposite walls, you will pass alternatingly through two rooms and return. Moreover, there are a total of six rooms around each corner, which suggest that Euclidean geometry is not suitable for this architecture.
In the hyperbolic plane, one can arrange six 60 degree hexagons around a vertex as above. The geodesics indicate which edges are to be identified, again implying silently that everything is rotationally symmetric.
Now in the above Euclidean model the identifications are done by Euclidean translations, defining what is called a translation structure. One can accomplish the same with other Euclidean polygons (that are still conformal images of a regular hexagon) like so:
In the above image we have a very short inner edge connecting the 240 degree vertices. There are a few more one can use, but they are not quite as pretty. In any case, they provide us with plenty of holomorphic 1-forms on the surface of genus 4 given by the algebraic equation w⁶=z⁶-1: This is, after all, a sixfold cover over the sphere, branched over the sixth roots of unity. The first model realized this geometrically by replacing the sphere by the double of a Euclidean hexagon.
I am not good with names. I recognize maybe a handful of Indiana wild flowers, but that’s it. In particular the small ones I tend to ignore. So, please take the names in this post with a grain of salt.
This one above, for instance, I believe is called Salt and Pepper (Erigenia bulbosa), and it is tiny (the petals are just 2mm long). So you see, I have been exercising my macro photography skills.
A little larger is the Bloodroot (Sanguinaria canadensis). Its white petals are extremely delicate.
The one above was again very small, and I have no clue what it is. The last one I also don’t know. The little buds or whatever they are were maybe 2mm in diameter. Very cute.
Recently, a local artist had an intriguing question. Suppose you have a hook in the ceiling (who hasn’t?), and two spot lights in front of the hook, slightly to the left and to the right. Suppose also that you have drawn two curves on the back wall. Can you bend a wire and suspend it from the hook so that the two projections match the drawings?
I first thought: Yes, this means we just have to determine the intersection of two cones, so this is possible but maybe tricky.
After playing around with it a little I realized that this is simpler than I thought: Of both curves have the same height, this is essentially always possible, and even completely explicit. In fact, this is almost as simple as using two perpendicular parallel projections.
For instance, below you see a single red wire that has two figure 8 curves as projections.
Then of course one wants to play with it and rotate the wire.
Clearly, there are two more rotational positions where one of the projections is again a figure 8, the one above and the one below.
Now we need to find somebody who can accurately bend wires for us.
This year everything seems to be late.
This is ok. Somewhat worse is that the prices for Darjeeling have gone up again. I can only hope that the workers benefit from it, too.
My favorite this year so far is the Upper Namring “Premium”. I don’t know whether these little epithets like “Premium”, “Wonder”, “Exotic” or “Supreme” have a qualifying meaning; I liked it better when they would just call it “Invoice 12”, counting the harvests. But clearly that requires explaining, while everybody seems to understand “Wonder”.
This “Premium” harvest ic clearly not completely uniform, but I like the mix of bright green leaves with the rolled darker ones, this gives the tea a slightly grassy note in addition to the floral character of a powerful Darjeeling.
Next time I will try to visually convey the taste differences between 2017 and 2018 Puttabong Moondrops.
Below is something rare. You see two minimal surfaces in an (invisible) box that share many properties, but also couldn’t be more different.
Let’s first talk about what they have in common: They share lines at the top and bottom of the box, and they meet the vertical faces of the same box orthogonally. This means you can extend both surfaces indefinitely by translating the boxed surfaces around, in which they become triply periodic surface of genus 3.
How are the different? The red one is a little bit more symmetric and belongs to a 2-dimensional deformation family of the Diamond surface that has been known for about 150 years. You can see how these surfaces deform in an earlier post.
The other one belongs to a different deformation family that is only a few weeks old, discovered by Hao Chen, and of which you can see here some wide angle pictures, with clearly different behavior.
These surfaces existed right under our nose, but nobody expected them to exist, because minimal surfaces are usually content with a single symmetric solution. Chances are that these surface hold the key in understanding the entire 5-dimensional space of all triply periodic minimal surfaces of genus 3.
Snow in April is a rare thing in southern Indiana.
The snow cover was very light and didn’t stay long on the warm ground.
But the trees were decorated with lots of tiny white accents from snow flakes and water droplets, creating an unusual winter landscape.
Instead of the typically harsh winter sun, everything was bathed in ambient light.
The pictures were all taken with the Lensbaby Velvet 85, fully open at 1.8. The stopped down images I took also look gorgeous, but have an entirely different, less surreal character.
More snow is predicted for Friday night. I can’t wait.
In the overwhelming book The Symmetries of Things by John Horton Conway, Heidi Burgiel and Chaim Goodman-Strauss there is an image of an object called Octahedral 3¹². You get an essential part of it by placing eight antiprisms over a triangle (or octahedral annuli) inside eight cubes like so:
The precise placement of the octahedra is so that the yellow dots divide the cube edges 1:3. Then you can repeat this periodically to get a triply periodic polyhedron where at each vertex exactly 12 triangles meet. This object is already invariant under the translations along the cube diagonals, so the quotient surface is an polyhedral surface of genus 4 (Euler!), tiled with 24 equilateral triangles, still 12 at each vertex. We can unwrap this package into the hyperbolic plane by replacing each Euclidean triangle with a hyperbolic 30º triangle:
The triangles corresponding to individual anti-prisms are colored with the same color, and the waist polygons of the anti-prisms become the shaded geodesics. That they close on the antiprism after passing through six triangles enforces the identification pattern of this 24-gon, indicated by letters. If you continue this, you can check that this hyperbolic version is in fact invariant not only under the 90º rotation about the center, but under the 30º rotation. In particular, we have also a 120º rotation, and the quotient of the surface by the generated cyclic group is a sphere tiled with 8 triangles — an octahedron, as is not hard to see. The same surface is also the basis for a famous triply periodic minimal surface. The inner 12-gon corresponds to the following minimal piece:
Copies of it can be assembled to larger portions, like so
Who am I?