Circles can also intersect perpendicularly in a more complicated way than discussed in Spheres IV. Like so:

This might look complicated, but is in fact just a transformed version of the easier to grasp dart disk:

To see how these two images are related, pretend the radial lines in the second image are in fact huge circles that all intersect in the center point. Then they will also intersect in another point, which is, in the case of lines, the ominous point at infinity, but, in the case of circles, becomes just another point in the plane. This other point and the origin are the common points of one family of circles, as you can see in the first image, and the second family of circles intersects the first perpendicularly. The first image can be transformed into the second by what is called an inversion.

If we want to repeat this in three dimensions, it is maybe best to start with the second image, replacing the radial lines by vertical green planes, and the circles by concentric blue spheres. Then, something curious happens. Lines and circles are in some sense the same thing, and so are planes and spheres. But if we look for a third family of surfaces that intersect the planes and spheres orthogonally, we need to step outside the plane/sphere paradigm. It turns out that we need vertical red cones to cut both the blue spheres and the green planes perpendicularly:

Now, coming back to the 3D version of the first image, we just need to invert the above cones, planes, spheres as to become this:

The red surface is called a cyclide. It has two cusps that correspond to the tip of the cone and the (still ominous) point at infinity.

Now imagine that you are inside that cyclide, looking around…

Special Needs

The Best Friends Animal Society has their head quarters near Kanab, Utah. In a world where people kill each other because of a joke, the people here work for the sake of animals whose only privilege it is to be not human. My daughter and I volunteered in Benton’s House for special needs cats (blind, incontinent, you name it) for a few hours to just socialize with animals that would have been euthanized in most animal shelters long ago. I have no more words.

The Shape of Time (Sand Art II)

No, these dunes are not pink. The Coral Pink Sand Dunes State Park is purposefully misnamed, but it is still a place worth visiting.

The cream-orange colored sand offers home to a variety of life forms, all of which seem to be eager to leave some sort of trace. Here, this is in vain, as the rough high altitude has slowed down time. Any efforts of growth are reduced, and feeble attempts of drawing in the sand have become minimalistic.

Often, it is impossible to discern whether the specimens are still alive or dead.

But, even if dead, there is still art that can be shaped.

Stronger forces are attempting to leave longer lasting traces.

Fortunately, the State Park officer is armed, and time will reduce these tracks quickly to their proper relevance.

Intelligent Design

Intelligent Design is the slightly provocative title of a small, overpriced book I wrote, containing black and white graphics that show simple geometric phenomena, with explanations.

The constraint for the design was that it had to be cut out by a die cutter. I had acquired a Silhouette Cameo which can import AutoCAD dxf files and cut these very accurately (from card stock, for instance). One can then use these cutouts as window art.

The process puts interesting constraints on the graphics. It needs to be connected (otherwise it will just fall apart), simple, and simultaneously intricate.

Under these constraints, one can still achieve a modest 3D effect by thickening parts that should be close to the viewer.

This is a 7-4 torus knot. Look at it from some distance.

Snow Trillium

In the midwest, there is a fifth season between winter and spring, when everything seems to be in limbo for about a month. The temperatures rise above freezing point, but it’s not warm enough for any serious vegetation to spring up.

This is the time for the courageous, and one of them is the snow trillium. It typically blooms in early March, earlier than all other native wild flowers.

It enjoys steep limestone slopes facing south.

When I went looking today at one of my favorite wildflower spots, the Cedar Bluffs Nature Preserve in Indiana, it didn’t look good. Apparently one day of intermittent warming last week had lured the trilliums into growth, and they were than hit by a hopefully final wave of sub zero temperatures and snow. The result is not pretty.

Luckily, trilliums are very resilient where they like it. They will be back next year, courageous as always.

Update: The image above is not that of a dead snow trillium, but rather of a hepatica plant. More about this in a later post.

The 600-Cell (Spheres III)

Various arrangements of touching spheres, with a fair amount of color, reflections, and light, can lead to startling views, like this one:

So, what are we seeing here? In short, this is the stereographic image of the 600 cell, with its vertices being represented by spheres so large that they touch in the 3-dimensional sphere.

As usual, an analogy helps. Let’s start with the ordinary cube in space. This appears to be a 3-dimensional object. We can also think of it as a tiling of the 2-dimensional sphere by spherical squares, of which one fell off here:

Now, still working in the 2-sphere, place a spherical disk at each vertex of the cube with a radius so large that all the disks just touch:

To view this in the plane instead of in the 2-sphere, we can apply a stereographic projection, and get a rather boring looking collection of eight touching disks.

Now we repeat the same procedure in one dimension higher. The cube is one of the five platonic solids in 3-space. In 4-space, there are six regular polytopes, and one of them is the 600-cell. It consist of 600 tetrahedra that we can use to tile the 3-sphere. It also has 120 vertices. Placing a small 2-sphere at each vertex and connecting adjacent vertices by thin tori in the 3-sphere, results (after stereographic projection) in the following model.

Now make the 120 spheres so large that they just touch. The first image shows a partial view of these spheres. The spheres are all reflective, and we are standing inside the 600 cell, so we see mostly reflections of (reflections of) spheres.

Pine Hills Nature Preserve

Shades State Park in Indiana has so many wonderful spots that it is easy to miss the little Nature Preserve at its boundary.
The 15 minute access trail is not really preparing the visitor for what happens at its end: A steep descent leads into the narrow Clift Creek valley, and you are greeted with steep, barren rock faces.

The creek meanders around backbones with promising names like Devil’s Backbone that are at some points less than two meters wide but offer vertical drops of 30 meters and more. Crossing them in winter requires care.

Even from below, these overhanging rock faces are vertiginous.

Usually, the best time to visit Indiana landscapes is during the Fall, but this place is so complex that it is almost made for a reduced color palette.

This ancient sandstone cliff looks tired. Who wouldn’t, after all these years.

These rocks were left for a forgotten purpose, waiting now for time to end.