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.

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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.

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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.