**The tautological clock**This humorous clock reminds me of the concept of a

**vector bundle** in mathematics. To form a vector bundle over a space X, we associate a vector space (such as k-dimensional Euclidean space) to each point of X in such a way that the spaces fit together to form another space of the same type as X. What this clock is doing is associating a clock to each time of day, and then fitting those together to form another clock.

More specifically, I think this clock is reminiscent of the concept of a

**tautological bundle**, which appears in the following context. The set of k-dimensional subspaces of an n-dimensional vector space forms a space called a

**Grassmannian**, whose points

*are* the k-dimensional subspaces. If we work over the real or complex numbers, the Grassmannian has the structure of a smooth manifold, which means that, within an n-dimensional space, it is possible to morph k-dimensional subspaces into each other in a smooth way. One can then form the tautological bundle over the Grassmannian by using the vector spaces given by the points of the Grassmannian itself! This sounds almost like a mathematical joke, and it is more or less the same joke that the clock in the picture is making.

The clock face shown is a discrete object in the sense that it only displays its configurations at intervals of one hour. However, vector bundles are continuous objects, and the analogy above suggests the following

**construction of a continuous version of the clock**. For this, we need a three-dimensional clock in which the small clock faces are inserted into the rim of the large clock face at right angles, instead of being placed flat on it. This could be used to show the continuous motion of the clock hands over a 12-hour period.

To make it easier to see what is going on, we could just keep track of the tips of each hand using fluorescent lighting. The continuous tautological clock would then have two coiled, illuminated rings running around its circumference: a larger, more tightly coiled ring showing the positions of the minute hand during a 12-hour period, and a smaller, more loosely coiled ring showing the positions of the hour hand during the same 12-hour period. It might not be a very useful timepiece, but I think it would look interesting as a work of art.

Is that the time? I should stop: it's almost 12-o'clock-o'clock.

**Relevant links**Grassmannians have applications to video-based computer recognition of faces and shapes, as well as to particle physics. More information on Grassmannians can be found here:

http://en.wikipedia.org/wiki/GrassmannianWikipedia on vector bundles:

http://en.wikipedia.org/wiki/Vector_bundleWikipedia on tautological bundles:

http://en.wikipedia.org/wiki/Tautological_bundleI don't know where this picture came from originally, and doing a reverse image search produces a large number of hits. I found it via

+Maximilian Montserrat.

#mathematics