We consider the problem of finding the probability that a random triangle is obtuse, which was first raised by Lewis Caroll. Our investigation leads us to a natural correspondence between plane polygons and the Grassmann manifold of 2-planes in real n-space proposed by Allen Knutson and Jean-Claude Hausmann. This correspondence defines a natural probability measure on plane polygons. In these terms, we answer Caroll's question. We then explore the Grassmannian geometry of planar quadrilaterals, providing an answer to Sylvester's four-point problem, and describing explicitly the moduli space of unordered quadrilaterals. All of this provides a concrete introduction to a family of metrics used in shape classification and computer vision.The issue of choosing a "random triangle" is indeed problematic. I believe the difficulty is explained in large measure by the fact that there seems to be no natural group of transitive transformations acting on the set of triangles.-Stephen Portnoy A Lewis Carroll pillow problem: Probability of an obtuse triangle Statistical Science, 1994In 1895, the mathematician Charles L. Dodgson, better known by his pseudonym Lewis Carroll, published a book of 72 mathematical puzzles called "pillow problems", which he claimed to have solved while lying in bed. The pillow problems mostly concern discrete probability, but there is a single problem in continuous probability in the collection:Three points are taken at random on an infinite plane. Find the chance of their being the vertices of an obtuse-angled triangle. This is a very appealing problem and a number of authors have tackled it in the years since. After a moment's thought, it is clear that the main issue here is that the problem is ill-posed-since there is no translation-invariant probability distribution on the infinite plane, the problem must really refer to a natural probability distribution on the space of triangles. But what probability distribution on triangle space is the right one? Portnoy [23] presented several different solutions to the problem involving distributions on triangle space invariant under various groups of transformations; Edelman and Strang [9] connect the problem to random matrix theory and shape statistics; Guy [10] got the answer 3/4 for a variety of measures, and the legendary statistician David Kendall got exact answers when the vertices of the triangle were chosen at random in a convex body [16]. Interestingly,

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Neurons in the gustatory cortex (GC) represent taste through time-varying changes in their spiking activity. The predominant view is that the neural firing rate represents the sole unit of taste information. It is currently not known whether the phase of spikes relative to lick timing is used by GC neurons for taste encoding. To address this question, we recorded spiking activity from >500 single GC neurons in male and female mice permitted to freely lick to receive four liquid gustatory stimuli and water. We developed a set of data analysis tools to determine the ability of GC neurons to discriminate gustatory information and then to quantify the degree to which this information exists in the spike rate versus the spike timing or phase relative to licks. These tools include machine learning algorithms for classification of spike trains and methods from geometric shape and functional data analysis. Our results show that while GC neurons primarily encode taste information using a rate code, the timing of spikes is also an important factor in taste discrimination. A further finding is that taste discrimination using spike timing is improved when the timing of licks is considered in the analysis. That is, the interlick phase of spiking provides more information than the absolute spike timing itself. Overall, our analysis demonstrates that the ability of GC neurons to distinguish among tastes is best when spike rate and timing is interpreted relative to the timing of licks.SIGNIFICANCE STATEMENTNeurons represent information from the outside world via changes in their number of action potentials (spikes) over time. This study examines how neurons in the mouse gustatory cortex (GC) encode taste information when gustatory stimuli are experienced through the active process of licking. We use electrophysiological recordings and data analysis tools to evaluate the ability of GC neurons to distinguish tastants and then to quantify the degree to which this information exists in the spike rate versus the spike timing relative to licks. We show that the neuron's ability to distinguish between tastes is higher when spike rate and timing are interpreted relative to the timing of licks, indicating that the lick cycle is a key factor for taste processing.

We consider the space M of Euclidean similarity classes of framed loops in R 3 . Framed loop space is shown to be an infinite-dimensional Kähler manifold by identifying it with a complex Grassmannian. We show that the space of isometrically immersed loops studied by Millson and Zombro is realized as the symplectic reduction of M by the action of the based loop group of the circle, giving a smooth version of a result of Hausmann and Knutson on polygon space. The identification with a Grassmannian allows us to describe the geodesics of M explicitly. Using this description, we show that M and its quotient by the reparameterization group are nonnegatively curved. We also show that the planar loop space studied by Younes, Michor, Shah and Mumford in the context of computer vision embeds in M as a totally geodesic, Lagrangian submanifold. The action of the reparameterization group on M is shown to be Hamiltonian and this is used to characterize the critical points of the weighted total twist functional.

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