Functional Phylogenies

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We now regularly use the genetic sequences of different creatures to make guesses about their common ancestors. A creature's genetic sequence is a long string of symbols. But how can we discuss common ancestors if, instead of knowing strings of symbols about each creature, we only have information about their shapes? Though this is a particularly difficult situation, it is very common. Many objects in the world around us evolve and change their shape but do not have a genetic sequence. For example, the shapes of consumer items, from toothbrushes to cars, are under continual shape evolution but no-one supposes they have genomes. The challenge of making inferences about the past evolution of shapes, and of making guesses about which shapes have recent ancestors in common, is well established. The standard approach to this challenge is to extract sets of numbers that describe the shapes of interest and to use these summary sets to make guesses about the past. This proposal takes a different approach. We aim to develop mathematical techniques that use the shapes themselves, rather than summaries of them, to make inferences about the past. This approach has some advantages: it uses more of the information that we have; it allows us to characterize the process that yields shape evolution; and it allows us to make guesses about the shapes of unseen ancestors (rather than guesses about a restricted set of their features).This proposal aims to advance the study of shape evolution by considering the evolution of mathematical functions. A functional phylogeny is akin to a genetic evolutionary tree where it is a mathematical function, rather than a genetic sequence, that changes through time. We will test our theory by: using computer generated data; performing `Spatial Chinese Whispers' experiments; and investigating how the pronunciation of words has evolved in Romance languages. Our `Spatial Chinese Whispers' experiments will involve the task of drawing curves, one after the other, so that small errors in copying yield a slow drift in curve shape. Our investigation of speech sound evolution first converts the sounds of modern speakers into shapes and then exploits the fact that we are confident about the evolutionary tree of Romance languages. Just as information about current genetic sequences allow us to make guesses about the sequences of past organisms, this approach might allow us to test hypotheses about the sounds of languages we can no longer hear.This work has relevance to those interested in designing shapes for the future, as well as those interested in past shapes. By understanding past shape evolution one can use this to generate reasonable shape transformations which might help in product design. This work aims to take sets of shapes and to make guesses about which is most related to which: this can be very useful in areas which have nothing to do with shape evolution. The ability to detect unusual or familiar shapes has relevance to numerous public and commercial challenges from spotting unusual vehicles to guessing the shapes of letters.