# D'Arcy and the Gnomons.

What does a mollusk called 'head-foot' and a Scotsman
with a penchant for ice-cream have in common? The answer lies in a
sunflower. Or the path of an insect as it flies toward a light. Or a
Greek column. Or a pineapple. Is mathematics to be found in a textbook
or a shell? When scientists model nature, are we inventing something
new, or simply discovering relationships as old as the universe itself?
And how can all this help design hearing implants?

ON MATHEMATICS AND NATURE

Some see mathematics as a product of the mind. As a thought, a human conception that lives in the mind and sleeps in a textbook. Surely the most abstract and esoteric corners of mathematics have no place in the world of ants and rocks--after all, where can we point to find a Picard Group (1) or a hypercube?

The peculiarly Pythagorean view of the world as number sees mathematics as the key to understanding the universe--a view implicitly shared by physicists of our day, albeit without the same mysticism. Pythagoras saw mathematics as an intrinsic part of nature, music and the stars; something magical to be discovered. Pythagoras' thinking clearly inspired Plato's Forms and Solids, (2) for he conceived a model of the universe based on the dodecahedron. Some 2000 years later, Kepler's revolutionary elliptic planetary model was directly influenced by Pythagoras' Spheres, (3) which has more recently inspired the spatial design of the [C.sub.60] Buckminsterfullerene molecule. (4)

Western philosophy and mathematics have been intimately entwined from their beginnings. Plato, Descartes and even Kant were foundationalists who sought purity and 'transcendent perfection' in mysteries both mathematical and natural. (5) The earliest philosophers were deeply concerned with geometry, astronomy and logic. Gottlob Frege challenged the dichotomous notion of insisting mathematical concepts were either physical or mental by describing them as 'abstract objects', which is taken a step further in some modern thinking by describing them in terms of a quasi-collective consciousness.

But while Platonism would say we cannot 'invent' mathematics, but only discover it, an empiricist would insist we only learn about mathematics through our senses as we interact with the world. (6) Reuben Hersh offers us an intriguing perspective: that of mathematics as humanistic pursuit, a socio-historic phenomenon, (7) invoking the Fregean notion of the abstract. He points out that mathematics as we know it today is a product of history and human activity, including its mistakes, false starts, incompleteness and limitations.

So while there is no clear consensus on the pure universality of mathematics, the question has not impeded the pursuit. On the contrary, modern society is literally built on the sometimes shaky foundations of mathematics. Despite Kurt Godel, (8) we still teach 'incomplete' algebra, and despite Georg Cantor and Bertrand Russell (9) we still make heavy use of set theory. And while some invoke a deity in rapturous wonder at mathematics, as quoth Mary Somerville:

other giants, such as logicist Russell, are content to do without when he states that he

Perhaps it is human nature to invoke a higher power when faced with something so awesome and wondrous that we struggle to conceive of it 'just happening'.

ON SPIRALS

The seafaring mollusk known as nautilus (Cephalopoda Nautiloidea, or 'head-foot sailor') carried a calcified rendering of a mathematical wonder for its shell millions of years before Descartes anointed it with the first of its many names. The logarithmic spiral is also known as the growth spiral or the equiangular spiral. The eminent mathematician Jacob Bernoulli was so enamoured with this spira mirabilis, the miraculous spiral, he wished it inscribed on his gravestone. (12)

The logarithmic spiral can be defined as a curve that exhibits a constant angle between the radius vector (a line from the centre to a point on the curve) and the tangent vector (a line oriented along the path of travel). Try this: put a bin in the middle of your office, stand next to it, then stretch out your left arm so that it is pointing approximately at 10 o'clock. (13) Walk backwards around the bin, moving further away from it while keeping your arm pointing directly at the bin. Stop before you run into your desk, and you will have traced out a logarithmic spiral. The tangent vector is pointing out your back, while your sore arm (14) is the radius vector.

The logarithmic spiral has many special properties that make it very useful in both nature and engineering: it is self-similar, in that its shape remains unaltered by scaling and angular growth; the distance between arms increases in a geometric progression; (15) any straight line passing through the origin makes a constant angle with the curve (Figure 1); a degenerate logarithmic spiral is a straight line at one extreme and a circle at the other; it can be produced using incredibly simple rules, such as 'move forward a bit, turn left 30 degrees'. If instead of moving forward, we simply 'grow and turn', we enter the domain of Lindenmeyer Systems: a formal set of simple rules capable of generating remarkably complex fractal figures, such as trees and ferns. (16)

[FIGURE 1 OMITTED]

Nature provides a cornucopia (17) of examples of the logarithmic spiral in action. The arrangement of seeds in a sunflower--the optimal arrangement for efficient packing. The path of an insect as it flies toward a light--arising from the structure of its compound eyes. The path of an eagle as it swoops on its prey--so it can keep a constant eye on the target. The arrangement of scales on a pineapple. The swirling rage of a tropical cyclone. Or the swirling mist of stars in a galactic spiral. And the shell of a mollusk--due to the accretive mode of its construction. You may even find one in your own backyard.

So what does a logarithmic spiral have to do with rabbits? For rabbits were the etude of the great Fibonacci: how fast could they breed in ideal circumstances? The sweat of his brow (and that of his rabbits) produced a fascinating result, and thus his eponymous (18) sequence. (19) That is, the total population of this idealised colony after each generation is given by 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 ..., where each successive number is the sum of the previous two.

[FIGURE 2 OMITTED]

While many people are understandably riveted by this ingenious model of bunny proliferation, the fun doesn't stop there. Fibonacci numbers crop up in a surprising number of places in nature around us: the number of petals on a flower; the number of spirals of seeds in a flower head or a pine-cone; the number of leaves per turn around a stem; cauliflower florets; and romanesque broccoli. (20)

But there are some things even more interesting than rabbits and cauliflower. For example, we can geometrically construct an approximation of our miraculous logarithmic spiral with rectangles sized according to the Fibonacci sequence.

[FIGURE 3 OMITTED]

The Golden Mean is another very special number with many names. It is a natural relationship, a special proportion revered by artists and mathematicians alike. Many consider it not only aesthetically pleasing but almost mystical, due in no small part to its intriguing properties and history. (21) Between two measures, it is the ratio of the whole to the larger, as the larger is to the smaller. That is, (see Figure 4). Also known by the Greek symbol phi or [phi], the

[FIGURE 4 OMITTED]

a + b / a = a / b

Golden Mean or Divine Proportion is associated with a somewhat controversial area of study known as 'Sacred Geometry', a particularly Pythagorean concept. Phi is ancient, and may be found variously in nature, architecture, art and music. It is frequently claimed that the facade of the Parthenon is laid out according to the Golden Mean, so too the dimensions of the pyramids, (22) yet these claims (and many others) have been debunked as 'wishful thinking'. There are, however, many genuine examples, (23) such as the work of Paul Serusier (1864-1927), Gino Severini (1883-1966) and the Cubists. The renowned architect Le Corbusier invented a comprehensive system of proportions called Modulor, which was based on a combination of the Fibonacci sequence, the Golden Mean, and the proportions of da Vinci's Vitruvian Man, designed to provide a 'range of harmonious measurements to suit the human scale, universally applicable to architecture and to mechanical things'. (24) Unfortunately, while there are many authentic manifestations of the Golden Mean to be found across human pursuits, it is often difficult to distinguish these from overly optimistic seekers who retrospectively juxtapose the ratio onto older works with insufficient evidence or justification.

Now it was in fact our friend Kepler who discovered that the ratio of successive Fibonacci numbers converges to [phi], the Golden Mean. (25) It wasn't until as recently as 1994, however, that two French mathematicians, Stephane Douady and Yves Couder, proved (26) that the optimum growth pattern for the most efficient packing in a plant occurs when the angle between successive primordia (points of growth) is the irrational Golden Angle, 2[pi](1-[phi]). Thus these three mathematical concepts--logarithmic spirals, Fibonacci numbers and the Golden Mean--are intimately related, and appear to form part of nature itself. It is perhaps no wonder that many mathematicians are moved to poetry when beholding such elegance, depth and beauty.

Now D'Arcy Thompson (1860-1948) liked to walk around St Andrews, Scotland with a parrot on his shoulder. He was also very fond of ice-cream, and translating classical Latin and Greek texts. But more than for his parrot, he will be remembered for his opus magnum, On Growth and Form (1917), wherein an entire chapter is devoted to the logarithmic spiral. (27) In these pages we find a poetic treatise on natural development, a thorough analysis of patterns of growth and models that became a cornerstone of modern morphometry (the study of the shape of living organisms). Thompson's unifying theory is that ontogeny (individual growth) is determined by the physical forces of nature, and shaped by environment. This powerful statement, to some extent at odds to Darwinian theory (and pre-dating the discovery of DNA), is now generally accepted as complementary to our modern notions of growth, development and biological processes.

Shells, such as the nautilus discussed earlier, grow by accretion; matter is gradually accumulated or deposited at the opening, enlarging as it winds around a central axis. (28) This means that only the opening ever grows; the original form never changes, so that a young shell has the exact same form as a larger one of the same species. Thompson marvels that 'this remarkable property ... is characteristic of the equiangular spiral, and of no other mathematical curve.' Such a shell can be described as a gnomon, which in its most general sense is 'any figure which, being added to any figure whatsoever, leaves the resultant figure similar to the original.' (29) The word has its origins nearly 2000 years ago with the mathematician Hero of Alexandria, in a day when people were more interested in the design of sundials; (30) the gnomon was the 'indicator', the part of the sundial that cast the shadow. Gnomons can be found in many areas of both mathematics and nature, shells being the most obvious exemplar.

In exploring just this one particular aspect of mathematics, we find richly intertwining relationships, and more examples than could possibly be surveyed in this manuscript. It is clear that these relationships are there to be discovered and applied to new areas of enquiry: all we need is the right mathematical language, and a way to determine how well our views fit our observations. We need models.

ON MODELS

Model (n): a system or thing used as an example to follow or imitate; a simplified description, esp. a mathematical one, of a system or process, to assist calculations and predictions.

--Oxford English Dictionary

Mathematical modelling is the judicious art of oversimplification.

--Unknown

As Rodney Brooks is fond of pointing out, 'the world is its own best model.' This philosophy, espoused in his 'subsumption architecture' for robotics, (31) argues that the classical symbol-system of artificial intelligence (32) is 'fundamentally flawed'. He posits that any attempt to recreate an internal model of the external world is inherently lacking in every way, and is therefore pointless. Brooks explicitly rejects a reductionist approach to artificial intelligence, and argues strongly for a biologically inspired approach. Despite this, many roboticists today opt for a pragmatic hybrid approach. For example, a robot does not need to distinguish the fine grain of the hardwood, or the sheen of the lacquer to avoid running into a chair; recognising it merely as an obstacle is perfectly sufficient. A purposive model is one that is tailored specifically for the task at hand, with no more detail or complexity than is needed to satisfy the goals.

This serves to highlight the distinction between 'modelling perfectly' and a 'perfect model'. It may be that modelling perfectly, as Brooks' earlier observation alludes to, would require a model equivalent to the subject itself, since anything less would necessarily be imprecise or incomplete, and therefore imperfect. The 'perfect model', however, is a model that completely satisfies all the requirements of the task at hand, and is therefore achievable. In all practical scenarios, with finite time and resources, we will most likely have to satisfy ourselves with a 'good model'; one which is as accurate as we can practically make it given all the constraints, and for which the limitations and shortcomings are understood.

The father of computer science, Alan Turing, shared Thompson's fascination with nature and numbers. Indeed, Turing was greatly influenced by On Growth and Form; as a child he spent hours sketching flowers, poring over pine-cones, examining seeds and searching for relationships and patterns. His unfinished work in the latter period of his life explored plant growth and the relationship between the Fibonacci sequence and sunflowers. (33) In his seminal 1952 paper on chemical morphogenesis, (34) Turing rather modestly described his proposed model as 'a simplification and idealization, and consequently a falsification'. While it has since become 'the definitive basis for modelling biological growth', (35) Turing rightly (and succinctly) questions the nature of the modelling process, and just how much faith we can place in these idealised forms. Fortunately, all is not lost.

So what makes a good model? Ideally our model would be:

* General: can be applied to a wide variety of normal subjects.

* Expressive: captures essential aspects of the subject in a meaningful way.

* Simple: no more complex than necessary to carry out the task.

* Verifiable: can be validated against observation.

Are mathematical models appropriate for modelling nature? If we take the word of the Platonists, most definitely. In practice, engineers and physicists use them all the time, cognisant of their inherent limitations. But is there an intrinsic mathematical relationship there for us to find in nature? An idealist would likely say there is, but whether or not we can find it is another question. How would we recognise it if we found it? That is not so easy to answer. When evaluating a model, we can only increase our confidence the more we successfully validate it against different subjects.

Are we not inherently limited by our mathematical toolbox? Mathematicians are constantly expanding the horizons of theory in pure and applied domains. Our knowledge is assuredly finite, but growing rapidly. We may not yet have the tools at our fingertips in order to find the 'perfect model', so we make do with what we have. Sometimes, we really do just see every problem as a (linear function) nail, and bang on it with our (linear algebra) hammer.

Modelling as an activity is after all purposive, in that we have a goal, a subject and a set of constraints. We should also have some means to measure how well our model fits the subject, and an idea of just what is 'good enough'. There is in general no single best solution--the efficacy of a model can be measured only in terms of its fitness for purpose. Good practice would have us apply Occam's Razor to all models; that we employ the simplest explanation. In the same vein, the principle of parsimony eliminates from a model any parameter that does not contribute significantly to the explanatory power of the model. Linear is better than non-linear, fewer parameters are better than more parameters.

With this simplicity and generality in mind, we turn our attention to a particularly interesting problem at the nexus of the above, seemingly disparate topics: producing a mathematical model of a very small spiral organ.

ON THE COCHLEA

Millions of people worldwide suffer from profound sensorineural hearing loss. One in a thousand babies is born with congenital hearing defects, while around forty percent of people over the age of seventy-five develop progressive loss. The cochlear ear implant has become the standard clinical intervention for nerve-impaired deafness, with over fifty thousand recipients in 120 countries worldwide. The cochlear implant restores hearing by sending tiny electrical impulses to the residual nerves (36) (Figure 5).

[FIGURE 5 OMITTED]

Your humble author is engaged in the development of a three-dimensional shape model of the cochlea that may ultimately assist clinicians in providing better treatment for people so affected. The research is aimed at analysing cochlea morphometry, deriving metrics to help quantitatively describe cochleae (where there are now predominantly qualitative descriptions), and constructing a mathematical model that can capture its essential characteristics. Such anatomical shape models will give us insight into normal and abnormal shape variation. They may ultimately be used for a variety of purposes: diagnosis, by providing quantitative measures that can be compared to known normal forms; simulation and training, by enabling clinicians to generate a variety of plausible forms; and surgery planning, by highlighting regions of unusual shape that may require attention.

The cochlea (Latin for 'snail shell') is the organ of hearing, a tiny 2[cm.sup.3] shell-like structure in the inner ear, embedded in the temporal bone of the skull. It is spiral in shape, and is often described as approximating a logarithmic spiral. A normal cochlea revolves through two and a half turns, from the basal turn (lower turn) up to the helicotrema (top of spiral). Three channels run the length of the cochlea: the scala tympani, scala media and scala vestibuli. The cross-sectional shape resembles a cardioid (rounded 'B' shape).

While detailed studies have been carried out on the morphology of the bony labyrinth (incorporating the cochlea), (37) the metrics are typically linear measurements or relative orientations--for example, the width of the basal turn of the cochlea, or the angle of the lateral semicircular canal in the sagittal plane, all of which are most relevant to phylogenetic studies. Most of the cochlea models described are spacecurves that either do not take into account cross-sectional shape at all, (38) or use an approximation such as a circle. Clearly there is a need for clinically relevant models of the cochlea, beyond what is currently described in the literature.

It may be tempting to apply Raup's shell models (39) to the cochlea. These biological models of shell shape are simple yet effective parametric models that reproduce a wide variety of shapes found in nature, as well as a wide variety that are not. This may simply imply that the parameter space of the model only partially overlaps with the parameter space of the system, or it may imply that the natural model is better described in some other way. Or it may simply imply that the natural model has implicit constraints on its parameters, which is consistent with Thompson's theory of phylogeny. Since it is simple and works very well for a wide variety of examples, we can say it is a good model.

However, assuming the Raup model was an accurate representation of the actual biological growth process for shells, there is still the question of whether it is appropriate, or semantically and biologically valid to apply an accretion growth model to the cochlea. First is the fact that the curve of Raup's shells grows from top to bottom (apex to base), growing outward. However, the cochlea itself grows from base to apex, curving inward. Indeed, it is fully formed by the age of seven months in the womb. Second is the fact that the cochlea does not grow by accretion, and its development is linked to the growth of the temporal bone itself. Since so much of our knowledge of the growth and development in utero is incomplete, we must focus on what we can best observe and measure, which is primarily CT scans of fully-developed adult temporal bones.

L T Cohen describes a 2D spiral model that was originally designed to model the path of the implant electrode itself (40) (and not of the otic capsule, the bone that encapsulates the cochlea). Sun Yoo et al. describe an extension of Cohen's model, reinterpreting it as defining the centreline of the cochlea. Yoo added an exponential height function to the 2D spiral, and described a method of fitting the spacecurve model to CT data. (41) Until recently it was believed that the shape of the cochlea was primarily due to an economy of space; however, a new theory suggests that the spiral shape of the cochlea is highly significant, and can have an amplifying effect of up to 20dB for low frequencies. (42) The mechanism is believed to be similar to the 'whispering walls' effect in St Paul's Cathedral in London, where one can whisper at one side of the gallery and be heard on the other. (43)

This author is investigating a 3D parametric shape model, based on the models of Cohen and Yoo, since they are well-known in the literature and provide a simple and convenient starting point. My model extends these into 3D, adding a full outer surface and an elliptical cross-section. From the outset, the goal of this work is to produce a model that features a clinically relevant parameter space, and incorporates or yields metrics that are directly relevant in the clinical domain. (44) The explanatory power lies in the parameter space and its embedding in the real world; otherwise any numbers derived from such a model would have no basis for biological interpretation. Metrics such as the average diameter of the basal turn or the curvature of the centreline of the otic capsule are far more clinically relevant for the purposes of planning for cochlear implant surgery than (for example) the linear measurements described above, or the coefficients of some convenient but effectively arbitrary trigonometric function.

The rigid model currently being explored (45) is useful as a first approximation as it incorporates the gross-scale metrics that can uniquely describe the shape of the cochlea. However, it is also inherently unable to capture fine-scale detail, such as local pathologies, deformities and so on, and may even be incapable of deforming to normal shapes (only extensive fitting to normal samples can confirm this). So ultimately this model may not be the most pure, elegant or abstract, or even the most accurate. But in terms of fitness for purpose, the model satisfies our criteria for simplicity (seven parameters), expressiveness (the parameters are clinically relevant), general, and verifiable (it is currently being fitted to a range of human CT scans). Knowing the properties and limitations of this parametric model, the model fitting process is being extended to incorporate a fully deformable model, which will not only provide a means for validation but can also be used to quantify the regions of the model that poorly fit the organ, so that we may further improve the model and its ability to be generalised.

Here then is the intersection of nature, spirals, mathematics and models: can we represent the cochlea with a logarithmic spiral in a clinically meaningful way? Does a logarithmic spiral actually capture the natural growth and form of the cochlea?

BEYOND

The research described herein invokes a certain degree of Pythagorean idealism in its search for mathematical truth and beauty in nature. But it is necessarily tempered with pragmatism, for a clear goal stands before us. The path ahead is long, but the early results are encouraging.

More than a model of nature, mathematics can be seen as the language of relationships in nature. As we add to our vocabulary, our mathematical toolbox becomes richer, allowing us to capture, express and explore more profoundly the theoretical aspects of the patterns of life before us. The limits of our models only serve to highlight the limits of our finite understanding of the universe, which is at once humbling and challenging.

ENDNOTES

(1) This of course refers to a particular type of multiplicative Abelian group in Number Theory, and not the television series Star Trek.

(2) Forms: perfect, abstract entities that exist independently of the world. Solids: five perfectly symmetrical three-dimensional geometric shapes composed of non-planar points.

(3) Robert Mankiewicz, The Story of Mathematics, Cassell & Co, London, 2000.

(4) Martin Kemp, 'Intimations and intuitions', New Scientist, no. 2568, 9 September 2006, 48-9.

(5) Robert Solomon, A Short History of Philosophy, Oxford University Press, New York, 1996.

(6) Ted Honderich (ed.), The Oxford Companion to Philosophy, Oxford University Press, Oxford and New York, 1995.

(7) Reuben Hersh, What is Mathematics, Really?, Vintage, Cary, North Carolina, 1998.

(8) Godel shook the foundations of mathematics when he proved that no axiomatic system (one built up from a series of rules) was complete, in that it was capable of proving all algebraic truths within that system. Fortunately our buildings still stand.

(9) Cantor's set theory got tied up in knots when Russell pointed out a paradox involving sets containing themselves, rather like the way in which boxes cannot.

(10) Hersh, 9.

(11) Nicholas Griffin (ed.), The Selected Letters of Bertrand Russell: I, The Private Years (1884-1914), Routledge, London, 2002, 404.

(12) Tragically, the engraver mistakenly produced an Archimedean spiral (resembling a coiled rope) instead.

(13) Where 12 o'clock is straight ahead, 9 o'clock is to your left, and 3 o'clock is about tea time.

(14) Hope you didn't trip over anything!

(15) A constant ratio between successive measures, and hence the origin of one of its names.

(16) Przemyslaw Prusinkiewicz and Aristid Lindenmayer, The Algorithmic Beauty of Plants, Springer-Verlag, New York, 1996.

(17) Cornu copiae, the 'horn of plenty', was a mythical horn able to produce whatever was wished for, a gift from Zeus to Amalthea from the goat upon whose milk he was raised. Horns, of course, are shaped like a cone twisted by a helical spiral.

(18) Despite the sequence being attributed to Fibonacci, it was in fact first described by Indian mathematicians in 1150, calculating the optimal way to pack a cart with boxes of unit sizes one and two. Donald Knuth, The Art of Computer Programming: Volume 1, Addison-Wesley, Reading, Mass., 1997.

(19) Ian Stewart, Nature's Numbers, Weidenfeld & Nicolson, London, 1995.

(20) Stewart, 138.

(21) Mario Livio, The Golden Ratio, Broadway Books, New York, 2002.

(22) Midhat Gazale, Gnomon: From Pharaohs to Fractals, Princeton University Press, Princeton, NJ, 1999.

(23) Livio, 168.

(24) Livio, 174.

(25) Livio, 101.

(26) Stewart, 139.

(27) D'Arcy Thompson, On Growth and Form, Cambridge University Press, Cambridge, 1992.

(28) This parallels the growth of plants, whereby the meristem (the growing cells at the tip of a plant) grows and turns at a constant angle.

(29) Thompson, 182.

(30) Gazale, 125.

(31) Rodney Brooks, 'Elephants don't play chess', Robotics and Autonomous Systems, vol. 6, 1990, 315-327.

(32) The idea that intelligence is a central reasoning engine operating on a set of symbols that represents the world.

(33) Alan Hodges, Alan Turing: The Enigma, Walker & Company, New York, 2000.

(34) As a complement to modelling the physical shape of a shell, this work described reaction-diffusion partial differential equations that might explain the pigments and patterns on the shell surface.

(35) Diran Basmadjian, The Art of Modelling in Science and Engineering, Chapman & Hill, Boca Raton, Fl., 1999.

(36) Philipos Loizou, 'Introduction to cochlear implants', IEEE Engineering in Medicine and Biology, January 1999, 32-442.

(37) Fred Spoor, 'A comparative review of the human bony labyrinth', Yearbook of Physical Anthropology, vol. 41, 1998, 211 -251.

(38) LT Cohen et al., 'Improved and simplified methods for specifying positions of the electrode bands of a cochlear implant array', Journal of Otology, 1996, vol. 17, 859-865; Sun Yoo et al, 'Three-dimensional geometric modelling of the cochela using helico-spiral approximation', IEEE Transactions on Biomedical Engineering, vol. 47, no. 10, 2000, 13921402.

(39) David Raup, 'Computer as aid in describing form in gastropod shells', Science, vol. 138, 1962, 150-152.

(40) Cohen, 859.

(41) Yoo, 1392.

(42) Daphne Manoussaki et al., 'Cochlea's graded curvature effect on low frequency waves', Physical Review Letters, vol. 96, no. 8, 2006, 1-4.

(43) A E Bate, 'Note on the whispering gallery of St Paul's Cathedral, London', Proceedings of the Royal Physics Society, vol. 50, 1938, 293-297.

(44) Gavin Baker, Stephen O'Leary, Nick Barnes and Ed Kazmierczak, Cochlea modelling: Clinical challenges and tubular extraction, 17th Australian Joint Conference on Artificial Intelligence, Cairns, 2004.

(45) Gavin Baker, Nick Barnes, 'Model-image registration of parametric shape models: Fitting a shell to the cochlea', The Insight Journal, October 2005. Available: www.insight-journal.org.

GAVIN BAKER / COMPUTER SCIENCE AND SOFTWARE ENGINEERING

ON MATHEMATICS AND NATURE

Some see mathematics as a product of the mind. As a thought, a human conception that lives in the mind and sleeps in a textbook. Surely the most abstract and esoteric corners of mathematics have no place in the world of ants and rocks--after all, where can we point to find a Picard Group (1) or a hypercube?

The peculiarly Pythagorean view of the world as number sees mathematics as the key to understanding the universe--a view implicitly shared by physicists of our day, albeit without the same mysticism. Pythagoras saw mathematics as an intrinsic part of nature, music and the stars; something magical to be discovered. Pythagoras' thinking clearly inspired Plato's Forms and Solids, (2) for he conceived a model of the universe based on the dodecahedron. Some 2000 years later, Kepler's revolutionary elliptic planetary model was directly influenced by Pythagoras' Spheres, (3) which has more recently inspired the spatial design of the [C.sub.60] Buckminsterfullerene molecule. (4)

Western philosophy and mathematics have been intimately entwined from their beginnings. Plato, Descartes and even Kant were foundationalists who sought purity and 'transcendent perfection' in mysteries both mathematical and natural. (5) The earliest philosophers were deeply concerned with geometry, astronomy and logic. Gottlob Frege challenged the dichotomous notion of insisting mathematical concepts were either physical or mental by describing them as 'abstract objects', which is taken a step further in some modern thinking by describing them in terms of a quasi-collective consciousness.

But while Platonism would say we cannot 'invent' mathematics, but only discover it, an empiricist would insist we only learn about mathematics through our senses as we interact with the world. (6) Reuben Hersh offers us an intriguing perspective: that of mathematics as humanistic pursuit, a socio-historic phenomenon, (7) invoking the Fregean notion of the abstract. He points out that mathematics as we know it today is a product of history and human activity, including its mistakes, false starts, incompleteness and limitations.

So while there is no clear consensus on the pure universality of mathematics, the question has not impeded the pursuit. On the contrary, modern society is literally built on the sometimes shaky foundations of mathematics. Despite Kurt Godel, (8) we still teach 'incomplete' algebra, and despite Georg Cantor and Bertrand Russell (9) we still make heavy use of set theory. And while some invoke a deity in rapturous wonder at mathematics, as quoth Mary Somerville:

Nothing has afforded me so convincing a proof of the unity of the Deity as these purely mental conceptions of numerical and mathematical science which have by slow degrees been vouchsafed to man. (10)

other giants, such as logicist Russell, are content to do without when he states that he

like[s] mathematics because it is not human and has nothing particular to do with this planet or with the whole accidental universe--because, like Spinoza's God, it won't love us in return. (11)

Perhaps it is human nature to invoke a higher power when faced with something so awesome and wondrous that we struggle to conceive of it 'just happening'.

ON SPIRALS

The seafaring mollusk known as nautilus (Cephalopoda Nautiloidea, or 'head-foot sailor') carried a calcified rendering of a mathematical wonder for its shell millions of years before Descartes anointed it with the first of its many names. The logarithmic spiral is also known as the growth spiral or the equiangular spiral. The eminent mathematician Jacob Bernoulli was so enamoured with this spira mirabilis, the miraculous spiral, he wished it inscribed on his gravestone. (12)

The logarithmic spiral can be defined as a curve that exhibits a constant angle between the radius vector (a line from the centre to a point on the curve) and the tangent vector (a line oriented along the path of travel). Try this: put a bin in the middle of your office, stand next to it, then stretch out your left arm so that it is pointing approximately at 10 o'clock. (13) Walk backwards around the bin, moving further away from it while keeping your arm pointing directly at the bin. Stop before you run into your desk, and you will have traced out a logarithmic spiral. The tangent vector is pointing out your back, while your sore arm (14) is the radius vector.

The logarithmic spiral has many special properties that make it very useful in both nature and engineering: it is self-similar, in that its shape remains unaltered by scaling and angular growth; the distance between arms increases in a geometric progression; (15) any straight line passing through the origin makes a constant angle with the curve (Figure 1); a degenerate logarithmic spiral is a straight line at one extreme and a circle at the other; it can be produced using incredibly simple rules, such as 'move forward a bit, turn left 30 degrees'. If instead of moving forward, we simply 'grow and turn', we enter the domain of Lindenmeyer Systems: a formal set of simple rules capable of generating remarkably complex fractal figures, such as trees and ferns. (16)

[FIGURE 1 OMITTED]

Nature provides a cornucopia (17) of examples of the logarithmic spiral in action. The arrangement of seeds in a sunflower--the optimal arrangement for efficient packing. The path of an insect as it flies toward a light--arising from the structure of its compound eyes. The path of an eagle as it swoops on its prey--so it can keep a constant eye on the target. The arrangement of scales on a pineapple. The swirling rage of a tropical cyclone. Or the swirling mist of stars in a galactic spiral. And the shell of a mollusk--due to the accretive mode of its construction. You may even find one in your own backyard.

So what does a logarithmic spiral have to do with rabbits? For rabbits were the etude of the great Fibonacci: how fast could they breed in ideal circumstances? The sweat of his brow (and that of his rabbits) produced a fascinating result, and thus his eponymous (18) sequence. (19) That is, the total population of this idealised colony after each generation is given by 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 ..., where each successive number is the sum of the previous two.

[FIGURE 2 OMITTED]

While many people are understandably riveted by this ingenious model of bunny proliferation, the fun doesn't stop there. Fibonacci numbers crop up in a surprising number of places in nature around us: the number of petals on a flower; the number of spirals of seeds in a flower head or a pine-cone; the number of leaves per turn around a stem; cauliflower florets; and romanesque broccoli. (20)

But there are some things even more interesting than rabbits and cauliflower. For example, we can geometrically construct an approximation of our miraculous logarithmic spiral with rectangles sized according to the Fibonacci sequence.

[FIGURE 3 OMITTED]

The Golden Mean is another very special number with many names. It is a natural relationship, a special proportion revered by artists and mathematicians alike. Many consider it not only aesthetically pleasing but almost mystical, due in no small part to its intriguing properties and history. (21) Between two measures, it is the ratio of the whole to the larger, as the larger is to the smaller. That is, (see Figure 4). Also known by the Greek symbol phi or [phi], the

[FIGURE 4 OMITTED]

a + b / a = a / b

Golden Mean or Divine Proportion is associated with a somewhat controversial area of study known as 'Sacred Geometry', a particularly Pythagorean concept. Phi is ancient, and may be found variously in nature, architecture, art and music. It is frequently claimed that the facade of the Parthenon is laid out according to the Golden Mean, so too the dimensions of the pyramids, (22) yet these claims (and many others) have been debunked as 'wishful thinking'. There are, however, many genuine examples, (23) such as the work of Paul Serusier (1864-1927), Gino Severini (1883-1966) and the Cubists. The renowned architect Le Corbusier invented a comprehensive system of proportions called Modulor, which was based on a combination of the Fibonacci sequence, the Golden Mean, and the proportions of da Vinci's Vitruvian Man, designed to provide a 'range of harmonious measurements to suit the human scale, universally applicable to architecture and to mechanical things'. (24) Unfortunately, while there are many authentic manifestations of the Golden Mean to be found across human pursuits, it is often difficult to distinguish these from overly optimistic seekers who retrospectively juxtapose the ratio onto older works with insufficient evidence or justification.

Now it was in fact our friend Kepler who discovered that the ratio of successive Fibonacci numbers converges to [phi], the Golden Mean. (25) It wasn't until as recently as 1994, however, that two French mathematicians, Stephane Douady and Yves Couder, proved (26) that the optimum growth pattern for the most efficient packing in a plant occurs when the angle between successive primordia (points of growth) is the irrational Golden Angle, 2[pi](1-[phi]). Thus these three mathematical concepts--logarithmic spirals, Fibonacci numbers and the Golden Mean--are intimately related, and appear to form part of nature itself. It is perhaps no wonder that many mathematicians are moved to poetry when beholding such elegance, depth and beauty.

Now D'Arcy Thompson (1860-1948) liked to walk around St Andrews, Scotland with a parrot on his shoulder. He was also very fond of ice-cream, and translating classical Latin and Greek texts. But more than for his parrot, he will be remembered for his opus magnum, On Growth and Form (1917), wherein an entire chapter is devoted to the logarithmic spiral. (27) In these pages we find a poetic treatise on natural development, a thorough analysis of patterns of growth and models that became a cornerstone of modern morphometry (the study of the shape of living organisms). Thompson's unifying theory is that ontogeny (individual growth) is determined by the physical forces of nature, and shaped by environment. This powerful statement, to some extent at odds to Darwinian theory (and pre-dating the discovery of DNA), is now generally accepted as complementary to our modern notions of growth, development and biological processes.

Shells, such as the nautilus discussed earlier, grow by accretion; matter is gradually accumulated or deposited at the opening, enlarging as it winds around a central axis. (28) This means that only the opening ever grows; the original form never changes, so that a young shell has the exact same form as a larger one of the same species. Thompson marvels that 'this remarkable property ... is characteristic of the equiangular spiral, and of no other mathematical curve.' Such a shell can be described as a gnomon, which in its most general sense is 'any figure which, being added to any figure whatsoever, leaves the resultant figure similar to the original.' (29) The word has its origins nearly 2000 years ago with the mathematician Hero of Alexandria, in a day when people were more interested in the design of sundials; (30) the gnomon was the 'indicator', the part of the sundial that cast the shadow. Gnomons can be found in many areas of both mathematics and nature, shells being the most obvious exemplar.

In exploring just this one particular aspect of mathematics, we find richly intertwining relationships, and more examples than could possibly be surveyed in this manuscript. It is clear that these relationships are there to be discovered and applied to new areas of enquiry: all we need is the right mathematical language, and a way to determine how well our views fit our observations. We need models.

ON MODELS

Model (n): a system or thing used as an example to follow or imitate; a simplified description, esp. a mathematical one, of a system or process, to assist calculations and predictions.

--Oxford English Dictionary

Mathematical modelling is the judicious art of oversimplification.

--Unknown

As Rodney Brooks is fond of pointing out, 'the world is its own best model.' This philosophy, espoused in his 'subsumption architecture' for robotics, (31) argues that the classical symbol-system of artificial intelligence (32) is 'fundamentally flawed'. He posits that any attempt to recreate an internal model of the external world is inherently lacking in every way, and is therefore pointless. Brooks explicitly rejects a reductionist approach to artificial intelligence, and argues strongly for a biologically inspired approach. Despite this, many roboticists today opt for a pragmatic hybrid approach. For example, a robot does not need to distinguish the fine grain of the hardwood, or the sheen of the lacquer to avoid running into a chair; recognising it merely as an obstacle is perfectly sufficient. A purposive model is one that is tailored specifically for the task at hand, with no more detail or complexity than is needed to satisfy the goals.

This serves to highlight the distinction between 'modelling perfectly' and a 'perfect model'. It may be that modelling perfectly, as Brooks' earlier observation alludes to, would require a model equivalent to the subject itself, since anything less would necessarily be imprecise or incomplete, and therefore imperfect. The 'perfect model', however, is a model that completely satisfies all the requirements of the task at hand, and is therefore achievable. In all practical scenarios, with finite time and resources, we will most likely have to satisfy ourselves with a 'good model'; one which is as accurate as we can practically make it given all the constraints, and for which the limitations and shortcomings are understood.

The father of computer science, Alan Turing, shared Thompson's fascination with nature and numbers. Indeed, Turing was greatly influenced by On Growth and Form; as a child he spent hours sketching flowers, poring over pine-cones, examining seeds and searching for relationships and patterns. His unfinished work in the latter period of his life explored plant growth and the relationship between the Fibonacci sequence and sunflowers. (33) In his seminal 1952 paper on chemical morphogenesis, (34) Turing rather modestly described his proposed model as 'a simplification and idealization, and consequently a falsification'. While it has since become 'the definitive basis for modelling biological growth', (35) Turing rightly (and succinctly) questions the nature of the modelling process, and just how much faith we can place in these idealised forms. Fortunately, all is not lost.

So what makes a good model? Ideally our model would be:

* General: can be applied to a wide variety of normal subjects.

* Expressive: captures essential aspects of the subject in a meaningful way.

* Simple: no more complex than necessary to carry out the task.

* Verifiable: can be validated against observation.

Are mathematical models appropriate for modelling nature? If we take the word of the Platonists, most definitely. In practice, engineers and physicists use them all the time, cognisant of their inherent limitations. But is there an intrinsic mathematical relationship there for us to find in nature? An idealist would likely say there is, but whether or not we can find it is another question. How would we recognise it if we found it? That is not so easy to answer. When evaluating a model, we can only increase our confidence the more we successfully validate it against different subjects.

Are we not inherently limited by our mathematical toolbox? Mathematicians are constantly expanding the horizons of theory in pure and applied domains. Our knowledge is assuredly finite, but growing rapidly. We may not yet have the tools at our fingertips in order to find the 'perfect model', so we make do with what we have. Sometimes, we really do just see every problem as a (linear function) nail, and bang on it with our (linear algebra) hammer.

Modelling as an activity is after all purposive, in that we have a goal, a subject and a set of constraints. We should also have some means to measure how well our model fits the subject, and an idea of just what is 'good enough'. There is in general no single best solution--the efficacy of a model can be measured only in terms of its fitness for purpose. Good practice would have us apply Occam's Razor to all models; that we employ the simplest explanation. In the same vein, the principle of parsimony eliminates from a model any parameter that does not contribute significantly to the explanatory power of the model. Linear is better than non-linear, fewer parameters are better than more parameters.

With this simplicity and generality in mind, we turn our attention to a particularly interesting problem at the nexus of the above, seemingly disparate topics: producing a mathematical model of a very small spiral organ.

ON THE COCHLEA

Millions of people worldwide suffer from profound sensorineural hearing loss. One in a thousand babies is born with congenital hearing defects, while around forty percent of people over the age of seventy-five develop progressive loss. The cochlear ear implant has become the standard clinical intervention for nerve-impaired deafness, with over fifty thousand recipients in 120 countries worldwide. The cochlear implant restores hearing by sending tiny electrical impulses to the residual nerves (36) (Figure 5).

[FIGURE 5 OMITTED]

Your humble author is engaged in the development of a three-dimensional shape model of the cochlea that may ultimately assist clinicians in providing better treatment for people so affected. The research is aimed at analysing cochlea morphometry, deriving metrics to help quantitatively describe cochleae (where there are now predominantly qualitative descriptions), and constructing a mathematical model that can capture its essential characteristics. Such anatomical shape models will give us insight into normal and abnormal shape variation. They may ultimately be used for a variety of purposes: diagnosis, by providing quantitative measures that can be compared to known normal forms; simulation and training, by enabling clinicians to generate a variety of plausible forms; and surgery planning, by highlighting regions of unusual shape that may require attention.

The cochlea (Latin for 'snail shell') is the organ of hearing, a tiny 2[cm.sup.3] shell-like structure in the inner ear, embedded in the temporal bone of the skull. It is spiral in shape, and is often described as approximating a logarithmic spiral. A normal cochlea revolves through two and a half turns, from the basal turn (lower turn) up to the helicotrema (top of spiral). Three channels run the length of the cochlea: the scala tympani, scala media and scala vestibuli. The cross-sectional shape resembles a cardioid (rounded 'B' shape).

While detailed studies have been carried out on the morphology of the bony labyrinth (incorporating the cochlea), (37) the metrics are typically linear measurements or relative orientations--for example, the width of the basal turn of the cochlea, or the angle of the lateral semicircular canal in the sagittal plane, all of which are most relevant to phylogenetic studies. Most of the cochlea models described are spacecurves that either do not take into account cross-sectional shape at all, (38) or use an approximation such as a circle. Clearly there is a need for clinically relevant models of the cochlea, beyond what is currently described in the literature.

It may be tempting to apply Raup's shell models (39) to the cochlea. These biological models of shell shape are simple yet effective parametric models that reproduce a wide variety of shapes found in nature, as well as a wide variety that are not. This may simply imply that the parameter space of the model only partially overlaps with the parameter space of the system, or it may imply that the natural model is better described in some other way. Or it may simply imply that the natural model has implicit constraints on its parameters, which is consistent with Thompson's theory of phylogeny. Since it is simple and works very well for a wide variety of examples, we can say it is a good model.

However, assuming the Raup model was an accurate representation of the actual biological growth process for shells, there is still the question of whether it is appropriate, or semantically and biologically valid to apply an accretion growth model to the cochlea. First is the fact that the curve of Raup's shells grows from top to bottom (apex to base), growing outward. However, the cochlea itself grows from base to apex, curving inward. Indeed, it is fully formed by the age of seven months in the womb. Second is the fact that the cochlea does not grow by accretion, and its development is linked to the growth of the temporal bone itself. Since so much of our knowledge of the growth and development in utero is incomplete, we must focus on what we can best observe and measure, which is primarily CT scans of fully-developed adult temporal bones.

L T Cohen describes a 2D spiral model that was originally designed to model the path of the implant electrode itself (40) (and not of the otic capsule, the bone that encapsulates the cochlea). Sun Yoo et al. describe an extension of Cohen's model, reinterpreting it as defining the centreline of the cochlea. Yoo added an exponential height function to the 2D spiral, and described a method of fitting the spacecurve model to CT data. (41) Until recently it was believed that the shape of the cochlea was primarily due to an economy of space; however, a new theory suggests that the spiral shape of the cochlea is highly significant, and can have an amplifying effect of up to 20dB for low frequencies. (42) The mechanism is believed to be similar to the 'whispering walls' effect in St Paul's Cathedral in London, where one can whisper at one side of the gallery and be heard on the other. (43)

This author is investigating a 3D parametric shape model, based on the models of Cohen and Yoo, since they are well-known in the literature and provide a simple and convenient starting point. My model extends these into 3D, adding a full outer surface and an elliptical cross-section. From the outset, the goal of this work is to produce a model that features a clinically relevant parameter space, and incorporates or yields metrics that are directly relevant in the clinical domain. (44) The explanatory power lies in the parameter space and its embedding in the real world; otherwise any numbers derived from such a model would have no basis for biological interpretation. Metrics such as the average diameter of the basal turn or the curvature of the centreline of the otic capsule are far more clinically relevant for the purposes of planning for cochlear implant surgery than (for example) the linear measurements described above, or the coefficients of some convenient but effectively arbitrary trigonometric function.

The rigid model currently being explored (45) is useful as a first approximation as it incorporates the gross-scale metrics that can uniquely describe the shape of the cochlea. However, it is also inherently unable to capture fine-scale detail, such as local pathologies, deformities and so on, and may even be incapable of deforming to normal shapes (only extensive fitting to normal samples can confirm this). So ultimately this model may not be the most pure, elegant or abstract, or even the most accurate. But in terms of fitness for purpose, the model satisfies our criteria for simplicity (seven parameters), expressiveness (the parameters are clinically relevant), general, and verifiable (it is currently being fitted to a range of human CT scans). Knowing the properties and limitations of this parametric model, the model fitting process is being extended to incorporate a fully deformable model, which will not only provide a means for validation but can also be used to quantify the regions of the model that poorly fit the organ, so that we may further improve the model and its ability to be generalised.

Here then is the intersection of nature, spirals, mathematics and models: can we represent the cochlea with a logarithmic spiral in a clinically meaningful way? Does a logarithmic spiral actually capture the natural growth and form of the cochlea?

BEYOND

The research described herein invokes a certain degree of Pythagorean idealism in its search for mathematical truth and beauty in nature. But it is necessarily tempered with pragmatism, for a clear goal stands before us. The path ahead is long, but the early results are encouraging.

More than a model of nature, mathematics can be seen as the language of relationships in nature. As we add to our vocabulary, our mathematical toolbox becomes richer, allowing us to capture, express and explore more profoundly the theoretical aspects of the patterns of life before us. The limits of our models only serve to highlight the limits of our finite understanding of the universe, which is at once humbling and challenging.

ENDNOTES

(1) This of course refers to a particular type of multiplicative Abelian group in Number Theory, and not the television series Star Trek.

(2) Forms: perfect, abstract entities that exist independently of the world. Solids: five perfectly symmetrical three-dimensional geometric shapes composed of non-planar points.

(3) Robert Mankiewicz, The Story of Mathematics, Cassell & Co, London, 2000.

(4) Martin Kemp, 'Intimations and intuitions', New Scientist, no. 2568, 9 September 2006, 48-9.

(5) Robert Solomon, A Short History of Philosophy, Oxford University Press, New York, 1996.

(6) Ted Honderich (ed.), The Oxford Companion to Philosophy, Oxford University Press, Oxford and New York, 1995.

(7) Reuben Hersh, What is Mathematics, Really?, Vintage, Cary, North Carolina, 1998.

(8) Godel shook the foundations of mathematics when he proved that no axiomatic system (one built up from a series of rules) was complete, in that it was capable of proving all algebraic truths within that system. Fortunately our buildings still stand.

(9) Cantor's set theory got tied up in knots when Russell pointed out a paradox involving sets containing themselves, rather like the way in which boxes cannot.

(10) Hersh, 9.

(11) Nicholas Griffin (ed.), The Selected Letters of Bertrand Russell: I, The Private Years (1884-1914), Routledge, London, 2002, 404.

(12) Tragically, the engraver mistakenly produced an Archimedean spiral (resembling a coiled rope) instead.

(13) Where 12 o'clock is straight ahead, 9 o'clock is to your left, and 3 o'clock is about tea time.

(14) Hope you didn't trip over anything!

(15) A constant ratio between successive measures, and hence the origin of one of its names.

(16) Przemyslaw Prusinkiewicz and Aristid Lindenmayer, The Algorithmic Beauty of Plants, Springer-Verlag, New York, 1996.

(17) Cornu copiae, the 'horn of plenty', was a mythical horn able to produce whatever was wished for, a gift from Zeus to Amalthea from the goat upon whose milk he was raised. Horns, of course, are shaped like a cone twisted by a helical spiral.

(18) Despite the sequence being attributed to Fibonacci, it was in fact first described by Indian mathematicians in 1150, calculating the optimal way to pack a cart with boxes of unit sizes one and two. Donald Knuth, The Art of Computer Programming: Volume 1, Addison-Wesley, Reading, Mass., 1997.

(19) Ian Stewart, Nature's Numbers, Weidenfeld & Nicolson, London, 1995.

(20) Stewart, 138.

(21) Mario Livio, The Golden Ratio, Broadway Books, New York, 2002.

(22) Midhat Gazale, Gnomon: From Pharaohs to Fractals, Princeton University Press, Princeton, NJ, 1999.

(23) Livio, 168.

(24) Livio, 174.

(25) Livio, 101.

(26) Stewart, 139.

(27) D'Arcy Thompson, On Growth and Form, Cambridge University Press, Cambridge, 1992.

(28) This parallels the growth of plants, whereby the meristem (the growing cells at the tip of a plant) grows and turns at a constant angle.

(29) Thompson, 182.

(30) Gazale, 125.

(31) Rodney Brooks, 'Elephants don't play chess', Robotics and Autonomous Systems, vol. 6, 1990, 315-327.

(32) The idea that intelligence is a central reasoning engine operating on a set of symbols that represents the world.

(33) Alan Hodges, Alan Turing: The Enigma, Walker & Company, New York, 2000.

(34) As a complement to modelling the physical shape of a shell, this work described reaction-diffusion partial differential equations that might explain the pigments and patterns on the shell surface.

(35) Diran Basmadjian, The Art of Modelling in Science and Engineering, Chapman & Hill, Boca Raton, Fl., 1999.

(36) Philipos Loizou, 'Introduction to cochlear implants', IEEE Engineering in Medicine and Biology, January 1999, 32-442.

(37) Fred Spoor, 'A comparative review of the human bony labyrinth', Yearbook of Physical Anthropology, vol. 41, 1998, 211 -251.

(38) LT Cohen et al., 'Improved and simplified methods for specifying positions of the electrode bands of a cochlear implant array', Journal of Otology, 1996, vol. 17, 859-865; Sun Yoo et al, 'Three-dimensional geometric modelling of the cochela using helico-spiral approximation', IEEE Transactions on Biomedical Engineering, vol. 47, no. 10, 2000, 13921402.

(39) David Raup, 'Computer as aid in describing form in gastropod shells', Science, vol. 138, 1962, 150-152.

(40) Cohen, 859.

(41) Yoo, 1392.

(42) Daphne Manoussaki et al., 'Cochlea's graded curvature effect on low frequency waves', Physical Review Letters, vol. 96, no. 8, 2006, 1-4.

(43) A E Bate, 'Note on the whispering gallery of St Paul's Cathedral, London', Proceedings of the Royal Physics Society, vol. 50, 1938, 293-297.

(44) Gavin Baker, Stephen O'Leary, Nick Barnes and Ed Kazmierczak, Cochlea modelling: Clinical challenges and tubular extraction, 17th Australian Joint Conference on Artificial Intelligence, Cairns, 2004.

(45) Gavin Baker, Nick Barnes, 'Model-image registration of parametric shape models: Fitting a shell to the cochlea', The Insight Journal, October 2005. Available: www.insight-journal.org.

GAVIN BAKER / COMPUTER SCIENCE AND SOFTWARE ENGINEERING

Printer friendly Cite/link Email Feedback | |

Title Annotation: | mathematics |
---|---|

Author: | Baker, Gavin |

Publication: | Traffic (Parkville) |

Date: | Jan 1, 2006 |

Words: | 4948 |

Previous Article: | Foreword. |

Next Article: | It ain't over 'til ...? |

Topics: |