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Dear Peter,

A shape is reflecting a materialized conception “the understanding”. Two objects may have similar shapes and different properties. A donut and a wedding ring might be considered identical by the observer if, both have the same shape.  In Pavlov’s experiment a trained dog identifies a bell as a food so, we could say that in that case the shape of the food is identical to the shape of the bell. As the bell and the food are different objects, we conclude that the shape is the ‘understanding’ – the SIGNAL that allows to the dog to salivate.

I think shape could be the materialized conception of a sensory signal “image/sound/...” or an object that activates an “understanding”.

Dear James, 

I am sorry for spoiling the discussion a bit just in the beginning, but if you tell me "shape", the very first that comes to my mind is a (GIS) shape file... Funnily, in my mother tongue, in Hungarian we call it also literally  "shape" (i.e., we use the English word for that), probably that is the reason. Of course, if I think a bit, I know what you mean, in general. My second thought would be something to do with the topology (sphere, torus, Moebius stip, Klein bottle and alike) as you suggested above.

Kind regards, Balázs 

A accumulation of constituents on a recognizable design can be called shape, and its directly relative to observer`s parameters. and in real no similarity between two object occurs due to their relative position in space.....anyhow similarities in functional properties shows that some time relative position can be ignored/neglected.

When do objects have similar shapes?

When they have the same differential form.

Dear Dr Peters, dear all,

This is a rather difficult question, as the word "shape" is itself a rather polyssemic one. So I rcurred to its etymology.

I found out it comes from Old English verb "scapan" (past participle = scieppan, past tense: scop) "to create, form, destine"; -- originally from Proto-Germanic "skapjanan", "to create, to ordain";

(cf. also: Old Norse skapa, Danish skabe, Old Saxon scapan, Old Frisian skeppa, Middle Dutch schappen "do, treat, create"; Old High German scaffan, German schaffen "shape, create, produce").

(Pardon me -- I love that stuff!)

It has acquired many different meanings, of varied denotation, connotation and nuances. (see below some meanings I could find).

In the short story: "The weapon that changed shape" the SF/Fantasy author Philp K. Dick describes a weapon that -- even though being able to change itself into many shapes -- still remained a weapon.

Therefore, for an objective answer to your question, I recurred still to Google´s definition (Use Google Chrome -> Define English as your main language->  type the following words -> Define shape ) ...

"1. the external form or appearance characteristic of someone or something; the outline of an area or figure. (...) Synonyms: form, appearance, configuration, formation, structure".

Considering this definition, each "geometric form" is in itself a shape: a rectangular form is always a rectangle, and therefore both a rule and a sheet of paper have the same general geometric shape, as they are both rectangles (even if the paper is square, for the rectangle is a specific case of the general shape named "square"). A donut is a ringed shape, the same of a wedding ring (let´s forget for a moment that some modern wedding rings bear emeralds or diamonds, and thus have a rather irregular shape). However, not all objects that have holes are alike in shape, as some can be square or even hexagonal in shape (like nuts and bolts).

Artistically speaking -- yes, any portrayed objects has an individual shape, being determined to some extent by "chiaro-scuro" (the amount of light or darkness in their representations). Some of them may be square, some ringed, hexagonal, or otherwise similar or dissimmilar to each other. A  material object may be thus represented by its shape in a painting or photograph.

Speaking of science, Medicine recognizes, for instance, "fascies lunar" -- the "moonshaped face", charachteristic of Cushing´s syndrome: Not only the faces of people bearing this syndrome tend to be round, but also they make analogy to the fact that, for the layman, the full moon seems to present the shape of a smiling face. (I can´t resist to remember that I consicer Medicine both as Art and Science, and "Ars requirit hominem totus", "the Art requires Man as a whole").

If I were to speak in "spiritual" terms (I´m not very spiritual myself), I might say I´ve never seen a ghost... but then perhaps that is because, in order to be seen, they would have to appear before me in some visible shape!...

Thanks for formulating that question, it was quite stimulating to think about that subject!

--------

Some meanings of the word "shape":

As a noun:

1.the quality of a distinct object or body in having an external surface or outline of specific form or figure.

2.this quality as found in some individual object or body form: a House´s shape, a lake´s shape.

3. something seen in outline, as in silhouette: A vague shape appeared through the mist.

4. an imaginary form; phantom (ghost).

5. an assumed (external) appearance; guise: a wolf in the shape of a lamb.

6. a particular or definite organized form or expression: He could give no shape to his ideas. (also: he could not shape his idea or project)

7. proper form; orderly arrangement. You´re in good shape.

8. condition or state of repair: The old house was in bad shape. He was sick last year, but is in good shape now.

9. the collective conditions forming a way of life or mode of existence: What will the shape of the future be?

10. the figure, physique, or body of a person, especially of a woman: A dancer can keep her shape longer than those of us who have sedentary jobs.

11. something used to give form, as a mold or a pattern. A cast.

12. Also called section. Building Trades, Metalworking. a flanged metal beam or bar of uniform section, as a channel iron, I-beam, etc.

13. Nautical. a ball, cone, drum, etc., used as a day signal, singly or in combinations, to designate a vessel at anchor or engaged in some particular operation.

verb (used with object), shaped, shaping.

As a verb:

1. to give definite form, shape, organization, or character to; fashion or form.

2. to couch or express in words: to shape a statement.

3. to adjust; adapt: He shaped everything to suit his taste.

4. to direct (one's course, future, etc.).

 Compare to the similar German Word:

Gestalt - noun, plural gestalts, gestalten [guh-shtahl-tn, -shtawl-, -stahl-, -stawl-] (Show IPA). (sometimes initial capital letter) Psychology.

1. a configuration, pattern, or organized field having specific properties that cannot be derived from the summation of its component parts; a unified whole.

2. an instance or example of such a unified whole.

Dear Peter,

''What we mean by shape?''

Depending on the context, this word means different things.  There are many domains of mathematics that provides some specific notion of shape, form, structure.  In ordinary language, the kindergarden teacher will use this word with different meanings related to teaching of simple mathematic or the executio of simple crafts or in the observation of nature.  Some design engineer or computer graphic specialist of animation specialists will use shape with other meanings.  The vision scientist will try to understand what are the basis of our natural perception of object shapes.  Books and books can be filled with answers provided in each of these different contexts.  My research interested have been about certain image structures relevant to vision science.  One feathure of my approach is that each image structure (shape) correspond to a node into a tree of shape.  Each shape or node or image structure.  A structure is created by gradually transforming an image without structure into that structure by complexifying it at each node of the tree.  So the path from the root to a node is the morphogenesis process of this particular shape.  So a shape has a built-in ancestry of shape like a species in biological evolution.  The similarity between two shapes correspond to their common ancestry.  When animal shape are mapped into that tree, their image shape morphogenesis correspond to their actual ontogeny of their external surface from conception to maturation.  So structural hiearchy correspond to history formation.

Shapes are similar structures to already seen structures. They live from association.

Dear Sir, Do i get it right ??? we say that two objects are similar if they have the same shape, but not necessarily the same size. This means that we can obtain one figure from the other through a process of expansion or contraction, possibly followed by translation, rotation or reflection.When two figures are similar, the ratios of the lengths of their corresponding sides are equal. 

@Hanno Krieger: Shapes are similar structures to already seen structures.

Would you say that shapes are independent of whether they are similar to already seen structures?

@James Peters,

I don´t believe that shapes could be independant of earlier observation. They need a definition followed by an abstraction and perfection. Just remember stones beeing abstracted to mathematical spheres or sticks abstracted to lines. 

@ James and Hanno,

I think what is described for blind people suggests that the sensory signals (sound information, the somatosensory system, etc.) could provide enough data to create materialized information (shapes).  I do not think that we need to have a library of visual data to recognize a shape.  As for Pavlov’s dog, the bell’s ring is sufficient to substitute the real food.  I think the foetus uses sensory signals form external noises, languages, etc. to create or elaborate shapes. 

Echolocation can be used to activate the visual cortex. It is best known in bats, who send out high-pitched sounds and then use the echoes to track their prey in the dark.  

Another view on perception and creation of shapes addressed by James, can be analyzed using the Charles Bonnet syndrome.  (CBS) is a common condition among people who have lost their sight. It causes people who have lost a lot of vision to see things that aren't really there, known as visual hallucinations.

A blind person that touches a table can recognize a table. No sound no vision but a shape is re-created.

http://www.livescience.com/14327-blind-people-echolocation-brains.html

http://www.livescience.com/23709-blind-people-picture-reality.html

@Vassilis,

of course shapes can be observed by all senses which allow a spatial recognition. The verb "see" was just used to show the most important way of observing.

According to my point of view on practical, pragmatic Occam's principle one could make a library of earlier seen shapes qualified in different contexts which gives certain expected shapes or may be occurences of unexpected ones pre-labelled in say clusters. Given such a library one could compare it with a new shape and to certain given or may be even found precision determine the shape of the object. Obviously , the shape of a table in a context of  a kitchen is a table according occam's principle of acceptance of the simplest explanation, but not overly simple as long as no contra-evidence is received. Here my invariant extension of MDL principle could be of used as a simple machinelearning gauging device for both the  selection of the object, the precision of the decision and even the precision of the quality of  parameters . But obviously there are many ways to reach such decisions by selection criteria.

It is a common truth that when a word serves, at the same time,  every day's  life experiences  and a scientific subjection this will inevitably lead to a confusion. Shape is one of these words: The hall of the American  football hall the same shape with  a medium melon of southern Greece but mathematically speaking thiw could be considered an inaccurate statement.

In brief , for  a 2-D and 3-D totally connected closed  objects  of Euclidean plane/space(for higher dimensions and even more so for  non Euclidean topology strange things could occur!) shape is usually used for their boundaries and a 1-1 surrjective    mapping  between them determine their shape  similarity. If we do not have connectivity  similarity  of  two such objects usually requires the above mapping to include their interior as well.

Your question has the statements

• shape is an external form or appearance of something
• whether a ruler and a sheet of paper have the same shape
• Wulff shape is an an equilibrium minimal surface for a crystal or drop
• capturing shapes is a central activity in painting
• an interest in external forms can be found in the works of Plato and Artistotle

The first and last statements are related. The others confuse the issue by lack of relation to the question. Mathematical or hypothetical shapes have nothing to do with actual physical shapes except as conceptual aids. Indeed, a mathematical point has no shape as it has no dimension and a mathematical line has no visible shape with one dimension. A two-dimensional shape can be filled in concept. A drawn line has a shape. An actual drawn line has three dimensions. 

A sheet of paper and a ruler are different in shape. Their two dimensional outlines are rectangles, but few would call the outlines similar, except broadly so.

Painting and drawing are illusions. Various techniques are used to create the illusion of shape. A thick line next of a narrow line gives the impression that the thick line is closer, hence a visual shape is made by the two. A drawing or painting has no mathematical shape except for the dimensions of the materials. The illusion has a shape as formed in the mind. An optical illusion is a shape change of the illusion, but there was no change in shape.

Shape has definitions in context. You cannot take one definition to another context and expect compatibility. The Oxford English Dictionary quote is that for the nominative case and the most usual nominative concept. Shape used as a verb has more definitions that are mangled by context. 

Dear Colleagues,

Good Day,

"Similar shapes

Similar figures are identical in shape, but not in size. For example, two circles are always similar. Two squares are always similar:

And two rectangles could be similar:

But will probably not be."...

Please, see the rest of the article for more information....

http://www.bbc.co.uk/schools/gcsebitesize/maths/geometry/congruencysimilarityrev1.shtml

Dear Colleagues,

Good Day,

"Geometric Similarity

If two objects have the same shape, they are said to be geometrically similar. By definition, the ratio of any two linear dimensions of one object will be same for any geometrically similar object. This is easiest to illustrate with simple geometric shapes:"...

Please, see the rest of the article for more information....

http://www.ableweb.org/volumes/vol-20/1-colton/scalingtutorial/geometric.html

Dear Colleagues,

Good Day,

"Similar Figures and Similar Triangles, Written by Supriya Tripathi

Similar Figures:

Similar figures are two figures having the same shape. The objects which are of exactly same shape and size are known as congruent objects. For example, both the front wheels of a car, both hands of a person etc. are examples of congruent objects.

When two figures have same shape but their sizes are different, then such objects are called similar figures. For example: Different sized photographs of a person i.e. stamp size, passport size etc. depict the similar objects but are not congruent.

Certain geometrical shapes or figures are always similar in nature. Consider a circle, if the radius of the circle keeps on changing, its shape still remains the same. Therefore, it can be said that all the circles with different radii are similar to each other. The figure given below represents the concentric circles whose radii are different but all of them are similar. Although these circles have the same shape but their sizes are different and therefore these are not congruent."....

Please, see the rest of the article for more information....

http://byjus.com/maths/similar-figures/

I apologize to all since in my last answe,r due to poor eyes, I missed to add" angle preserving" at each boundary pair of  corresponding points  for the mapping beteen boundaries!

An interesting question indeed - over a long period in my life I have tried to identify "shapes" & "patterns" in charts of stock and commodity prices in order to get a feeling of "what to do next". The entire investment industry depends on pattern recognition - despite the mantra of efficient markets as celebrated by Noble Prize awarded academics. In a world of randomness "shapes" give us something to hold on - a certain kind of confidence and orientation. Very often "shapes" turning up in random set-ups are considered equivalent despite different scale-factors.

As a next step one might try to train machines to recognize "shapes" on behalf of humans. An interesting aspect of this question would be how to combine randomized, obfuscated perception of "shapes" with a topological framework to identify and classify them. 

The shape of a rainbow is always the same.  It corresponds to a circle or a semicircle. The rainbow or the spectra (image/schema) of the light (which reflects an activity) refracted differently through a prism has NO a specific topological reference.  A topological reference can be attributed to the water drop or the prisms, the sun and the observer. If, A is the Shape of the rainbow and B is the topology of the water-drop then the observer in C understands/perceives the A – for a phenomena occurred in B.  

To my opinion the shape may permit to perceive physical events/activities with a materialized conception.  I may permit to say that the shape enable us to use Euclidean perception (geometry) that might cover our understanding of Space organization to describe events addressed also by quantum theory or statistical analysis and more.

See: Is perceptual space inherently non-Euclidean?

In my opinion, one can assume the relation between physical objects and their shapes to be analogous to that between objects and (natural) numbers. For instance, 2 may refer to two airplanes, as well as to two flowers. In this sense, a shape includes a set of objects.

In geometry it is admitted that (geometric) shapes can be constructed on a sheet of paper, or represented on a computer screen. Euclid showed how to construct congruent triangles, and defined similar rectilinear figures.

To show congruence of triangular shapes he apparently used rigid motions he had not defined. Thus, the task of establishing shape similarity is affected by the same problem.

According to Klein's Erlangen Program, geometry can be based on transformation groups. Then, if the geometry admits similarity transformations, it makes sense to speak of similar shapes. Non-euclidean geometries don't possess similarity transformations, though.

From this viewpoint, to establish if two objects do have a similar shape, perhaps one needs an underlying geometry.

In my ,pragmatic occam's principle I missed to mention the description of shapes in terms of topology and deformation, symmetries, say scaling and rotational, but  these properties often can be pre- parameterized and categories of shapes might be pre-classified or at least clustered. Our extended invariant MDL principle is indeed invariant towards observational transformations . Therefore, I believe this obstacle does not hurt too much in the sense that if the problem has a sensory resolution by humans or AI , it must(could) have a corresponding invariant bayesian description . Such an invariant  bayesian description is exactly the starting point of the invariant extended MDL principle.

I found this is a very interesting topic, in my humble opinion basically in order to compare the shape first we see the boundaries than we go into other details.

I am little confused if we can say that d, p and b have same shape or not because it depends on our point of view, for example d and b are the mirror image of each other and p is the image formed by turning b upside down ?  

Shape in 3D will contain inside volume shape too making things more complex. Staying in 2D helps a lot. One issue is very important: Scaling invariance. The shape must be the same although a linear coordinate transformation is made.

In my view there are theoretical mathematical shapes which are ideal like sphere, triangular, square or cone etc. The real world approximates to these shapes. In reality any object which is solid defines its own exact shape like a human being or animal and is having its own identity. The mathematical shapes are dictated by exact equations not any real object. 

 Dhanu Chettri, according to my invariant MDL principle and also practical Occam's principle such invariance can be compared even if they are different observer's with diffferent data as long  as  the context and objects are classified(able) with data included simply by choosing the optimal, minimal complexity as measured by the message length of data and model with parameters as measured in terms of the invariant volume element in parameter space (may be even nonparameterized bayesian statistics in terms of induced discretized shifts of centers of basis functions at optimal resolution , i believe as in adaptive multigrid or wavelets?) as given by the Fisher Riemannnian tensor in the space of  parameters.  If any of you guys or James  P.  are interested , I might have some more general ideas/notes about how to obtain a good description of the IN_MDL principle from more general bayesian statistics and data. than hitherto described. There is also the problem of constraints on the dual paramterspace of hyperparameters which might not have been sufficiently discussed before.??

My point of view of shapes and  parameters and models is that in physical objects at least the shape of things in equilibrium at least, cannot be anything , but has to be induced by a background model description. Even in nonequilbrium open sytems hte induce paramters is often limited in both type and resolution due to a transport and  dissipation and boundary geometry/processes.   But, i respect  the idea that in an  open, nonequilibrium systems almost any shape can be formed. However, in practice this is an idealization in geometry and  math . I want  to point out that given the process and context of formation of shapes the parameters might be limited both in resolution and scale.  :  The basic local models comes in flavors of : elliptic, parabolic, wave or hyperbolic or hamiltonian with a symplectic geometry or quantum bracket operator induced geometry. in space of observables. . These basic models are obviously complicated more if you have boundaries, and  a network of micro, meso and macro interactions which might/ might not be given a multiscale or even a statistical description. These last  type of mixed network , multiscale interactions easily give rise to either  i)homogenized multiscale differential- integral descriptions or 2) simple thermodynamic landscape type descriptions . In any case ,the description and the process of the formation of shapes in say nature  might seem infinite dimensional and ideal , but is in fact limited both in scale and resolution. Even if what you want to compare is inifinitely resolved, ti makes sense to start with a coarse description , distinguish say shapes to some resolution  and then interatively sharpen it.

Vijay, both of us know these exact equations  very well so no dispute is necessary about that. I just wanted to  point out that in discussion about what shapes is , there is also a related question about how the shapes of say different topology etc.  can be compared and observed and even distinguished. Then you have to have reasonable observables for shapes. Given that neither the observer nor nature have infinite resolution . to distinguish shapes etc, I believe that my discussion of a Practical Occams principle in tems of invariant complexity is  a practical resolution of how  to resolve, describe and distinguish shapes even in a sequence of observations approaching  the limit of infinite resolution.I am very well aware of that my part in the discussion might be a side track for most of you ( possibly even irrelevant) more interested in the topological part of shapes etc.  It does not mean that my part is irrelevant per se or should be underestimated. 

In fact, many fo the problems which James mention in his intro , could be resolved to some degree by my point of view of invariant complexlty by description length in bits by  units of cells of invariant volumes of  observed shape parameters  since they to some extent all depends only on invariants towards observers/shape position etc, but not only that -also on invariant resolution of holes and heterogeneity of size and scale in the parameterization . . Really, interesting things might happen if the scale in one direction starts to vary with say another positional parameters say in a torus wth variable width  since then  even the torus as a concept might disappear wrt  resolution in transversal or longitudinal direction. Similar situations might be described in many other situation.

To the question about holes and rings... "Syllogism is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two or more propositions or assumed to be true".Our favorite formal logics:

Every family is the unit of society,

Society may be happy or unhappy.

"All happy families are alike,

Every unhappy family is unhappy in its own way"Tolstoy

 https://en.wikipedia.org/wiki/Syllogism

To Einstein, "Logic will get you from A to B. Imagination will take you everywhere".

"Ship of Fools", the Enduring Metaphor" Levi Asher, "To Socrates, a wise captain must guide the ship of fools". V.V.,to our proverb, "from the dirty to the knyaz"?

To Brant,

"We are full lade and yet forsoth I thynke

A thousand are behynde, whom we may not receyue

For if we do, our nauy clene shall synke

He oft all lesys that coueytes all to haue

From London Rockes Almyghty God vs saue

For if we there anker, outher bote or barge

There be so many that they vs wyll ouercharge."

http://www.litkicks.com/ShipOfFools

I would like to add some information to the already nice one:

As one can see from the following books, shape theory uses essentially methods of algebraic topology. Homology, etc. are basic tools.

 D. G. KENDALL,D. BARDEN, T. K. CARNE  Shape and Shape Theory,   WILEY, 1999

 Chapter 1 Shapes and Shape Spaces

Chapter 2 The Global Structure of Shape Spaces

Chapter 3 Computing the Homology of Cell Complexes

Chapter 4 A Chain Complex for Shape Spaces

Chapter 5 The Homology Groups of Shape Spaces

Chapter 6 Geodesics in Shape Spaces

Chapter 7 The Riemannian Structure of Shape Spaces

Chapter 8 Induced Shape-Measures

Chapter 9 Mean Shapes and The Shape of the\Means

Chapter 10 Visualising The Higher Dimensional Shape Spaces

Chapter 11 General Shape Spaces

=================================

 Christopher G. Small The Statistical Theory of Shape, springer, 1996

 “In general terms, the shape of an object, data set, or image can be defined as the total of all information that is invariant under translations,

rotations, and isotropic rescalings. Thus two objects can be said to have the same shape if they are similar in the sense of Euclideangeometry. For

example, all equilateral triangles have the same shape, and so do all cubes”

“One of these, the Kendall school of shape analysis, uses a variety of mathematical

tools from differential geometry and probability, and is the subject

of this book.”

==============================================

For, Borsuk's approach to shape theory, see,

Jerzy Dydak,Jack Segal, Shape Theory_An Introduction, Springer LNM #688

=================================

THE SHAPE OF INNER SPACE-String Theory and the Geometry of the Universe’s

Hidden Dimensions, Basic books, 2010

Dear Costas Drossos,

Many thanks for the excellent reference on  shape theory.

Dear Jim,

I just find out your interesting question about shapes, which get already very interesting answers. I will however be very happy to add some new points of view on this topic not coming from mathematics, physics, art and philosophy, but from social science.

Men are living into a country whose shape may be of interest to consider, when calculating the distribution of distances between their inhabitants. It may also affect their migrations into a country. However such a population is not uniformly distributed into this country and we will have also to consider a variable density of population all around it, like Archimedes developing the theory of the centre of mass.

From the beginning of my career I was interested by this topic. In my PhD which was published under the title Les champs migratoires en France in 1970, I proposed a spatial view of internal migration which introduces the shape of the country. The paper written with Joel Cohen on Modeling distances between humans using Taylor’s law and geometric probability is accepted for publication but not yet published (I can send a copy of it for those who are interested).

In this last paper, we demonstrate that for any family of similar shapes, both the mean distance between randomly chosen points and its standard deviation scale linearly with rescaling (for example, by changing the radius of a circle or the side of a regular polygon, and similarly for any shape): this is Taylor’s law developed in animal ecology since 1961, now verified in cosmology, ecology and in social sciences. The space may be continuous or discrete and the total number of points may be finite or infinite, subject to a few conditions.

Geometric probability originated with Buffon (1733), who studied the spatial distribution of different kinds of randomly distributed objects, and Crofton (1885), who calculated the probability that a figure formed by n points randomly distributed on a given surface possesses a specific property independently of any overall translation or rotation. We reviewed in this paper some distributions of distances between two randomly chosen points and calculate their expected value or variance to show Taylor’s law, now assuming that the population is uniformly distributed across the territory: segment of the real line, disk, sphere, square, annulus, etc.

In the last part of the paper we introduce populations distributed non-uniformly over a discretized territory. In order to apply this to real populations we used geographic grids of 1 square km introduced in several countries to provide comparable data on human population distributions. You will find here some comparisons of the theoretical distribution using some simple shapes with the observed one for Reunion Island and France.

References

Buffon (de), G.-L. (1733). Solutions de problèmes qui regardaient le jeu du franc carreau, Résumé by Fontenelle B., in Histoire de l’Académie Royale des Sciences de Paris, Boudot J. (ed). Paris: Imprimerie Royale, 43-45.

Cohen, J., Courgeau, D. (2017). Modeling distances between humans using Taylor’s and geometric probability, Mathematical Population Studies, in press

Courgeau, D. (1970). Les champs migratoires en France. Paris: Presses Universitaires de France.

Crofton, M. W. (1885). Probability, in Encyclopaedia Britannica (9th Edition, vol. 19), T.S. Baynes (ed). Edinburgh: A & C Black, 768-788.

Sorry I forgot to give the titles of the figures in my last post:

Figure 4. Probability distribution of distances obtained for an annulus of radii 10 and 30 with uniform distribution (solid line up to 52 km, dotted line beyond), with the values observed for inhabited areas of Réunion Island (points indicated by a cross “x”).

Figure 6. Sampled frequency distribution of distances (indicated by crosses “x”) between 1 km × 1 km squares for France compared with the theoretical probability density function of distances between random pairs of points from a square with side 728 km (solid line up to 730 km, dotted line beyond).

Figure 7. Frequency distribution of distances in km between pairs of individuals observed for Réunion Island (crosses) compared with the theoretical distribution obtained for an annulus (solid line) with uniform population density

Figure 9. Frequency distribution of distances (km) between pairs of individuals in France for a grouping into 5-km classes (crosses), compared with the theoretical distribution (solid line up to 730 km, dotted line beyond) of a square with estimated side 798 km with uniform population density.

They are necessary if you want to understand them.

@Daniel Courgeau: ...for any family of similar shapes, both the mean distance between randomly chosen points and its standard deviation scale linearly with rescaling (for example, by changing the radius of a circle or the side of a regular polygon, and similarly for any shape)....

What do we mean by shape and when do two objects have similar shapes? - ResearchGate. Available from: https://www.researchgate.net/post/What_do_we_mean_by_shape_and_when_do_two_objects_have_similar_shapes#view=58ba9d564048544850037492 [accessed Mar 4, 2017].

Your idea is very good.   I am reminded of Rene Descarte's view of the stars in Le Monde, 1633.  Descartes imagined whirlpools or vortices of second element, rotating around their respective central stars to sweep along their planets like boats in a strong current (‘WATERWORLD’: DESCARTES’ VORTICAL CELESTIAL MECHANICS, page 45).  

http://download.springer.com.uml.idm.oclc.org/static/pdf/723/chp%253A10.1007%252F1-4020-3703-1_3.pdf?originUrl=http%3A%2F%2Flink.springer.com%2Fchapter%2F10.1007%2F1-4020-3703-1_3&token2=exp=1488634920~acl=%2Fstatic%2Fpdf%2F723%2Fchp%25253A10.1007%25252F1-4020-3703-1_3.pdf%3ForiginUrl%3Dhttp%253A%252F%252Flink.springer.com%252Fchapter%252F10.1007%252F1-4020-3703-1_3*~hmac=e61d95ba60b293e4a6c0d6d5596f7ad992c5e012c6698e2bb0328cd60a7bda09

Descartes introduced tessellation of a plane surface using the positions of the starts to constuct celestial polygons with measurable properties (see, e.g., the attached 1633 drawing by Descartes, which was part of his treatise on a vortex uniiverse).

Descarte's work on tessellating planar views of the stars is tremendously important and is brought to fruition by Voronoi and Delaunay during the first three decades of the 20th century.   For more about this, see chapter 1 in my Computational Proximity book:

https://www.researchgate.net/publication/301790590_Computational_Proximity_Excursions_in_the_Topology_of_Digital_Images

Book Computational Proximity. Excursions in the Topology of Digital Images.

@James Peter ...Descartes introduced tessellation of a plane surface using the positions of the stars to constuct celestial polygons with measurable properties...

I am so happy about your citation of Descartes "Le monde", written in 1633 but published only after his death, as the condemnation of Galileo in 1633 did  not permit him to publish a book about heliocentrism. I agree with you that his work on tesselating planar view of the stars is important and that it reminds not only the work of Voronoi and Delaunay but also our work on distances between humans.  

By basing their dimensions, altitude, lattitude and in all aspects which are same, then they are having same dimensions.

@Daniel Courgeau: I agree with you that [Descartes'] work [in "Le monde", written in 1633] on tesselating planar view of the stars is important....

Remarkably, Descartes' view of vortex geometry has a considerable relevance nowadays in delineating shapes in digital images. The basic approach is to view what Pauil Alexandroff in 1926 called a nerve complex in a triangulated space as a virtual vortex that "pulls" the parts of an image shape into the nucleus of the nerve. Briefly, a nerve complex is a collection intersecting triangles on a triangulated shape. The nucleus of a shape nerve is that part of a nerve that is common to all triangles in the nerve. It is the nucleus of a nerve complex that tells us something about each of the parts of the nerve.

In more recent work on nerve complexes, the boundaries around holes and the cyclic paths around surrounding holes in a shape are studied in the text of homology groups. For an introduction to homology theory, see

Article P. J. Hilton and S. Wylie, Homology Theory, an introduction ...

For examples of nerve complexes (intersecting balls), see the attached segment from

Conference Paper Persistent homology: a step-by-step introduction for newcomers

For recent work on nerve complexes, see, for example:

Article Delta Complexes in Digital Images. Approximating Image Object Shapes

and

Conference Paper Maximal Nucleus Clusters in Pawlak Paintings. Nerves as appr...

@James

Thank you for the references to these papers. I will come back to you after reading them.

Joel Cohen and Mark Brown have published a new paper on our topic of distance between humans. You will find it here enclosed.

Daniel

@ James

Thank you for your interesting information on homology theory. I wonder however if it may answer the more general following question: how to compare human migration between areas of different shapes and sizes, in countries which have also different shapes and sizes?

I found in 1973 a quite general index, which was later called by demographers Courgeau’s k, which captures very simply the effect of space (shape, size and distance) on the level of mobility between these areas. We later in 2012, with Muhidin and Bell, used it for cross-national comparisons of changes of residence.

These findings are the results first of theoretical reasoning, and second of empirical application of these results in a great number of countries.

The theoretical basis of this method is as follows: if, as we know to be true, there is a relationship between propensity to move and distance, there must also be a relationship between the level of mobility by zonal system and the number of zones dividing the territory.

In a number of simple cases (square territory divided into little squares, for example) and with a law of migration following a Pareto model, it can be shown that the crude migration intensity, measured by the number of migrations which cross at least one boundary between n² zones, divided by the total population at risk is proportional to ln(n). Unfortunately I was not able to show that this result may be verified for every shape, size and distance.

However many empirical results show us that this relationship is verified for a great number of countries. Here is the example of United States and France (see the joint figure).

Do you think that a theoretical explanation can be found of such an empirical result, with your computational proximity or with the homology theory?

References:

Courgeau, D. (1973). Migrations et découpage du territoire. Population, 28(3), pp. 511-537.

Courgeau, D., Muhidin, S. (2012). Estimating changes of residence for cross national comparison. Population English Edition, 67 (4), pp. 631-652.

@Daniel Courgeau: how to compare human migration between areas of different shapes and sizes, in countries which have also different shapes and sizes?

First, many thanks for your very detailed answer. Very helpful and very interesting.

There a couple of basic terms that may help further our discussion. The two terms are space and shape space. A space is any nonempty set of points. A shape space is a collection of sets of point X and each particular configuration (arrangement of the points) in a subset of X defines a shape. Shapes of spaces were introduced by A. Zomorodian, p.3 in

https://www.cambridge.org/core/books/topology-for-computing/1171035B570105A57865CEA390BA5E74

For a very nice introduction to homology by Prof. Zomorodian, see

http://graphics.stanford.edu/courses/cs468-04-winter/notes/06.pdf

Two other basic terms are topology and proximity theory. The focus in a topology of any space is on the nearness of points to sets. For example, we could introduce a topology on a space defined by a migratory population (each point in the set would be defined by the location of a particular population member at an instant in time). The sets in our case would a set of population members or the shape of a migratory population.

The focus in a proximity space view of a shape space is on the nearness of one subshape to another subshape. By subshape, I mean a segment or fragment of a shape in a shape space. For example, we can begin to consider the nearness of one shape to another shape in shape space defined by a migratory population.

@James Peters

First thank you for your quick answer which I think may permit us to go further in our very interesting discussion. I will answer to the two points you raised.

First, I am considering a space which is a non empty set of points X (individuals) in a general territory: France, U.S.A, Canada etc. Also, I am considering a shape space (the zonal system) which is a collection of set points X and each particular configuration in a subset of X defines a shape.

Second, I focus in a typology on a space defined by a migratory population. The sets are in my case a set of population members on the shape of a migratory population. So that the focus may be on the nearness of one shape to another shape in the shape space defined by a migratory population.

So that, it appears that the basic terms you are using are very near to the terms I used to define my approach. In order to go further I am sending you in the joint file some of the main hypotheses and results in the simple case I considered, for a square territory. I think that the classical method I used is not sufficient to obtain a more general result for every shape space of the territory.

Do you think that with your more general approach we can go further? I will be very happy to continue this work with you.

@James Peters

I would like to add more information on this topic.

First, as I told you nine months ago, the paper I wrote with Joel Cohen demonstrate for the first time that, for any family of similar shapes, both the mean distance between randomly chosen points and its standard deviation scale linearly with rescaling. I think that this may be a first step for the demonstration of the value of Courgeau’s k, as this distribution of distances plays a major role in this demonstration.

Second, if in numerous cases Courgeau’s k captures quite correctly the effect of space (shape, size, distance) on the level of mobility between areas, in other rare cases the level of capture is not so good.

Stillwell et al. (2014) proposed to create multiple intermediate geographies by random aggregation of spatial units (Image studio software). With this software we are able to start from an observed zonal system with N areas, create intermediate geographies with n areas and calculate the mean number of migrations observed under this new zonal system. In this case, even if the capture for the observed zonal systems is not so good, the linearity of the curves obtained with the new zonal system issued from an observed one is very good. In order to show you more clearly these results you have in the first figure the observed data for France and the observed and mean simulated data from the observed ones for Denmark (Bell et al., 1985) in the second one observed and mean simulated data from the observed ones for Switzerland.

We can conclude that one observed point (current migration rate according to the nb households / nb zones) is a random estimation of a corresponding mean value which is unknown.

Bibliography:

Bell M., Charles-Edwards E., Ueffing P., Stillwell J., Kupiszewski M., Kupiszewska D. (2015). Internal migration and development: comparing migration intensities round the world. Population and Development Review, 41(1), pp. 33-58.

Stillwell J., Darras K., Bell M., Lomax N. (2014). The IMAGE studio: A tool for internal migration analysis and modelling. Applied Spatial Analysis and Policy, 7(1), pp. 5-23.