A Point-Theory of Morphogenesis

Abstract

Building on earlier work with generalised conic sections, we use the superformula to introduce ultra-flexibility instead of rigidity as encoded in the geometry of Euclid and Descartes. By considering Points as ultra-extensible primitives, we define Points endowed with shape, size, and historical continuity. This Point-Theory of Morphogenesis addresses multiple challenges for a mathematical theory of morphogenesis for both natural and abstract shapes. The theory is formalised by a minimal set of one definition, two axioms, and two postulates.

1. A Theory of Morphogenesis: The Challenges

On natural forms, D’Arcy Thompson wrote: “So the living and the dead, things animate and inanimate, we dwellers in the world, and the world in which we dwell—πάντα γα μὰν τὰ γιγνωσκόμενα (for all things indeed, that are known)—are bound alike by physical and mathematical law” [1]. Morphogenesis is the study of forms and their development and evolution. From a mathematical point of view, it amounts “to construct an abstract, purely geometrical theory of morphogenesis that is independent of the substrate of forms and the nature of the forces that create them” [2].

For physics, this programme has already taken some steps, with models in the language of mathematics, but extending this programme to biology is a major challenge. In response to E. Wigner’s “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”, I.M. Gelfand stated, “There exists yet another phenomenon which is comparable in its inconceivability with the inconceivable effectiveness of mathematics in physics noted by Wigner—this is the equally inconceivable ineffectiveness of mathematics in biology” [3].

According to Marcel Berger, ‘‘Present models of geometry, even if they are quite numerous, are not able to answer various essential questions. For example: among all possible configurations of a living organism, describe its trajectory (life) in time” [4]. The challenge can even prove insurmountable: when observing a plant, it is impossible to find in an algorithmic or otherwise mathematically precise way the mathematical system that created it. “Plants” can be replaced by rocks, galaxies, animals, clouds…

A geometrisation of physics seems to be an easier task than a geometrisation of biology, where history plays a crucial role, be it in development or in an evolutionary context. A biologist works with the discrete (animals, plants, microbes, cells, and DNA, RNA, proteins…) but recognises the continuous in evolution and development. René Thom once said that we can only experience the continuous (space and time), but we need the discrete in order to understand (categories, measurements…).

An illustrative example of the difference between a mathematical and a biological mind is the Cantor set. The iterative algorithm, where parts of a line are removed to arrive at the Cantor dust, is mathematically very sound. From a biologist’s point of view, however, the discarded pieces of line neither disappear nor are they lost. They remain firmly connected to all other pieces by historical or other threads (in another place or dimension) and from this perspective continuity is preserved.

Our focus is on a geometrical theory of forms and their development in the broadest possible sense, including their history. Such theory of forms and morphogenesis must overcome several fundamental challenges.

First, it concerns both natural and abstract forms. Natural forms include everything from the small and smallest (molecules, atoms, quarks, to the infinitely small) to the large and largest (stars, galaxies, black holes, the universe, to the infinitely large), and to the meso-scale (plants, animals, termite mounds, bacteria, mountains, oceans…). Natural forms can be extended to everyday forms and thus also include man-made forms (cars, bridges, windows, music, stories, myths…). They also include more abstract forms, such as inspirations, thoughts and dreams, and sensory (and non-sensory) perceptions, not only of humans, but of all sentient beings.

Abstract forms include mathematical objects such as geometric shapes, manifolds, categories, or structures in the broadest sense. They may also include some expected developments such as the extension of the notion of manifold [5]. Abstract forms also include non-existent objects such as golden mountains and square circles [6].

Second, it is not only about the pure description of forms, but also about their emergence and their entire history, regardless of whether they are viewed through a discrete or continuous lens, one of the oldest conflicts in human thought. We need the discrete to understand (categories, measurements…), but experience space and time as continuous. Bifurcations appear discrete to us, but the underlying manifolds are continuous [2], similar to the biologist’s view of the Cantor set.

Thirdly, the challenge is to deal with individual instances of natural and abstract forms, knowing that mathematics is the science of patterns. When modelling plants or other natural phenomena, one starts from a particular model and applies stochastic methods to obtain more realistic forms, but this is at best a very rough approximation. One of the most elegant methods to describe shape and form is Ulf Grenander’s Pattern Theory [7,8,9,10], but this has the disadvantage of focussing on patterns, not on individuals.

When studying a particular plant, it is impossible to find the mathematical model that generated it in an algorithmic or otherwise mathematically precise way [11]. If you study a thousand plants, they will all differ in some way, because their phenotypes are a temporal sequence of adaptations to internal and external stress factors acting on the plants. Again, you can replace “plants” with rocks, galaxies, animals, clouds… However small the differences may be, there will be differences, and one cannot understand the true diversity of nature by focussing only on patterns. A geometric theory must take into account the individual plants, animals, crystals, galaxies, atoms….

Fourth, the relationships between a real plant (or any other object under study) and its model (i.e., any approximation, with any accuracy) combine various aspects of the first three challenges. Accuracy in science is always finite (${10}^{987654321}$ may seem a big number, but it is hardly any closer to infinity than ${10}^0$), but however small or negligible certain effects may seem, one must consider Dirac’s extension of Archimedes: “If you pick a flower on Earth, you move the farthest star.”

The Point-Theory of Morphogenesis aims to overcome all these challenges and match exactly with the object under investigation. Our strategy will be in line with René Thom’s endeavour [2], and the Unique Rational Science of Gabriel Lamé’s (1795–1870): first find a best-fitting coordinate system, then solve the relevant boundary value problem (to gain an understanding of the conditions that give rise to the shape) [12].

To this end, we need to decouple form and shape (and their morphogenesis) from any numerical model (decoupling the geo from the metric in geometry). The assignment of categories such as dimensions, time, space, maps, grids, and number systems can be performed in a separate step. The only thing we need to ensure in the first step is the commensurability of forms and shapes, a continuous transformation from one form to any other form. Unity in diversity and diversity from unity. Before Pythagoras’ All is Number should come Barbara McClintock’s “Basically everything is one. There is no way in which you draw a line between things. What we normally do is to make these subdivisions, but they are not real. Our educational system is full of subdivisions that are artificial, that should not be there” [13].

This will establish the existence of phenomena, that are independent of our observations and independent of the methods we use to study shapes and phenomena, developed from a human perspective, creating an additional challenge.

2. A Plurality of Geometries for the Natural Sciences

The motivation for our geometric methods lies in our observations of the world, and so geometry began. Euclidean geometry in dimension two essentially uses the circle as a base or fundamental figure to determine isotropic distances from a human perspective. The Greeks used commensurability, rather than numbers, as the key concept in geometry (Elements, Book X). Much later, René Descartes introduced the real numbers into geometry.

In the 19th century, doubts about Euclid’s fifth postulate, Gauss’ investigations of surfaces in two dimensions, and Riemann’s generalisation to the multidimensional case led to Riemannian geometry, which enabled great progress in the study of natural phenomena. The specific choice of rules and metrics in geometry corresponded to specific invariants, and these invariants corresponded to the conservation laws in physics.

In almost all developments, the circle is the main concept in the mathematical vocabulary for the study of nature, as the following three examples show:

• The circle as a measure of curvature by Isaac Newton. The extension to higher dimensions led to the General Theory of Relativity.
• Superposition of circles to describe complex phenomena. Ptolomy’s epicycles and the Fourier methods, which are ubiquitous in modern analysis, are mathematically equivalent [14].
• The introduction of complex numbers into this scheme led to quantum mechanics (“In this sense, quantum mechanics is a complexification of Ptolemy’s epicycles” [15]).

The circle and the concept of isotropy continue to dominate the study of natural phenomena, often in disguise. For Richard Feynman, it is one of the most fundamental mysteries in science: “We have in our minds a tendency to accept symmetry as some kind of perfection. In fact, it is the old idea of the Greeks that circles were perfect, and it was rather horrible to believe that the planetary orbits were not circles, but only nearly circles. The difference between being a circle and being nearly a circle is not a small difference; it is a fundamental change so far as the mind is concerned. There is a sign of perfection and symmetry in a circle that is not there the moment the circle is slightly off. That is the end of it, it is no longer symmetrical. Then the question is why it is only nearly a circle—that is a much more difficult question… So, our problem is to explain where symmetry comes from. Why is nature so nearly symmetrical? No one has any idea why” [16].

In fact, when investigating natural forms and phenomena, it is by no means self-evident that other beings, living under different conditions, would arrive at the same geometry with the Euclidean circle as the basic figure.

This is one of the main ideas of Finsler geometry, in which the basic figure can also be ellipses or other shapes. Riemann had already established that an extension to the fourth power was possible. In the words of Chern, Finsler geometry is Riemannian geometry without the quadratic restriction [17]. The simplest Riemann–Finsler geometry is given by $ds={\left[\sum {(dx}_i^n)\right]}^{1/n}$ and Minkowski metrics by $s={\left[\sum {(x}_i^n)\right]}^{1/n}$ with the Euclidean case for $n=2$.

Our blind faith in the circle has led to considerable success in science, but it can also hinder progress. Indeed, science can always progress further when conic sections (circle, ellipse, parabola, and hyperbola), or their generalised forms, provide very compact and less complex models.

• Galilei and Kepler used conic sections (projectively equivalent to the circle) for the trajectories of projectiles and the orbits of planets.
• The generalisation of conic sections to higher-order, with superparabolas and supercircles (Lamé curves), adds one additional parameter to classic conic sections.
• The further generalisation of Lamé curves to any symmetry via the superformula has its roots in biology.

The discoveries of Galilei and Kepler inspired Newton’s laws and methods (based on lines and circles). The generalisation of conic sections to higher-order (in the exponents) involves power laws, which are all higher-order parabolas. These are widely used in the natural sciences for the study of allometry, first systematically investigated in biology by Galilei.

Supercircles and superellipses and more generally Lamé curves have received much less attention, although the equations were known (e.g., Fermat’s Last Theorem and Barrow’s quasi-circular curves [18]). Gabriel Lamé was the first to systematically study this class of curves, and his motivation was “to apply Descartes’ beautiful geometry to crystallography” [19]. The extension to Finsler geometry should therefore be called Riemann–Finsler–Minkowski–Lamé geometry.

Just as Euclidean geometry in dimension two essentially derived from the circle as a basic figure, Lamé’s supercircles ($\left|x^n\right|+\left|y^n\right|=R^n)$ underlie the simplest definite Minkowski–Finsler geometries in which 4-fold anisotropies occur, when considered from a human point of view [20]. For each supercircle, the associated trigonometric functions can be defined, and a Pythagorean theorem applies to any supercircle based on these functions. In recent years, superellipses have been successfully used to model tree rings of conifers [21,22,23], plant leaves [24,25], and cross-sections of square bamboos [26,27], corroborating Lamé’s ideas.

The generalisation of supercircles to any symmetry [26] can act as a generic geometric transformation on planar functions (Equation (1)) [28]:

$$\varrho \left(\vartheta ;A,B,n_1,n_2,n_3\right)={\displaystyle \frac{1}{\sqrt[n_1]{{\left|\frac{1}{A}\cos \left(\frac{m}{4}\vartheta \right)\right|}^{n_2}+{\left|\frac{1}{B}\sin \left(\frac{m}{4}\vartheta \right)\right|}^{n_3}}}}·f\left(\vartheta \right),$$

(1)

with $A,B,n_1\in {\mathbb{R}}_0^{+}, m,n_2,n_3\in \mathbb{R}.$ This results in so-called Gielis curves for dimension two and Gielis (hyper)surfaces for dimension three (and more) [20,29,30], as basic figures to describe most natural m-fold anisotropies for any $m\in \mathbb{R}.$ It is the simplest way of representing shapes, both geometrically and topologically [29]. It has led to generalisations of surfaces of constant mean curvature surfaces [31] and Euler’s elastic curves [32] for anisotropic cases. Equation (1) and the notion of a flexible radius inspired the generalisation of the stretched Laplacian [33,34,35,36], so that relevant boundary value problems can be solved on any normal polar domain using the classic Fourier projection method, even for multivalued functions [33]. Equation (1) has been studied in the framework of Finsler geometry to model forest fires [37] and seismic wave propagation [38], and of metric spaces [39]. In a more general sense, the unit circle of a starfish can also be considered a measure of curvature [29]. Recently, Equation (1) was used for defining superelliptic inner and vector products and a generalisation of quaternions [40,41].

From the point of view of the natural sciences, the application of Equation (1) to the “most natural” curves and surfaces of Euclidean geometry (e.g., for dimension two: the circles with $f\left(\vartheta \right)=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t},$ and the logarithmic spirals $f\left(\vartheta \right)=e^{a\vartheta })$ among the closed and non-closed curves, respectively) leads to many of the shapes that we can observe in nature—in biology, crystallography, physics, and chemistry [30,42]. In the last ten years, the superformula has established itself as a solid scientific method. For example, all bamboo leaves and all bird eggs can be defined with arbitrary precision. In total, over 40,000 biological shapes have been tested and successfully modelled [43] with Equation (1). The superelliptic shapes of tree rings are a simplified form of Equation (1), and only two parameters are sufficient to describe the changes in the tree rings over the years [21,22,23], a simple, low dimensional solution to what we would call a multidimensional optimisation problem. As a low dimensional optimiser, Equation (1) has been used advantageously in electromagnetics [44,45,46,47], mechanics [48,49,50], photonics [51,52,53], among others.

3. Continuous Transformations and Flexible Radii

Equation (1) aims at a unified description of natural and abstract shapes with one generic geometric transformation. It offers new methods to construct curves, surfaces, and solids, and enables continuous transformations from circles to squares or starfish, from spheres to spindle tori to tori, and more. At the same time, this notion of continuous transformation can also be understood as flexible radius or flexible distances. For both cases, examples are given below.

3.1. Continuous Transformations

Certain instances of the superformula have parameters $\left[\right(A, B, m) ;(n_1,n_2,n_3\left)\right]$ with specific numerical values. The first group $(A, B, m)$ defines size $(A, B)$ and symmetry $\left(m\right)$ and the second group are the shape parameters $(n_1,n_2,n_3)$. A number ϱ$\left[\right(A, B, m) ;(n_1,n_2,n_3\left)\right]$ is called an n-tuple, a number with different parameters. The n-tuple may be denoted as ϱ $[f\left(\vartheta \right);(A, B, m) ;(n_1,n_2,n_3\left)\right]$ for the transformation of the plane curves $f\left(\vartheta \right)$.

Some examples include the following:

• The inscribed square is denoted as ϱ$\left[\right(1;1;4\left)\right(1;1;1\left)\right]$.
• A unit supercircle is denoted as ϱ$\left[\right(1;1;4\left)\right(n_1=n_2=n_3\left)\right]$.
• The starfish is denoted as ϱ$\left[\right(10;10;5\left)\right(2;2;7\left)\right]$.
• The circle is denoted a ϱ$\left[\right(1;1;0\left)\right(2;2;2\left)\right]$.

Proposition 1.

A circle can have any symmetry $m\in \mathbb{R}.$

If A = B and $n_2=n_3$ are all 2, ϱ is always equal to 1.

Some examples include the following:

• A circle can be a shape without angles ($m=0, \mathrm{o}\mathrm{r} m\to \infty )$ but also a square ($m=4$), a pentagon ($m=5)$, or a pentagram ($m=5/2$).
• A continuous transformation of a circle to a square or a pentagram ($m=5/2$) only requires a change in the shape parameters $\left(n_2,n_3\right)$.
• The transformation of a circle into a square or pentagram does not require a change in the symmetry parameter $m$ and no symmetry is broken.
• It is not limited to closed shapes, as the associated trigonometric functions lead to completely new possibilities for generating sounds and waves in general [54]. Self-intersecting curves (m is a rational number) generate polyphonic sounds [55].

The generalisation to 3D or higher can be achieved in different ways, for example, in parametric coordinates (Equation (2)).

$$\left\{\begin{array}{c}{x=\varrho }_1\left(\vartheta \right) \cos \vartheta .{\varrho }_2\left(\phi \right)\cos \phi \\ y={\varrho }_1\left(\vartheta \right) \sin \vartheta .{\varrho }_2\left(\phi \right)\cos \phi \\ z={\varrho }_2\left(\phi \right) \sin \phi \end{array},\right.$$

(2)

where ${\varrho }_1$ and ${\varrho }_2$ are curves defined by Equation (1), that lie in two planes perpendicular to each other. One can be regarded as a basic figure (cross-sectional shape), which is rotated around the second figure (the base line). If both ${\varrho }_1$ and ${\varrho }_2$ are circles, the result is a sphere. When ${\varrho }_2$ becomes larger than ${\varrho }_1,$ the sphere gradually transforms into a spindle torus and a torus. Thus, we have a continuous transformation from any circle into any torus, and from a genus $g=0$ surface to a genus $g=1$ surface with one hole. This can be extended to all orientable surfaces with $g\ge 0.$ For example, rose curves as base lines ($cos\left(k\vartheta \right)$, $k\in \mathbb{Q}$) generate toroidal surfaces with different holes. The number of holes of a surface and its orientability are topological invariants but surfaces can be understood as continuous transformations.

Proposition 2.

A sphere can be continuously transformed into any compact orientable topological surface.

In cylindrical coordinates, a generalised cylinder is obtained, whose cross-sections can be circles, polygons, or starfish. They can change along the height of the cylinder, together with the changes in the parameter values in the n-tuple. An example is the stem of a cactus, which changes slightly in each cross-section. Figure 1 shows three double cones, whose cross-sections are defined by specific n-tuple of parameters (Equation (1)). The shape remains constant along the height but the size changes.

When moving along the height axis, the cross-section increases or decreases until it reaches zero size where both cones meet. During this movement, only one set of parameters in the n-tuple changes, namely the size. $A,B\in {\mathbb{R}}_0^{+}$ but considering the numerator 1 in Equation (1) as a unit circle, the size can be determined by $f\left(\vartheta \right)=R$ for any circle. All other parameters of the n-tuple $(m;n_1,n_2,n_3)$ (remember: a single superformula number) remain fixed. Size and shape are independent. This leads to the following:

Proposition 3.

Information is preserved for each$f\left(\vartheta \right)$, even if $f\left(\vartheta \right)$ = R = 0.

Despite the apparent shrinkage of the shape to a size of zero in the double cones in Figure 1, the structural information remains encoded in the parameters of the superformula. The superformula remains invariant even as individual parameters change. The number of dimensions does not increase either; only the values of the n-tuple evolve over time. No information is lost or destroyed. It is like lenses and our visual system.

3.2. Flexible Radius

The classical application of geometric methods in the natural sciences is based on rigid geometries, with rules for flexibility, through transformation rules in geometry or topology. The shapes generated by Equation (1) can be imagined as the result of using a flexible ruler.

To draw a starfish or a square (Figure 2), one can use a rubber band or a spring, and Equation (1) tells you how and when to stretch them. It will be a unit circle in its own metric and only to an outside observer will the length of the elastic radius change. An inner observer will not notice this because they are also stretching. Furthermore, the observers live in their own circles. It is only a square, if you look at it from the outside, from the human point of view. A prime example of such elastic contractions are transformations in Special Relativity Theory.

Proposition 4.

Lorentz-Fitzgerald transformations are a special case of Equation (1) for (${A=B=1; n}_1=n_2=n_3=2 and m=4)$.

In his work on relativity, Henri Poincaré used ${\displaystyle \frac{1}{\sqrt{1-{\epsilon }^2)}}} ($with $\epsilon$ as a number between 0 and 1). The Pythagorean theorem gives $\rho ={\displaystyle \frac{1}{\sqrt{1-{\mathit{sin}}^2\vartheta }}}$ for Equation (1). Figure 3 shows that this leads to the equality of the red and green (Area = 1) zones.

Remark 1.

The formal structure of Equation (1) is quite general. The space-time metrics of the Robertson–Walker type, which are based on deformations of the theorem of Pythagoras, are formally like supertransformations (Equation (1)) of Euclidean circles [30].

Another example of this flexibility are the trigonometric functions on supercircles:

$${}_n\cos \left(\vartheta \right)=\mathit{cos}\left(\vartheta \right)·{\displaystyle \frac{1}{\sqrt[n]{{\left|\mathit{cos}\left(\vartheta \right)\right|}^n+{\left|\mathit{sin}\left(\vartheta \right)\right|}^n}}}$$

(3)

$${}_n\sin \left(\vartheta \right)=\mathit{sin}\left(\vartheta \right)·{\displaystyle \frac{1}{\sqrt[n]{{\left|\mathit{cos}\left(\vartheta \right)\right|}^n+{\left|\mathit{sin}\left(\vartheta \right)\right|}^n}}}$$

(4)

$${}_n\tan \left(\vartheta \right)=tan\left(\vartheta \right)$$

(5)

Consider a right-angled triangle inscribed in a unit circle with one end of the oblique side coinciding with the centre of the circle and the other end lying on the circle. One can stretch this oblique side so that the end coincides with a Point on a supercircle with exponent n. The horizontal and vertical sides of the triangle, the cosine and sine, are stretched to ${}_n\cos \left(\vartheta \right)$ and ${}_n\sin \left(\vartheta \right),$ respectively, and the oblique side is stretched to

$${\displaystyle \frac{1}{\sqrt[n]{{}_{n}c{os}^n\left(\vartheta \right)+{}_n{sin}^n\left(\vartheta \right)}}}$$

(6)

This leads to a generalised Pythagorean theorem ${}_{n}c{os}^n\left(\vartheta \right)+{}_n{sin}^n\left(\vartheta \right)=1$. When a circle with radius $R$ in transformed into a supercircle, the same relationships apply. The radius is stretched, but so are the horizontal and vertical projections, ${}_n\cos \left(\vartheta \right)$ and ${}_n\sin \left(\vartheta \right),$ respectively.

The Clootkrans proof of Simon Stevin can be used to further illustrate this type of elasticity or flexibility (Figure 4). This proof showed the impossibility of a perpetuum mobile [16,56] and laid the foundation for forces and vectors. In the example of Figure 4, the number of cloots (balls) on AB is four, with two on BC. So, if angle B in Figure 4 is a right angle, the minimum number of balls on side AC is five. In a circle, A corresponds to the centre, AB to the horizontal axis, BC to the vertical axis, and AC to the oblique side of a right-angled triangle. When ST is stretched to fit on a supercircle, the Clootkrans is stretched, but the number of cloots (equivalent to the markings in Figure 2) remains unchanged.

4. From Rigid to Ultra-Flex

With Equation (1) we can transform a circle into a starfish, a flower, or a seed, and with extensions of Equation (1) [57], many more shapes can be created. But in our imagination, we can go even further and morph a circle or a starfish into anything, including a unicorn or an explosion. That is what we want to model. Then we must go from flexible to ultra-flexible, with ultra as in Nec Plus Ultra, the highest possible, the ultimate flexibility. How this works can be understood by juxtaposing it with Euclid.

In Euclid’s Elements, a Point is defined as that which has no size or dimension (and implicitly no shape). The first postulate states that a line can be drawn between any two Points, regardless of how close or far apart they are. A line can extend all the way to infinity (the second postulate).

Imagine, however, that we define a Point as infinitely extensible. Then we can extend a Point to any length, and it is still the same Point. We can stretch this Point-Line to a finite or infinite plane, and this plane to a space. The same Point can be stretched into any curve, disc, plane, or surface, in any dimension. It can form continuous manifolds but also discrete grids, nets, and meshes.

An example from physics is gold, a very ductile (stretchable) material, that can be drawn into very fine wires. A block of 1 cm³ gold, for example, can be stretched into a gold wire of almost 13 km, if the wire thickness is 10 micrometres. Moreover, with a wire thickness of 1 micrometre, the gold cube could be stretched to a length of 100 km. Metals such as copper and platinum are even more ductile than gold.

Now imagine that we have an infinitely stretchable material at our disposal, and we reduce the 1 cm³ to a very small size, a Point. We can then stretch this Point into a finite or infinitely long wire (this is our straight line). This line can in turn be stretched into a plane, a (hyper-)surface or a (hyper-)solid. Our geometry then consists entirely of a stretchable Point (or vice versa: any plane or surface can be reduced to a single Point).

For a more formal approach to our Point-Theory of Morphogenesis, we follow the advice of one of the great geometers of the 20th century, A.N. Alexandrov: “Retreat to Euclid!” For Alexandrov, Euclid’s geometry provided a universal, visually intuitive foundation that connected mathematics to the cultural and intellectual traditions of ancient Greece. For the study of forms, we can now return to the elegance and universality of these methods, by aiming for a minimal set of definitions, axioms, and postulates (“Examples, problems, and solutions come first. … In developing and understanding a subject, axioms come late. Then in the formal presentations, they come early” [58]).

5. A Formal Point-Theory of Morphogenesis

5.1. One Definition, Two Axioms, and Two Postulates

Euclid begins with axioms, stating that things which are equal to the same thing are also equal to one another, and if things coincide with one another, they are equal to one another. However, when considering equal, greater, or less, we should keep Figure 2, our external glasses and our God’s Eye views in mind.

In our Point-view, all things are commensurable in the same way that all supercircles and supercurves are commensurable. They can be measured (compared) with a common ruler, namely Equation (1) or an ultra-flexible ruler. The fifth axiom in the Elements states that the whole is greater than the part, but in a Point, everything can be both whole and part, including any characteristic.

In our minimal set, we need one definition (D1), two axioms (A1, A2), and two propositions (P1, P2):

One definition:

Definition 1.

A Point has size and shape.

Two axioms:

Axiom 1.

No two Points are equal (Axiom of Distinguishability).

Axiom 2.

Any Point can be morphed into any other Point (Axiom of Connectedness).

Two postulates:

Postulate 1.

A Point is that which has extension.

Postulate 2.

A Point is that which has intension.

5.2. Remarks and Motivations

A Point is not necessarily a form or shape in the classical mathematical sense. Points can also be equations, (operator) algebras as in non-commutative geometry (where classical Points have no meaning), or categories.

Ex- (P1) and In- (P2) refer to Points increasing in size (extension) or decreasing in size (in- as the opposite of ex-). The extension and intension are understood as Nec Plus Ultra (infinitely large and infinitely small, and any larger or smaller is not possible). The use of -tension in both postulates also refers to physical tension. Gabriel Lamé was the first to study superellipses systematically, and his work on curvilinear coordinates led Elie Cartan to name Lamé one of the cofounders (with Gauss and Riemann) of Riemannian geometry. He is also one of the founding fathers of elasticity theory [59]. Although any stretching is possible in our Point-Theory, under natural (terrestrial or cosmological) conditionzs, both the flexible radius of natural superellipses or supershapes and the drawing of gold wires are finite.

Another motivation for P2 (intension) is semantic, namely the classical meaning of intention. The intention of a Point-Seed is to grow into a plant, and a Point-Termite that has developed from a Point-Termite_Egg intends to contribute to the well-being of the Point-Termite_Mound. A counterargument is that the explosion of fireworks is not intentional in the fireworks, but it is intentional in those who handle (or mishandle) the fireworks. A critical look at axiomatic systems also reveals intent. Euclid’s axioms and postulates are rules, with the intention that others apply or follow those rules. The geometer who uses Euclid’s axioms and theorems to study abstract or natural shapes has the intention of following these rules.

This remark also serves another purpose, namely, to reveal the implicit conventions in the language of science (in the broadest sense). Golden mountains and square circles have served as examples of non-existent objects under certain rules and requirements, such as having the properties square and round at the same time [4]. For the square circle, however, these are additional constraints, where square and round are defined from a human point of view. Supercircles are Points that give existence to square circles. Similarly, a Point-Gold_Nugget developing into a golden mountain gives existence to this abstract object.

6. Numbers and Grids

In a next step, additional elements or constraints such as space and time can be introduced to the Point-Theory of Morphogenesis. Time, for example, requires the notion of a totally ordered set [60]. Euclid’s axioms and the exclusive requirement of ruler and compass lead to Euclid’s Elements, and from there to the various generalisations that followed, with the circle and n-sphere based on our human thoughts. Viewing nature through a circular lens leads to questions like Feynman’s enigma: “Why is nature so nearly symmetrical? No one has any idea why.” The answer lies in a more general consideration of unit circle or unit ball.

The introduction of concepts such as space, time, (a)symmetry, (a)nisotropy, and dimensions depends on our human perspective. From there, we develop theories and methods and use them as the basis for our study of nature, with our external glasses and our God’s Eye view. Mathematicians and scientists use many such implicit assumptions, and the Axiom of Indistinguishability, hidden in plain sight in quantum theory, is one of the best examples. Nature is not nearly symmetrical, it is fully symmetrical, with superparabolas, superellipses, and supercurves in nature, with their own inherent symmetries.

The postulates P1 and P2 are the first step of our Point-Theory, but D1 (Points with shape and size) already prepares for the next step. The assignment of grids or meshes and numbers to Points can be carried out with continuous maps or discrete grids, and one can choose between a variety of number systems (in considerations of the infinite, one should separate the pure (Nec Plus Ultra) extension (P1) and intension (P2) of Points from the introduction of numbers and numerical systems to Points). In principle, any current method in applied mathematics can be used, but new developments will open new opportunities.

To combine differential and difference calculus from the start, for example, Time Scales introduced by Stefan Hilger [61,62] can be used. Such Time Scales can include real and rational numbers, but also any continuous or discrete scale, including Cantor dust. A proposed method for combining Points are R-functions [49]. They are directly related to supercircles and are naturally suited to multivalued logic [63,64], as an extension of simple binary logic. The Fourier projection method based on the stretched Laplacian [34,35,36] can be used to solve relevant boundary value problems.

Looking further, Equation (1) will lead to improved understanding of biological and ecological systems [65,66], with focus on individual leaves, trees, and tree rings, instead of probabilistic methods for ensembles [67]. Other examples include extensions to anisotropic cases [32,68], generalisations of quaternions [40,41], new types of signals [54], or new trigonometries [69].

7. Conclusions

One of the advantages of Equation (1) recognised by geometers is that it describes the shape and its development starting from a central Point. With stretchable radii, the (Nec Plus Ultra)-flex extension allows for all possible and conceivable transformations from a single Point.

Using the idea of an infinitely flexible Point, a Point-Theory of Morphogenesis is constructed that completely and uniquely identifies the object (or subject) under study. Model $\equiv$ Object, or Model $\equiv$ Subject. In this sense, it is ontologically perfect. One example is the geometrical continuum. Moreover, our Point-Theory is about reality, both natural and abstract, since “Reality presents itself to us as phenomena and shapes… Science must come back to this essential goal, to understand reality” [2].

One definition, two axioms, and two postulates capture this formally. Points have size and shape (Definition D1), no two Points are identical (A1—Axiom of Distinguishability, which amounts to the strong(est) version of Leibniz’s Identity of Indiscernibles) and all Points can be transformed into any other one (A2—Axiom of Continuity or Connectedness).

The Point-Theory of Forms and their Genesis addresses all four major challenges for such theory, and its manifestations include square circles (Aristotle’s οὺθὲν γἀρ ἂν οὕτωϛ θαυμάσειεν ἀνὴρ γεωμετρικὸϛ ὠϛ εὶ γένοιτο ή διάμετροϛ μετρητή. For nothing would a geometrician wonder so much as if the diagonal became measurable (with the side) [70]), Darwin’s endless forms most beautiful, but also mathematical structures, thought patterns, and thinking kernels [71]. All things, indeed, the known and the unknown. $\mathsf{\Pi }$άντα γὰρ μὰν τὰ γιγνωσκόμενα καὶ τὰ ἀγνοούμενα.