About propaedeutics of geometry

"Who does not know geometry, let not enter the Academy."

(The inscription above the gate)

The geometry propaedeutics coming from Euclid is tied to the abstractions of the plane and the triangle as a planar figure. In this sense, it has been taught to everyone since childhood as an engineering science, but not as a science of the foundations of natural forms, although it is presented as such. Planimetry begins with a triangle (which was seriously supported in the post-Euclidean period as an archetype by the dogma of the Christian trinity), but not with a diangle, which is hardly even talked about in schools. But it is precisely the diangle and its stereometric spherical shapes that can most often be found in nature (especially living) in the form of the eyes, lips, leaves, shells of mollusks, as well as projections of similar objects on surfaces (including flat ones) in the form of a shadow, and the stereometric figures of the osohedron and dihedron formed by the diangle, which are also obviously observed in the forms of many fruits and cereals. Whereas a flat and smooth triangle, mostly rectangular, to see in the natural environment, that is, not in the world of artifacts, is much more difficult. Yes, and in the world of the most ancient artifacts created by man, we also see two-angled forms, but we do not think and do not call them such simply because we were not taught to see, name and understand them like that. These are boats and weaving shuttles, the simplest domes formed by the edge arches for tent structures, as well as some maces. Without such a more fundamental, although obvious in everyday discretion, concept of geometric forms, we cannot think and use their geometric capabilities to create mechanical, that is, physical, systems and solutions, believing such solutions in ancient times to be a fortuitous random find or outdated technological archaic, based on a very superficial aesthetic judgment. Meanwhile, forms based on such constructions are characterized by great strength, reliability and flexibility.

In schools, the beginnings of geometry, as a rule, are presented in the form of the ideas of Euclid and Descartes, that is, very derivative of certain principles and model, hidden not only from neophytes, but in many cases from the teachers themselves, since not all of them are sophisticated in history of their subject, and far from always present the entire list of conditions imposed on the discussion of geometry, while neophytes themselves are able to observe the noted alternatives in a natural way. These derivative ideas, being great, but representing more complex things, are hardly not universally stated in violation of the Cartesian principle of ascent from simple to complex.

Meanwhile, the diangle optimally combines closed and open, straight and rounded. It also represents a simple oscillation period of a string fixed at two opposite points – an element of one of the first physical devices through which attempts were made to comprehend the basis of all physical processes. And it is precisely just it, and not the triangle, that determines the moment of connection of the most elementary geometry with the most elementary mechanics. In this connection, the diangle shows (again, in the simplest, albeit unnoticed, form) that the transition from a certain dimension to a higher one is carried out precisely in the process of oscillation, since the diangle is conceived as a transversely oscillating segment of a straight line, which is essentially two-dimensional and is formed by maxima of the antinode of this segment in all directions in two coordinates out of three available during the oscillation and two – before it.

In its physical expression, the diangle represents the idea of ​​coincidentia of tension and compression, and its stereometric analogy is directly considered in the work of R.B. Fuller "Tensegrity" and in his co-author's work "Synergetics exploration...". Also, the osohedron and dihedron, the elemental forms of which represent the same figure formed by two diangles, differ in their geometrical ideas in the difference between the axially and equatorially exaggerated asymmetries of a circle or ball also considered by Fuller in "Tensegrity": the axial is an elongated osohedron, while the equator – flattened dihedron. "American Leonardo" was fascinated by the triangle, it is no coincidence that his other scientific epithet is "the man who invented the triangle." However, both planimetrically, and stereometrically, and arithmetically, a triangle is preceded by a diangle, especially if both of these figures are considered as hemispheres of a dihedron.

It is noteworthy that the convex diangle in its usual representation as an element of the osohedron is also a special case of a diangle in general, since Hippocratic dimples (which were proposed, by the way, by Hippocrates of Chios before Euclid) are also diangles. It’s they, in a couple, that exhaust the universe of convex diangles, known to the inhabitants of three-dimensional space, and together with those that have one concave (inverse) angle and both concave (inverse) angles – the universe of diangles in general. Unless, of course, a strange figure representing open arcs opposed to each other by external vertices is not considered as this: if it is so (although where are the angles?), then only in the sense of its special definition through a function in the Cartesian coordinate system (or in which yet). And if you do not consider the diangle in the case of 3+n dimensional continuum.

Another important property of a diangle, as well as the property of almost all polygons, is its ability to be closed by external corners to a spherical figure, and then it will represent a curved "eight" – two oneangles connected at one vertex. Thus, it seems that the diangles is not the most elementary planimetric figure, because it is preceded by a oneangle that forms the shape of a drop or loop. Just like diangles, oneangles can be biconvex (ogival) or convex-concave (hole-shaped or crescent-shaped), and together with those that have a concave (inverse) angle, they exhaust the universe of oneangles in general. A concave-corner oneangle is also called a "cordioid", but more often it is considered in the sense of one circle moving relative to another, and not in the sense of "oneangleness". Thus, the universe of diangles is exhausted by four elementary figures, the universe of oneangles - by three.

And if the vertex of the corner is interpreted as the vertex of the graph (which is very common and acceptable), then such a figure marks a reflexive relation. Here, an even more important conclusion is revealed: the circle shape is more fundamental than the classical Euclidean triangle, because, representing the curvature in itself, it represents the idea of ​​a line of a different dimension (relatively more straightened): the optimal shape of the only side of the oneangle must be round and convex at least at some part of its length – otherwise case it will already be a figure with a large number of angles. A triangle is formed by three circles (which both Fuller and his student Snelson have in different variations), but a circle created by a simple compass, if you think of it as a polygon with an endlessly-countless number of angles, can only be geometrically obtained with an infinite number of triangles. The same oneangle marks the geometric embodiment of the idea of ​​coinsidentia of intention and extension, substance and length. And if the tetrahedron, according to Fuller, is a "quantum of energy" as such, since "energy has shape", then a diangle is the shape of a quantum as such. Any angle begins with a oneangle, the loop makes up its geometric essence.

Also, the angle of a oneangle can be thought of as a connection or intersection of the ends of a single line forming an arc, and the corners of a diangle as formed by connecting or intersecting the ends of two arcs opposite in one way or another. But if the angle is nevertheless thought of precisely as a loop of negligibly small radius forming two sides (and even more so if it is thought of in this way), then the solution of the question of geometric primacy in favor of the oneangle before the diangle is possible only within the framework of a consideration of one scale. And the very correlation of a oneangle and a diangle turns out to be both inter-dimensional (projective) and inter-scale, in which sense the relationship between the categories of measurement and scale is observed at the level of elementary geometry. Anyway "Everything in Universe is divisible by two. There will always be two poles to any system. Unity is two". In this connection, it seems that the proper oneangle is always one way or another reduction or aspect of the diangle, including as a projection. And that Snelson with his quantum-mechanical exercises of speculation "a-la De Broglie" was nevertheless closer to these pre-triangular things than Fuller.

But what are the mechanical properties that make up the application utility of a oneangle? With a diangle, it’s clear: it is useful in the sense of lens bearing systems and bellows springs. If we take the shape of the loop – that is, the mechanism of a closed cable-stayed, then here too it is clear. What about the taut realization of its shape? Obviously, from the simplest and most well-known artifacts, it can be considered as a bow closed with its ends, assuming sufficient flexibility of its shoulders for such a closure. Mechanically, one- and diangles are predominantly springs (which, with minimal flexibility, become levers). So, their first function is the accumulation and transfer of effort. Since these are ring structures, but not quite circles, their ability to transform transverse waves into longitudinal waves is interesting here - in contrast to rings that are round in cross section and convert any waves communicated to them from the outside into transverse ones. And here it is important to understand how the application of transverse forces to them or their transverse oscillation creates a longitudinal oscillation on their part, and in what structural order it is better to provide it.

Why all this? To the beginning of the conversation: it is precisely such things (and even, probably, precisely these) that should form the basis of teaching geometry to ensure the freedom of an individual person in the physical world, and not to integrate it into existing engineering systems for the division of labor - including in secondarily in order to provide these systems with adaptability and optimality. For it is precisely these things and forms that are the most demanded in nature and the simplest and easiest to reproduce on the basis of its resources, despite all the difficulties of their comprehension, expressed in the language of higher mathematics.