DO WE REALLY UNDERSTAND GEOMETRY?

© I.L. Vikentyev, "TRIZCHANCE" System
(author's edition 2002)
Editor's note: We are publishing the introduction to "Index of Geometric Effects," first published in the collection "Rules of a Game Without Rules" (Petrozavodsk, Karelia, 1989). Several hundred effects were collected in this index, connected in some way with the shape of the body.
In the first instance this material is intended for those working on TRIZ and FVA (Functional Value Analysis).
Someone asked Euclid, the "Father of Geometry" to teach him this art, and asked the fully natural question: "What practical use will I get from studying these theorems?" Euclid turned to his slave and said: "Give him a penny, the poor man has come to find a use!"
This is a legend. However, it illustrates an important point  for centuries people studied the geometrical qualities of the body, but not connected with its physical side, but rather its technical. "Geometry is abstracted from all the properties of the body, apart from dimensional properties" tones the encyclopaedia. So what, says another reader, surely its important? Yes. You see we live in the conditions of a technical civilisation, where any machine is not free from its geometric form. Is all this taken into consideration? Numerous surveys in creative arts schools have shown that modern students, school pupils and engineers know less about the technical applications of geometry than they do about the applications of physical phenomena.
Here is such a characteristic example. 40 Higher Education students were asked which technical characteristics of a hyperboloid they knew. Each remembered the essence of the mathematical definitions, some even referred to the construction of V.G. Shukhov's tower…but that was all. But the hyperboloid, like any other geometric figure, has a wealth of useful creative applications in technology.
Fig. 1.1 
Let's say, cylinders with equal volume, but with different base diameters, located concentrically on one axis, the geometrical position of the points of opposite edges is a hyperboloid of revolution (look at Figure 1.1, and all will become clear). So, for example, it is easy to compare the volume of liquid in different ampoules, measure off preparations, Patent Certificate 1051383. (here and hereafter Pat Cert. N.). 
Surely "geometric effects" (hereafter  GE) are rare? No. Confirmation of this is the large number of inventions published in the official "Bulletin of Inventions." So, what's the problem then? Alas, at the present time there is no single GE information fund. As a consequence of this many solutions are duplicated in different sectors of technology; on each occasion inventors, expressing themselves poetically, start from zero, rediscovering the discovered and not developing the achievements of their predecessors.
You can be convinced of this yourself. For example  Pat. Cert.N. 441421: "Dowelless joint" the face of two shafts are cut at an angle and joined together by a cylindrical casing  fig. 1.2.
Fig. 1.2 
One shaft starts to rotate  the second shaft begins to turn immediately. An effective, simple solution. But in 10 years (!) four analogous solutions have appeared: Pat. Cert.N. 1133439, 1206514, 1225951, 1250749. 
This means a reference book is needed, an index of geometric effects, similar to the index of physical effects. And this reference book is being published for the first time, it is in front of you. It has no comparison. For this reason the main materials for its composition were drawn from not only books and articles by other authors, but also designs and patents. Despite the different absorbing examples the reader should remember that he is reading a serious reference book that demands pensive reading. Its aim is not just to familiarise you with technical solutions, but to provide a tool for inventive and rationalizing work.
So, good luck, dear reader, in learning and applying geometric effects!


FROM THE COLLECTION OF GEOMFORMS  TO THEIR DESIGN


So, we have established, that there are a lot of geometric effects (GE), that are either unknown to technical workers, particularly the younger ones, or are actually reproduced in other technical sectors. Therefore, to improve the coefficient of useful action "geometric knowledge" of an engineer, the inventor rationalist must form an index of geometric effects. But how can this be done?
There is experience of such a work. This is G.S. Altshuller's creation of the Theory of Invention Problem Solving (TRIZ). Here one must be governed by three simple principles, but as experience shows, these are not always taken into consideration by designers and teachers of technical design methods:
 research must be based on a large collection of technical and patent information;
 the existence of different qualitative levels of the technical solution must be taken into consideration;
 recommendations should be tested on a significant number of tasks at a high level (so far as to imitate the best and not merely average models).
Let's examine these requirements in greater detail. In fact, since 1960, for every one hundred thousand inventions in the national patent fund, there must be 1520 that are not completely trivial solutions, of which the spiral to mind. However, for each of the ellipsoid, hyperboloid and paraboloid type shapes, this figure does not exceed 23. This means, that it does not make sense to create a GE Directory using a complete survey of even several tens of thousands of inventions. Large volumes are necessary for the procedure. This work has been carried out (and is still in progress) by the author.
Over 1,200,000 national and foreign formulae for inventions and patents were examined in total during the manuscript's preparation. (The engineering and technology behind this examination is a separate and interesting matter).
Practical testing of recommendations for the GE Directory is carried out in technical design schools and in a number of qualifications' and skills' institutes, during implementation of the function and cost analysis phase (FCA).
Now comes the following question: how should the GE Directory best be organised? Let's tackle this question using the example of how one item in the Directory is written.
An ageold method exists that can be used to remove a ring relatively simply from your finger that is too tight. A thread is pushed under the ring from the side of the hand and wound thickly around the finger in a spiral, proceeding from the ring, then to the joint and then to the top of the finger. Next, when you reach the top of the finger, once pushed under the ring, the thread starts to unravel. The ring them comes off easily as a result of the movement and the pressure there on the finger...
So here we have Pat. Cert. N. 899277: ''A compression device for porous items with an aperture''. Previously, to obtain apertures in granules, made for stamping and punching, a rod of the required diameter was inserted. After the granules had been stamped and had caked together, they took out the rod, or to be more exact  tried to take it out. In reality the porous items buckled and broke, as when the rod was being removed considerable effort was applied….The author's evidence brought a protected idea to mind: a spiral is wound around the rod and then after compression has been applied, it is unwound spirally. Using this procedure the items themselves would remain undamaged. Like in the example of removing the ring, force is applied only on that part of the structure that has much less relation to the substance of the item, and not to the structure as a whole. We note that it is also possible to produce curvilinear apertures using this method.
Lets turn our attention to a different sector of technology: the manufacture of asbestoscement components. These are rolled by a roller specially designed for asbestos. In contrast to asphalt asbestoscement sticks to the roller, and therefore the rolled surface has dents and potholes. A solution to this problem is outlined in Pat.Cert.N 188874: one end of the roller is reinforced with a spiral, and the other end of the roller is left free, bending round the roller several times. During rolling the spiral unwinds in its place, sharply changing its curvature and discharging superfluous material… So let's note here that these solutions, similar to the above, are also mentioned in Pat. Cert. N. 174974, 334006, 558736, 1040066, 1068291, 1082504 etc.
So what does this mean? It means that once that you have analysed a number of inventions it is possible to draw out some common principles; so say specialists in methods of technical design and devices. In this instance they would say: "If one has to remove harmful interaction from macrosurfaces, a flexible body is wound round one of them, following which force is applied to each of its points."
In the current GE Directory sections are completed according to the principle "Geometrical shape and its technical function."
The index at the end of the directory is set out in the reverse principle: "Technical function  and geomforms bringing it to life." (This material was only published in the book and is not on the site  note I.L.Vikentyev).
Lets illustrate how to use this table. Supposing we need to find a form that will enable us to change its limited volume several times. We would open the "Index of possible GE applications", find the function  regulation of volume, alongside reference 5.12, 10.7. We read these points on the GE Directory and realise that the telescoping of the spiral from the roll and (or) the binding of the singlestrip hyperboloid of revolution may help us. If we want to find out more about the technical details of these solutions, then we must refer to the literature, a list of which is outlined in each section.
It should be stressed however that you can find the basic principles and solution ideas using the GE Directory, but not the desired construction itself and all its details. It is important to understand the position of GE in the TRIZ information fund.
As opposed to chemical effects, which enable you to obtain some substances from others by the absorption or isolation of energy, or physical effects that enable you to transform one form of energy into another, GE usually redistributes flows of energy and substances that are already present.
The creation of a GE directory is an important but interim phase of the project. It is now already clear that future work for its development will be carried out in the following areas:
We'll stop at nothing in each of these areas.
Analysis of the patent fund has shown that, strange as it may seem, geometry in technical systems is developing not in the form of "even more" geometry but in the direction of synthesis with energy and technical control systems. For example, it would seem, in order to increase the massheat volume the washed surface must be increased and increased, in particular, to transfer from spiral walls in apparatus to spiralwaffles. But this is simply "geometric" solution is unrealistic, or has been made without taking energy considerations into account: in the given instance expenditure for friction increase significantly…The inventor of ¹ 242 "Conformity of changing heat exchange on channel walls with discreet turbulization of current at the required convection" found a waffle profile and their mutual location which would not bring about significant energy losses while simultaneously increasing heat exchange significantly.
The "Geometric Spatial Operator" designed by I.L. Vikentyev repels the transformation of traditional forms in accordance with physical processes in the technical system and external environment, enabling more complete use of system resources (for more details see: Vikentyev I.L. Geometric Operator // Theory and practice of teaching technical design, Chelyabinsk, 1988, pages 5153).
The following direction  material field resources and GE.
The fact is that form is a creative resource that is practically everywhere, but is extremely rarely used effectively. Usually, the use of form enables us to maximise the redistribution of the flow of energy and substances that are already present in the technical system or the external environment.
Lets confirm this by using examples. The first is historical. The road that runs along the slope to the gates of the mediaeval castle in Tallinn, was designed so that the approaching traveller had to turn his righthand side to the castle. This was done with the purpose of "removing" the right hand of attackers, who would have needed it to hold their shield…Another example. When a ship is launched from the slipway the end may go under the water and strike the bottom. One of the common methods for solving this problem is to attach cumbersome and expensive pontoons to the body of the boat, which provide additional buoyancy. A more attractive solution is possible: according to Pat.Cert.N 441195, wings can be fastened to the bow of the vessel. This creates hydrodynamic lifting power from the moment it comes into contact with the surface of the water. By increasing the launch speed the lifting power of the wings increases, thereby preventing any mishaps.
The synthesis of nonclassical geometric forms.
One can indicate three main mechanisms through which "duplication" of the forms occurs:
à) geomform1 ± geomform2, when one of the forms can be a cavity.
Fig. 2.1 shows a "wedge + wedge" structure, which in contrast to an ordinary wedge allows displacement in three directions (Pat. Cert.N. 312974).
Fig. 2.1 Fig. 2.2 
Question: how do you control the hyperbolic or parabolic profile? You have to make a corresponding template... Run to the workshop, give an order to the metal worker... According to Pat.Cert.N.491462 this can be much simpler: make a cone and cut a mould from this. If the surface of the cut is parallel to the model you have a parabol, if it is perpendicular to the base  then you have a hyperbol (Fig. 2.2); 
b) geomform + revolving and/or forward movement
Fig. 2.3 
It seems that if circles or ellipses are placed on the axis at an incline and mirrored in relation to each other, then the zigzag trail that is formed when they wobble creates an additional support for carrying methods of transport Fig. 2.3 (Pat Cert N. 414144, 573382); 
c) geomform + physical (or chemical) effect
Examples for combining geometric forms with physical effects have been outlined sufficiently in the Directory itself, therefore we will limit our examples here to just one solution.
According to Pat.Cert.N. 1116218 the working wheel of a pump is proposed where the blade acts as the edge of the section between the wet and dry parts of the disc surface. (see Fig. 2.4).
Fig. 2.4 
Hydraulic resistance in a direction perpendicular to the section is greater than along it, therefore the liquid will disseminate along the section edge, i.e. the edge acts as a blade with infinite small dimensions. Of course the productivity of a pump of this kind is small, but the author would like to mention that this example is reminiscent of the combination of geometry and physics. 
And the final area  the problem of classifying geometrical effects.
At present this comes under the functional sign. A more precise principle for classification has not yet been found. Besides, questions of ordering classification are an eternal problem for any form of information system, in so far as the speed of "reducing" information usually lags behind the speed of its accumulation. The design of the GE Directory is far from complete, and we invite all those who wish to participate.
In conclusion let us note that the most complete mathematical properties of various geometric figures are outlined in [1] and [3]. In books [2] and [4] the interested reader will find information on the history of geometry and discover its connections with other sciences.


Bibliography:


1. Virchenko N. À., Lyashko I. I., Shvetsov K.I. The Graphics of Function: Directory, Kiev, "Naukova Dumka", 1979 ã.
2. Levitin K.E. Geometrical Rhapsody, M. "Znanie", 1984.
3. Savelov À. À., Crooked Planes. Systems, properties and applications: Directory, Ì., "Fizmatgiz", 1960.
4. D. Pideau, Geometry and Art, Ì., "Mir", 1979 ã.
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Igor L. Vikentyev
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