The term model has a variety of meanings and is used for many different purposes. We use modeling clay to form small replica of physical objects; children – and sometimes also adults – play with a model railway or model aeroplane; architects build (scale) model houses or (real-size) model apartments in order to show them to new clients; some people work as photo models , others take someone for a model, many would like to have a model friend. Models can be much smaller than the original (an orrery, a mechanical model of the solar system) or much bigger (Rutherford’s model of the atom). A diorama is a three-dimensional representation showing lifelike models of people, animals, or plants. A cutaway model shows its prototype as it would appear if part of the exterior had been sliced away in order to show the inner structure. A sectional model is made in such a way that it can be taken apart in layers or sections, each layer or section revealing new features (e.g. the human body). Working models have moving parts, like their prototypes. Such models are very useful for teaching human anatomy or mechanics.
The ancient Egyptians put small ships into the graves of the deceased to enable them to cross the Nile. In 1679, Colbert ordered the superintendents of all royal naval yards to build an exact model of each ship. The purpose was to have a set of models that would serve as precise standards for any ships built in the future. (Today, the models are exposed in the Musée de la Marine in Paris.) Until recently, a new aeroplane was first planned on paper. The next step was to build a small model that was placed in a wind-tunnel to test its aerodynamics. Nowadays, aeroplanes are designed on computers, and sophisticated simulation models are used to test different aspects of them.
The various meanings of model in the previous examples all have a common feature: a model is an imitation, a pattern, a type, a template, or an idealized object which represents the real physical or virtual object of interest. In the ship example, both the original and the model are physical objects. The purpose is always to have a copy of some original object, because it is impossible, or too costly, to work with the original itself. Of course, the copy is not perfect in the sense that it reproduces all aspects of the object. Only some of the particularities are duplicated. So an orrery is useless for the study of life on Mars. Sectional models of the human body cannot be used to calculate the body’s heat production. Colbert’s ship models were built so accurately that they have supplied invaluable data for historical research, this was not their original purpose. Apparently, the use made of these ship models has changed over time.
Besides these physical models, there are also mental ones. These are intuitive models which exist only in our minds. They are usually fuzzy, imprecise, and often difficult to communicate.
Other models are in the form of drawings or sketches,
abstracting away many details. Architects do not generally construct scaled-down models. Instead, they draw up exact plans and different two-dimensional projections of the future house. Geographers use topographical models and accurate maps to chart terrain.
Cave drawings are sketches of real animals; they inspired Picasso to draw his famous torro (see Figure 1). Picasso’s idea was quite ingenious: how to draw a bull with the least number of lines? This exemplifies an essential characteristic of the notion of a model which is also valid for mathematical models, and for those in all other sciences: simplicity, conciseness and aesthetics. (More about these concepts see Chapter 2 of [10, 7].)
Certainly, what simplicity means depends on personal taste, standpoint, background, and mental habits. But the main idea behind simplicity and conciseness is clear enough: How to represent the object in such a way that we can “see” or derive its solution immediately? Our mind has a particular structure, evolved to recognize certain patterns. If a problem is represented in a way corresponding to such patterns, we can quite often immediately recognize its solution.
A illustrative example is the intersection problem (from preface written by Ian Stewart in [2]). Suppose you have to solve the following problem (Figure 2): Connect the small square A with F, B with D, and C with E inside the rectangle by lines in such a way that they do not intersect.
At first glance, it seems that the problem is unsolvable, because if we connect A with F by a straight line, it partitions the rectangle into two parts, where B and C are in one part and D and E in the other. There is no way then, to connect B with D or C with E, without intersecting the line A-F. Well, we were not asked to connect A and F with a straight line. So let us connect A with F by a curved line passing between say, B and C. But, then again we have the same problem: The rectangle is partitioned into two parts, B being isolated. The same problem arises when the line from A to F is drawn between D and E. But the problem can be presented a little bit differently: Imagine that the rectangle surface is spanned by a thin rubber. So pick D and C (by two fingers) and rotate the rubber with the two fingers by 180 degrees around the center of the rectangle in a continuous way such that the places of C and D are interchanged (Figure 3).
Now the problem is easily solvable. Connect the squares as prescribed. After this, return the rubber to the initial state again (Figure 4).
(A completely different approach to solve this problem can be found in my My Books – which is a good (advanced) exercise in mathematical modeling, see also the implemented model at intersec.)
Let’s summarize:
The model is the result of a mapping (or a transformation) of the real or virtual object to be considered.
the model is an abstraction, i.e. only some aspects of the objects of interest are mapped, all others are ignored (abstrahere = leaving out).
the aspects that enter the model are of special interest for an intelligent actor (the model-builder and the model-user): The model is used for a certain purpose.
a model has some simple, concise (and aesthetic) structure that captures the focused aspects of the object.
Normally, the model is a simplification, but it must be rich enough to capture the problem at hand. Hence, it is something relative, there is no single absolute representation of the problem at hand. It is not only a model of something, but also for someone and for some purpose.
A good example is modern physics: Neglecting the Planck constant (that is: ℏ = 0), the gravitational constant (G = 0) and the speed of light (c = ∞), physics reduces to classical mechanics. Considering the gravitational constant (G = 6.6741 × 10^{11}[m^{3}∕(kg ⋅ s)]) in addition, we must use Newtonian mechanics; Considering the speed of light (c = 2.9979 × 10^{8}[m∕s] instead, we must apply special relativity, and so on (see [3]). It is not true, that the models of classical mechanics are learnt, because it is necessary for the “more advanced” theory as Quantum Field Theory. It is useful and applicable for its own, namely when the speed is small, when quantum effects are negligible, and when gravitation is not important – that is in our every day life. Classical mechanics has plenty of applications on its own. Each model in physics has its own eligibility and purpose, it will do to know its limitation.
Hence, we saw that a problem can be approached with different models. The contrary is also true: A (formal) model might have various interpretations. The intersection problem was built of a rectangle with 6 locations in it (A, B, C, D, E, and F) and connecting lines. So it is basically a geometric interpretation. Another interpretation is as following: Replace “rectangle” with “square electric board”, “locations” with “endpoint of wires”, and “lines” with “wires”. Now we have the same model for a different problem: connect the endpoints of the wires on the 2-dimensional board in such a way that they do not cross. Another interpretation is: a oval closed room (with only one entry-point) has corresponding (curved) walls (from a point A to F, B to D and C to E, see the red lines in Figure 3) such that the whole room can be visited. Still another interpretation is: in an area we have six airports arranged in a similar way as the 6 locations in our original problem. Planify flight paths in such a way that the aeroplanes never cross each other. (We suppose that only the corresponding airports are linked, of course.)
An different question is whether the intersection problem is the right model for the flight-colliding problem. This could probably be solved in a more satisfactory manner by introducing different flying altitudes for the different connections. The wire-connection problem could probably be solved by insulating the wires and letting them intersect.