Mathematical Models

MATHEMATICAL MODELS. The child's box of bricks is probably mankind's earliest acquaintance or contact with mathematical conceptions. The concrete forms of the cube which go to make up the puzzle pictures of the nursery, or the more complete selection of geometrical solids comprising cubes, prisms and cylinders which make up the "Building Sets" of the same period, must, however, in some measure, appeal to the latent mathematical faculty of the child mind, just as the abacus or "counting bead frame" may have stirred some little impulse in the arithmetical complex.

At an early stage in the child's career it is instructed that the cube, prism, etc., have many special properties which may, when used in right proportions, render them amongst the most pleasing forms of architecture. The simple doubled cube for example provides an exquisite form of pedestal and cross and the inherent beauties of the rectangular prism furnish a valuable architectural theme; also by means of models, it is possible to illustrate to the practical man a conception which may be perfectly clear to a gifted or trained mathematical mind.

A knowledge of plane geometry acquired without any reference to models may be said to flatten out the mind and to engender habits of thought which make it difficult at a later stage of mathematical education to explore space of three dimensions.

Plane

early editions of Euclid had diagrams intended to be cut and folded, and a work by Cowley of 1752, New and Methodical explanations of the Elements of Geometry, included pieces of cardboard for the building up of various mod els. So talented a thinker and philosopher as Herbert Spencer appreciated the advantage of a model, for, writing to his father in January 1839, and speaking of his self-set task of regular, daily and systematic study of mathematics, he says "I have found out the grand principle of the projection of shadows and I feel almost certain of its correctness. To make myself still more satisfied I have made a model in pasteboard and I find that the real shadow is as exactly as possible what I had made it by projection." Intuitive valuable aid to the training of the young mathematical mind is to cut out, in cardboard, a number of equilateral triangles of the same size; the single triangle rep resenting the plane figure; several piled one on top of the other with corresponding edges coplanar and corresponding corners collinear illustrate the solid figure of the triangular prism, while four of them placed together with pairs of edges coinciding, one of them being used as a base, gives the first of the regular solids, viz., the tetrahedron. The student may thus, by inductive process, quickly arrive at the historical selection of the five regular solids.

Simple models of this nature may be used to demonstrate common practical problems involving important principles relat ing to regularity and maximum and minimum values ; as for example : I) Three straight lines of given total length enclose the greatest area when the lines form an equilateral triangle.

2) Four planes of a given total area enclose the greatest volume when the planes form a regular tetrahedron.

This statement could be varied by saying that for a given volume enclosed by four planes the surface is a minimum when the planes form a regular tetrahedron.

From these examples it will be remarked that regularity of shape is clearly connected with economy of bulk or volume, and where such regular forms occur in nature as in, say, crystal forma tions, we may naturally look for some explanation of maximum and minimum properties.

An important application of this style of model can be made by drawing out in the first place a regular hexagon of say, one inch edge. Set out upon each edge a further series of similar hexagons. Cut out with a sharp knife the first or inner hexagon and round the i8 lines of the outer edge of the figure, i.e., the boundary lines. Next cut along and through one only of the radial lines; then cut halfway through, and fold back or crease the remaining radials common to each hexagon. The paper may now be folded and provides a medium for the illustration of some interesting problems. First fold over one hexagon upon another when the "space" becomes pentagonal. Folding two we get the square; three, the triangle; the four fold giving a mathematical "solid of no depth." If a number of such developed surfaces be cut out of different colours and made up permanently by gumming the folds, practically the whole series of semi-regular polyhedra may be worked up in effective manner. Of particular interest in its physical application is the "two-fold," i.e., that giving a square and four hexagonal faces. Two of these units suitably connected at the joints by adhesive paper give the solid decatetrahedron of the Catalan Collection by Delagrave (1877); P1. I. fig. i shows a polyhedron of 14 faces (6 square, 8 hexagon) which may be looked upon as a transition form between the cube and octahedron and which ten years or so later (1889) Lord Kelvin recognised as a shape providing minimum partitional area for cells of given volume, naming it the tetrakaidecahedron. (See SOLIDS : Geo metric.) Pl. I. fig. 2 shows a somewhat similar construction of the "development" for the dodecahedron.