Geometric pattern

“ Undoubtedly, too, many enter on this phase of study without adequate preparation, having little appreciation of the underlying geometric construction, not only in design but in all form.
This, on investigation, can always be found, and either the general mass or sub-divisions identified with some simple geometric shape enclosing the more intimate details. Correct approximation and placing are essential to the accuracy and success of the drawing. Too often through lack of training the tendency is to draw detail right away, without the necessary preliminary of determining the mass shapes. ”

“ For, evidently, we cannot conceive a geometric cube without its eight vertices, nor can we pretend that a figure requiring eight distinct points as the terms of its dimensions is free from composition. Now, if an empty geometric volume cannot be simple, what shall we say of a volume full of matter? Wherever there is real extension, there are real dimensions, of which the beginning, and the end, and all the intermediate terms are really distinct from one another. Hence in a material extension there should be as many distinct material terms as there are geometric points within its limits. ”

“ Without a geometric plan of some sort, however, we may easily get into difficulties with awkward leading lines, gaps, or masses, that tumble down, and are only perceived when the paper is printed and hung. ”

“ These simple geometric plans are the most generally useful as plans of designs intended for repetition and extension over space, and they are always safe and sound systems to build upon, since a geometric plan is certain to join comfortably if our measurements are right. ”

“ By joining V ́ A ́ and drawing T ́ S ́ parallel to V ́ F ́, then T ́ S ́ is obtained which will be the length of half the side of the square for drawing the inner spiral. The method for obtaining the inner spiral is the same as for the outer.
Fig. 37 There is no geometric means of drawing a perfect catenary curve; at best we can only obtain{196} it by an approximation in geometry. The curve is formed by suspending a chain from two points and pricking points along the curve of the chain. These
Fig. ”

“ It should always be remembered that mental discipline, whatever the phrase may mean, can as readily be obtained from a genuine application of a theorem as from a mere geometric puzzle. On the other hand, it is evident that not more than 25 per cent of propositions have any genuine applications outside of geometry, and that if we are to attempt any applications at all, these must be sought mainly in the field of pure geometry. ”

“ If the angle Red were smaller, and the angle Green larger, would the adjacent angles still be equal to two right angles?
Then,—
Any two adjacent angles are equal to two right angles.
If we draw the straight line c d where the edges of the square corners come together, what kind of angles will a c d, d c b, be? ”

“ For, directly we endeavour to construct a decorative design—that is, a design intended to adorn or to express an object or surface—we find that we must build it upon some sort of a plan, or geometric controlling network or scaffolding, so as to give it unity, rhythm, and coherence—especially so in the case of repeating designs. Even in an isolated panel or picture the necessity of this linear basis will be felt, since one cannot draw a line or define a form without demanding an answer—that is, a corresponding, re-echoing line or mass. ”

“ We do not teach all the names relative to the geometric figures, giving only those of the most familiar forms, such as square, circle, rectangle, triangle, oval. We now call attention to the fact that there are rectangles which are narrow and long, and others which are broad and short, while the squares are equal on all sides and can be only big and little. These things are most easily shown with the insets, for, though we turn the square about, it still enters its frame, while the rectangle, if placed across the opening, will not enter. ”

“ I suppose 33 must mean something in Geometry; what does it mean?” “Nothing at all,” replied I, “not at least in Geometry; for Geometry has only Two Dimensions.” And then I began to shew the boy how a Point by moving through a length of three inches makes a Line of three inches, which may be represented by 3; and how a Line of three inches, moving parallel to itself through a length of three inches, makes a Square of three inches every way, which may be represented by 32. ”

“ Geometrical lines and figures are properly nothing more than numbers, or the fundamental arithmetical notions fixed in space and invested with a corporeal shape, and thereby become visible. It is, however, not unusual to regard the first principles of geometry—such as the point, the right line, the square, and the triangle, out of which all else is compounded—as independent of experience, and existing absolutely in and by themselves. ”

“ THE GEOMETRICAL DIVISION OF DESIGN is when the subject is designed entirely with triangular, rectangular, square, oval, elliptical, or circular shapes, or where the outlines follow straight lines which may go at right angles to each other or in oblique directions. ”

“ In plane geometry, reckoning the line as a curve of the first order, we have only the point and the curve. In solid geometry, reckoning a line as a curve of the first order, and the plane as a surface of the first order, we have the point, the curve and the surface; but the increase of complexity is far greater than would hence at first sight appear. ”

“ An artisan represents a solid by drawing its projection upon two planes at right angles to each other, and a map maker (cartographer) represents the surface of the earth by projecting it upon a plane. A photograph of the class is merely the projection of the class upon a photographic plate (plane) , and when we draw a figure in solid geometry, we merely project the solid upon the plane of the paper.
There are other projections than those formed by lines that are perpendicular to the plane. The lines may be oblique to the plane, and this is the case with most projections. ”

“ In modern analytical geometry we find methods in use which are exceedingly ingenious, and give forms curiously like those of multiple algebra, but which, at least if logically carried out very far, are excessively artificial, and that for the expression of the simplest things. The simplest conceptions of the geometry of three dimensions are points and planes, and the simplest relation between these is that a point lies in a plane. ”

“ Teachers will naturally see that just as we may measure the plane angle by taking the ratio of an arc to the whole circle, and of a dihedral angle by taking the ratio of that part of the cylindric surface that is cut out by the planes to the whole surface, so we may measure a polyhedral angle by taking the ratio of the spherical polygon to the whole spherical surface. ”

“ Any science is called a “geometry” if it investigates the theory of the classification of a set of entities (the points) into classes (the straight lines) , such that (1) there is one and only one class which contains any given pair of the entities, and (2) every such class contains more than two members. In the two geometries, important from their relevance to existent space, axioms which secure an order of the points on any line also occur. These geometries will be called “Projective Geometry” and “Descriptive Geometry.” ”

“ Geometrical drawing is the art of representing, to the eye, the problems “worked out” by geometricians, and the importance of a knowledge of geometrical drawing is paramount. The student will find that the figures delineated and explained in the next few pages constantly occur in mechanical drawing. Says Walter Smith, State Director of Art Education in Massachusetts, “I have never known a case where a student did not progress more satisfactorily in his studies after a course of practical geometry.”
The elementary conceptions of geometry are few:
1.—A point. ”

“ We thus see that the geometry of the ancients will always have, by its nature, a primary part, absolutely necessary and more or less extensive, in the complete system of geometrical knowledge. It forms a rigorously indispensable introduction to general geometry. ”

“ Expression of Equations by Surfaces. Considering next this fundamental conception in the inverse point of view, we see in the same manner that every equation of three variables may, in general, be represented geometrically by a determinate surface, primitively defined by the very characteristic property, that the co-ordinates of all its points always retain the mutual relation enunciated in this equation. This geometrical locus will evidently change, for the same equation, according to the system of co-ordinates which may serve for the construction of this representation. ”

“ First make a geometrical plan A at the angle required. Then draw its perspective K. Divide line 4b into four equal parts, and continue these measurements all along the base: from each division draw lines to V, and draw the hexagon K. Having this one to start with we produce its sides right and left, but first to the left to find point G, the vanishing point of the 182 diagonals. Those to the right, if produced far enough, would meet at a distant vanishing point not in the picture. But the student should study this figure for himself, and refer back to Figs. ”

“ The debt which analytic geometry owes to synthetic geometry. The reaction of pure geometry on [pg 117] analytic geometry is clearly seen in the development of the notion of the class of a curve, which is the number of tangents that may be drawn from a point in a plane to a given curve lying in that plane. If a point moves along a conic, it is easy to show—and the student is recommended to furnish the proof—that the polar line with respect to a conic remains tangent to another conic. ”

“ Definitions.—The axioms of Euclidean Geometry are obtained from inspection of existent space and of solids in existent space,—hence from experience. The same source gives us the notions of the geometrical entities to which the axioms relate, viz. solids, surfaces, lines or curves, and points. A solid is directly given by experience; we have only to abstract all material from it in order to gain the notion of a geometrical solid. ”

“ It is a matter of common knowledge that the mathematical geometrician defines the position of any point in space with reference to three planes, which in the simplest case are all mutually at right angles to each other like the faces of a cube, and which intersect in three rectangular axes a, b, c, the third c being the vertical axis, b the lateral one, and a the front-and-back axis. ”

“ Thus the locus of a point so moving in a plane as constantly to be at a given distance from a fixed point in the plane is a circle. In analytic geometry we usually speak of the locus of a point, thinking of the point as being anywhere on the locus. Some teachers of elementary geometry, however, prefer to speak of the locus of points, or the locus of all points, thus tending to make the language of elementary geometry differ from that of analytic geometry. ”

“ THE type of geometry we obtain is dependent, as we know, on our definition of congruence. If we define as congruent displacements the displacements of those bodies which in ordinary life we consider rigid and undeformable, we obtain Euclidean geometry. If, on the other hand, we define as congruent the displacements of those bodies which in ordinary life we should regard as of changeable shape, we generally obtain some non-Euclidean type of geometry. ”

“ From these authors we learn that the property of the right-angled triangle had been found out, the principles of geometrical analysis laid down, the restriction of constructions in plane geometry to the straight line and the circle agreed upon, the doctrine of proportion, for both commensurables and incommensurables, as well as loci, plane and solid, and some of the properties of the conic sections investigated, the five regular solids (often called the Platonic bodies) and the relation between the volume of a cone or pyramid and that of its circumscribed cylinder or prism discovered. ”

“ They need to be defined for young people, but most of them are not basal, that is, they are not used in the proofs of propositions. Some of these terms, such as magnitudes, curve line, broken line, curvilinear figure, bisector, adjacent angles, reflex angles, oblique angles and lines, and vertical angles, need merely a word of explanation so that they may be used intelligently. ”

“ The introduction to solid geometry should be made slowly. The pupil has been accustomed to seeing only plane figures, and therefore the drawing of a solid figure in the flat is confusing. The best way for the teacher to anticipate this difficulty is to have a few pieces of cardboard, a few knitting needles filed to sharp points, a pine board about a foot square, and some small corks. With the cardboard he can illustrate planes, whether alone, intersecting obliquely or at right angles, or parallel, and he can easily illustrate the figures given in the textbook in use. ”

“ At a first thought it might appear as if these eight movements had nothing in common with trigonometry, but a very fundamental connection may be seen as follows: If the vertex of the angle A is the center of a square and the initial line of A coincides with a line of symmetry of the square, the operations of taking the complement and the supplement of A correspond to movements transforming the square into itself. ”

“ Now, the most elementary geometry teaches us that every triangle having these angles will have the lengths of its sides in the same invariable proportions one to another whatever may be the size of the triangle itself, and it has been found convenient to calculate these proportions once for all, not merely for angle 20°, but for every angle up to 90°. ”