Polygon

“ You may do this: you may put the word hexagon or dodecagon, or any other word describing a polygon in the place of Circle in your proof, and the proof would be just as good as before. Does not this satisfy you that you cannot have proved a property of that special figure—a circle?
"Or you may do this: calculate the side of a polygon of 24 sides inscribed in a circle. I think you are a Mathematician enough to do this. You will find that if the radius of the circle be one, the side of this polygon is .264 etc. ”

“ The pentagon gives rise to one star polygon, the hexagon gives none, the heptagon two, the octagon one, and the nonagon two. In general, the number of star polygons which can be drawn with the vertices of an n-point regular polygon is the number of numbers which are not factors of n and are less than én. ”

“ Suppose we have a number of equal circular counters, then the number of counters which can be placed on a regular polygon so that the tangents to the outer rows form the regular polygon and all the internal counters are in contact with its neighbours, is a “polygonal number ” of the order of the polygon. If the polygon be a triangle then it is readily seen that the numbers are 3, 6, IO, 1 5 and generally en (n + 1) ”

“ The construction of any regular polygon on a given side may be readily performed with a protractor or scale of chords, for it is only necessary to lay off from the extremities of the given side lines equal in length to the given base, at angles equal to the interior angle of the polygon, and repeating the process at each extremity so obtained, the angle being always taken on the same side ”

“ Fourteen propositions are preserved by Pappus and Theon. Here are a few of them: Of isoperimetrical, regular polygons, the one having the largest number of angles has the greatest area; the circle has a greater area than any regular polygon of equal periphery; of all isoperimetrical polygons of n sides, the regular is the greatest; of all solids having surfaces equal in area, the sphere has the greatest volume. ”

“ VI. 19.
therefore the polygon ABCDE is to the polygon FGHKL in the duplicate ratio of the side AB to the homologous side FG.
Wherefore, similar polygons &c. q.e.d.
Corollary 1. In like manner it may be shewn that similar four- sided figures, or figures of any number of sides, are to one another in the duplicate ratio of their homologous sides; and it has already been shewn for triangles; therefore universally, similar rectilineal figures are to one another in the duplicate ratio of their homologous sides. ”

“ If the circumference of a circle be divided into n equal parts, and tangents be drawn at the points of intersection, the n tangents will form a polygon if the radius of the circle is small enough, but if the radius is large enough they will never meet. ”

“ In the figure, these are the upper and lower sides. The height is a perpendicular from base to base. As this figure is easily made, the length of the bases and the height may be given: thus, "Make a trapezoid whose height shall be one inch,and whose bases shall be an inch and a half and two inches." When no two of the sides are parallel the figure is called a Trapezium.
4.⁠Make a six sided polygon of unequal sides.⁠ (fig.4.)
The word polygon means many-angled. To make a polygon, the best method is, first to place dots at the angles and then draw right lines from dot to dot. ”

“ Although popularly every one called a Circle is deemed a Circle, yet among the better educated Classes it is known that no Circle is really a Circle, but only a Polygon with a very large number of very small sides. In proportion to the number of the sides the Polygon approximates to a Circle; and, when the number is very great, say for example three or four hundred, it is extremely difficult for the most delicate touch to feel any polygonal angles. ”

“ An exterior angle of a triangle is one that is formed by any side and the continuation of another side.
Hence a triangle has six exterior angles; and also each exterior angle is the supplement of the adjacent interior angle.
The Polygon.
xxvi. A rectilineal figure bounded by more than three right lines is usually called a polygon.
xxvii. A polygon is said to be convex when it has no re-entrant angle.
xxviii. A polygon of four sides is called a quadrilateral.
xxix. A quadrilateral whose four sides are equal is called a lozenge. ”

“ Just as we may produce the sides of a regular polygon and form a regular cross polygon or stellar polygon, so we may have stellar polyhedrons. Kepler, the great astronomer, constructed some of these solids in 1619, and Poinsot, a French mathematician, carried the constructions so far in 1801 that several of these stellar polyhedrons are known as Poinsot solids. ”

“ When more than three surfaces of discontinuity in a fluid system meet in equilibrium along a line, with respect to the surfaces and masses immediately adjacent to any point of this line, we may form a polygon of which the angular points shall correspond in order to the different masses separated by the surfaces of discontinuity, and the sides to these surfaces, each side being perpendicular to the corresponding surface, and equal to its tension. ”

“ Graphical Statics.—A graphical method of reducing a plane system of forces was introduced by C. Culmann (1864) . It involves the construction of two figures, a force-diagram and a funicular polygon. The force-diagram is constructed by placing end to end a series of vectors representing the given forces in magnitude and direction, and joining the vertices of the polygon thus formed to an arbitrary pole O. The funicular or link polygon has its vertices on the lines of action of the given forces, and its sides respectively parallel to the lines drawn from O in the force-diagram ”

“ A single known force in a polygon determines the direction of all the others, as these must all correspond with arrows pointing the same way round the polygon. Let the arrows be placed on the frame round each joint, and so as to indicate the direction of each force on that joint; then when two arrows point to one another on the same piece, that piece is a tie; when they point from one another the piece is a strut. ”

“ When the primitive surface has a triple tangent plane or one of higher order, the triangle in the tangent plane formed by joining the points of contact, or the smallest polygon without re-entrant angles which includes all the points of contact, will belong to the derived surface, and will represent masses consisting in general of three or more phases. ”

“ We first rabatt the plane α (fig. 46) as before so that P1 comes to P, hence OP1 to OP. Let the given polygon in α be the figure ABCDE. We project, not the vertices, but the sides. To project the line AB, we produce it to cut α′ in F and OP in G, and draw GG1 perpendicular to α′; then G1 corresponds to G, therefore FG1 to FG. In the same manner we might project all the other sides, at least 710 those which cut OF and OP in convenient points. It will be best, however, first to produce all the sides to cut OP and α′ and then to draw all the projecting rays through A, B, C ... ”

“ They were also familiar with the construction of a polygon equal in area to a given polygon and similar to another given polygon. This problem depends upon several important and somewhat advanced theorems, and testifies to the fact that the Pythagoreans made no mean progress in geometry.
Of the theorems generally ascribed to the Italian school, some cannot be attributed to Pythagoras himself, nor to his earliest successors. The progress from empirical to reasoned solutions must, of necessity, have been slow. ”

“ We should see that the first germ of this idea is found in the procedure constantly employed by the Greek geometers, under the name of the Method of Exhaustions, as a means of passing from the properties of straight lines to those of curves, and consisting essentially in substituting for the curve the auxiliary consideration of an inscribed or circumscribed polygon, by means of which they rose to the curve itself, taking in a suitable manner the limits of the primitive ratios. ”

“ If perpendiculars be let fall on the sides of a polygon from any point, dividing each side into two segments, the sum of the squares on one set of alternate segments is equal to the sum of the squares on the remaining set. ”

“ If from any point in the circumference of the circle, circumscribed about a regular polygon of n sides, lines be drawn to its angular points, the sum of their squares is equal to 2n times the square of the radius. ”

“ Blind, Deaf, and Dumb, Patient Effort Wins for Her Culture and Rare Womanhood. “I AM trying to prove that the sum of the areas of two similar polygons, constructed on the two legs of a right triangle, is equal to the area of a similar polygon constructed on the hypotenuse. ”

“ In general, a regular polygon of n sides is left unchanged as a whole by just movements of its plane, viz., movements around its center and rotations through 180° around its lines of symmetry, in addition to the identity. The first movements together with the identity clearly form a group by themselves. Such a group within a group is known as a subgroup. This category of groups of operations is known as the system of dihedral rotation groups or the system of the regular polygon groups. ”

“ Each face is a regular polygon, hence the angles of the faces at any vertex must be angles in equal regular polygons, must be together less than four right angles (XI. 21) , and must be three or more in number. Each angle in a regular triangle equals two-thirds of one right angle. Hence it is possible to form a solid angle with three, four or five regular triangles or faces. These give the solid angles of the tetrahedron, the octahedron and the icosahedron. The angle in a square (the regular quadrilateral) equals one right angle. ”

“ Moreover, since, on account of the continual increase of density in descending through the atmosphere, the light would be continually turned out of its course more and more, it would therefore move, not in the polygon represented in the figure, but in a corresponding curve line, whose curvature is rapidly increased near the surface of the earth. ”