Regular polygon

“ 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 process would then stop.
I will, however, suppose that the original boundary is everywhere rectilinear. It is clear then that, after every cutting, the boundary is still rectilinear. If the creases be at right angles to one another, the ultimate figure may be an irregular polygon, having its four quarters alike, such as may be inscribed in an oval; or it may have its sides so many and so small, that the ultimate appearance shall be that of an oval. But if the creases be not at right angles, the ultimate figure is a perfectly regular polygon, such as can be inscribed in a circle ”

“ 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. ”

“ 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. ”

“ Fig. 17 Within a given circle to describe any Regular Polygon—say a Pentagon.
Draw the diameter A F and divide it into the same number of parts as the required polygon is to have sides—in this case it will be five parts. To divide the diameter into the number of equal parts, draw a line A X any angle to A F. Set off any convenient measurement five times on this line. Join point 5 to F, and draw the lines 4, 4 ́, 3, 3 ́, &c., parallel to 5 F to meet the diameter. With A and F as centre and A F as radius describe arcs intersecting at L. From{181}
Fig. ”

“ The Regular Dipyramidal or Quadrilonchial Ground-Form.—The ground-forms whose geometrical type is the regular double pyramid are characterised by a vertical main axis which possesses equal poles, and which is crossed at its centre by several equal transverse axes. The horizontal equatorial plane is therefore a regular polygon, and divides the body into two equal regular pyramids. The simplest and commonest form of this group is the quadratic octahedron, the ground-form of the quadratic crystallographic system; its equatorial plane is a square. ”

“ 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. ”

“ Problem. To find the numerical value of the ratio of the circumference of a circle to its diameter.
As already stated, the usual plan of the textbooks is in part the method followed by Archimedes. It is possible to start with any regular polygon of which the side can conveniently be found in terms of the radius. In particular we might begin with an inscribed [Pg 277] square instead of a regular hexagon. In this case we should have
and so on. ”

“ Another inset shows the equivalence of the decagon and a rectangle which has one side equal to the perimeter of the decagon and the other equal to half of the altitude of each triangle composing the decagon. Small triangles divided horizontally in half can be fitted into this figure, with one of the upper triangles divided in half lengthwise.
Thus we demonstrate that the surface of a regular polygon may be found by multiplying the perimeter by half the hypotenuse.
SOME THEOREMS BASED ON EQUIVALENT FIGURES
A. All triangles having the same base and altitude are equal. ”

“ 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. ”

“ On the suppositions, we must be always cutting down, at all parts of the boundary; but it has been shown that we can never come nearer than by four inches to the intersection of the axes.
But it does not follow that the process will go on for ever. We may come at last to a state in which both the creases are axes of symmetry at once; and then the process stops. If the paper had at first a curvilinear boundary, properly chosen, and if the axes were placed at the proper angle, it would happen that we should arrive at a {15}regular curved polygon, having the two axes for axes of symmetry. ”

“ When is a polygon said to be inscribed in a circle ? 15.
When a tangent is drawn from a given point, how many points of a circle can it touch ?
16.
What is the proportion between the radius of a Circle and one side of a hexagon inscribed in it ?
17.
What is meant by unequal circles ? (p. 20.)
18.
What is meant by a regular Polygon ? (p. 21.)
19.
What is a regular hexagon ?
20.
What is a regular pentagon ?
21.
What is a regular octagon?
22.
How will you circumscribe a triangle with a circle ?
23.
How will you find the Centre of a circle to be in- scribed in a triangle ? ”

“ The ascent of the posterity of the Circles in the social scale is not restricted, as it is among the lower Regular classes, by the Law of Nature which limits the increase of sides to one in each generation. If it were so, the number of sides in a Circle would be a mere question of pedigree and arithmetic, and the four hundred and ninety-seventh descendant of an Equilateral Triangle would necessarily be a Polygon with five hundred sides. ”

“ A is a hexagon, and if one of its sides be measured, it will [138] be found that it measures the same as along line 1 from O B to the diagonal line 45 degrees, which distance is shown by a thickened line.
At 1-1/2 is shown a seven-sided figure, whose diameter is 3 inches, and radius 1-1/2 inches, and if from the point at 1-1/2 (along the thickened horizontal line) , to the diagonal marked 49 degrees, be measured, it will be found exactly equal to the length of a side on the polygon. ”

“ 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. ”

“ The inscription of a pentagon in a circle is effected by inscribing an isosceles triangle similar to that constructed in prop. IO, bisecting the angles at the base and producing the bisectors to meet the circle. Euclid then proves that these intersections and the three vertices of the triangle are the vertices of the required pentagon. The circumscription of a pentagon is effected by constructing an inscribed pentagon, and drawing tangents to the circle at the vertices. This supplies a general method for circumscribing a polygon if the inscribed be iven, and conversely. ”

“ 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 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. ”

“ 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 ... ”

“ 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. ”

“ The term "isosceles trapezoid" is used to mean a trapezoid with two opposite sides equal, but not parallel. The area of such a figure was incorrectly given by the Ahmes papyrus as 1⁄2 (b + b') s, where s is one of the equal sides. This amounts to taking s = a.
The proposition is particularly important in the surveying of an irregular field such as is found in hilly districts. It is customary to consider the field as a polygon, and to draw a meridian line, letting fall perpendiculars upon it from the vertices, thus forming triangles and trapezoids that can easily be measured. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”