Hexagon

“ A regular hexagon, or six-sided figure, is easy enough, for everybody knows that all you have to do is to describe a circle and then, taking the radius as the length of one of the sides, mark off the six points round the circumference. But a pentagon is quite another matter. So, as my puzzle has to do with the cutting up of a regular pentagon, it will perhaps be well if I first show my less experienced readers how this figure is to be correctly drawn. Describe a circle and draw the two lines H B and D G, in the diagram, through the centre at right angles. ”

“ Reid, "three possible figures of the cells which can make them all equal and similar, without any useless interstices. These are the equilateral triangle, the square, and the regular hexagon. Mathematicians know that there is not a fourth way possible in which a plane shall be cut into little spaces that shall be equal, similar, and regular, without useless spaces. Of the three figures, the hexagon is the most proper for convenience and strength. Bees, as if they knew this, make their cells regular hexagons. ”

“ The angle at the base of each cell, the apex of the pyramidal cavity, is buttressed by the ridge formed by two faces of the hexagon of another cell. The two triangles, or extensions of the hexagon faces which fill one of the convergent angles of the cavity enclosed by the three rhombs, form by their junction a plane angle on the side they touch; each of these angles, concave within the cell, supports, on its convex side, one of the sheets employed to form the hexagon of another cell; the sheet, pressing on this angle, resists the force which is tending to push it outwards ”

“ 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. ”

“ 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. ”

“ Thus we find that the most general possible homogeneous partitioning of space with plane interfaces between the cells gives us fourteen walls to each cell, of which six are three pairs of equal and parallel parallelograms, and the other eight are four pairs of equal and parallel hexagons, each hexagon being bounded by three pairs of equal and parallel straight lines. This figure, being bounded by fourteen plane faces, is called a tetrakaidekahedron. ”

“ Section 66 All the angles of a square, summed together, equal 360 degrees. If one draws a line diagonally from one corner of the square to the other corner the square has been divided into a triangle. If so, it is shown to contain 180 degrees twice. A pentagon contains 180 degrees three times, for it can be divided into three triangles. A hexagon contains 180 degrees four times. The general rule is that if one subtracts two sides from the total number of sides of any shape the result is the number of times in contains 180 degrees. ”

“ Each of the thirty-six edges may be straight or crooked or curved; the pairs of opposite edges, whether of the quadrilaterals or hexagons, need not be equal and parallel; neither the four corners of each quadrilateral nor the six corners of each hexagon need be in one plane. ”

“ Dictyastrum hexagonum, n. sp. (Pl. 43, fig. 7) .
Rhopalastrum hexagonum, Haeckel, 1880, Atlas (pl. xliii. fig. 1) . Arms nearly triangular, one and a third times as broad at the distal end as long, and three times as broad as at the base. Central disk about the same diameter. In each arm six simple broad chambers. If we connect the six corners of the truncated distal ends by straight lines, we get a regular hexagon.
Dimensions.—Radius of each arm 0.17, basal breadth 0.06, terminal breadth 0.17. ”

“ The same house may appear in one map as a hexagon, in another as an octagon or decagon, and the late Hollar group differs from its predecessors in using a completely circular form. But there is confirmation in the Paris Garden manor map of 1627, which shows the ground-plan of the Swan decagonal, and in the statement of Mrs. Thrale that the ruins of the Globe still visible in the eighteenth century were hexagonal without and round within. ”

“ Mark off the length of the radius of the given circle B D F six times on the circumference as at D E F, &c. Draw the three diameters A D, B E, and G F, and produce them a little beyond these points. Join the points G, D, E, F, &c., by straight lines to produce the hexagon within the given circle. (2) Bisect the angle K O H, the line of bisection will cut the circle at point R. Through R draw H K parallel to B C. With O as centre and O H as radius describe a circle cutting the produced diameters at K, L, M, {187}&c. ”

“ The hexagon may further degenerate into a triangle, giving the theorem (Fig. 24) The lines joining the vertices to the points of contact of the opposite sides of a triangle circumscribed about a conic all meet in a point. ”

“ 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. ”

“ Inscribe a circle in a regular hexagon. (fig. 22.) This problem is the inverse of the 36th. First draw
the hexagon, then describe the circle, touching it on all sides. The centre of the circle may be found by rais- ing perpendiculars on the middle of any two sides until they cross each other. The point where they cross, is the centre. ”

“ Now put leaves into this hexagon, Fig. 31, and you will see how charmingly the rhododendron has got out of its difficulties. The next cycle will put a leaf in at the gap at the top, and begin a new hexagon. Observe, however, this geometrical figure is only to the rhododendron what the a in Fig. 25 is to the oak, the 30 icy or dead form. To get the living normal form we must introduce our law of succession. ”

“ Hence, if the central sphere rotates, it will not necessarily drive those around it so that their motions will be antagonistic to each other, but the velocities will not arrange themselves in a systematic manner.
In four-dimensional space the figure which forms the next term of the series hexagon, tetrakaidecagon, is a thirty-sided figure. It has for its faces ten solid tetrakaidecagons and twenty hexagonal prisms. Such figures will exactly fill four-dimensional space, five of them meeting at every point. ”

“ In Fig. 66, which is an elevation of a hexagon, the surface c is in shade, and to represent this surface correctly, it must be made darker than the others. This darkening of the surface is effected by drawing the shading lines heavier or closer together, or by both of these means combined. The surface b is in light, but the rays fall upon it obliquely; the shading lines on this surface will therefore be lighter and more widely spaced than on c. The surface a is also in light, and receives the rays normally, that is, the [51] direction of the rays is normal to the surface. ”

“ The little Hexagon meditated on this awhile and then said to me: "But you have been teaching me to raise numbers to the third power; 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 show 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 ”

“ 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. ”

“ From a central point with a chain, or line, construct, and lay down with pickets six equilateral, and contiguous triangles, the bases of which will form the required hexagon.
[249]
4. Octagonal redoubt.
Construct a square (vide No. 1) , from the centre of each side of which erect perpendiculars outwards, in length proportional to the side as 13 to 60 (nearly 1 to 5) ; join the extremities, or termination of the perpendiculars, to the angles of the square, which will determine the sides of the octagon. ”

“ In the first combination, all the angles were equal; in the second, two of the angles only were equal; but in the present combination, none of them are equal. The field of view, represented in the figure by D E H L M P, is a truncated rhomb, consisting of no fewer than thirty-one images of the aperture A O B. The figure is composed of two hexagons D E F B R C, R B K L M N, every division of the hexagon consisting of two reflected images, and of two rhombs C R N P, F B K H, each of which is composed of four reflected images. ”

“ The angles of these hexagons, at each of which three hexagons meet, are of course angles of 120°. So far as the form is concerned, so long as we are concerned only with a morphological result and not with a physiological process, the result is precisely the same whatever be the force which brings the bodies together in symmetrical apposition; it is by no means necessary for us, in the first instance, even to enquire whether it be surface tension or mechanical pressure or some other physical force which is the cause, or the main cause, of the phenomenon. ”

“ Sometimes they take their square slates, and fit them with one of the angles uppermost, so that each slate looks something like the ace of diamonds in a pack of cards. Sometimes they are still more ambitious, and certainly succeed in producing a better effect, by cutting the slates in hexagons instead of squares, and fixing them as shown in the right-hand figure of the illustration. Putting aside the familiar hexagons of the honeycomb, and the apparent hexagons of an insect’s compound eye, we have in the common Tortoise an example of hexagonal plates that exactly resembles the slate roofing. ”

“ Six planes of symmetry of two kinds intersect in the hexad axis. The hexad axis is uniterminal and all the forms are open forms. The general form {hikl} consists of twelve faces at one end of the crystal, and is a dihexagonal pyramid. ”

“ If we suppose the impulsive influence to be powerful enough to bring (sensibly) the entire peripheries of all the rings into mutual contact, the rings bending (from their elasticity) for this purpose, then it would appear that the whole must form a system of hexagons, as the annexed diagram (fig. 1) may serve roughly to illustrate. For the conditions of equilibrium would not allow any thing else. ”

“ Water crystallises like quartz, in the trigonal system, its normal forms being the hexagonal prism and the rhombohedron. A slab of lake ice is generally a huge crystal plate perpendicular to the trigonal axis, or in the case of disturbed growth an interlacing mass of such crystals, all perpendicular to the optic axis, the axis of the hexagonal prism and of trigonal symmetry. ”

“ Little concave depressions will appear, regularly interspaced, and separated by convex folds. The little concavities being of equal size (unless the system be otherwise perturbed) each one will tend to be surrounded by six others; and when the process has reached its limit, the intermediate boundary-walls, or raised folds, will be found converted into a regular pattern of hexagons.
But the analogy of the mechanical wrinkling of the coat of a seed is but a rough and distant one; for we are evidently dealing with molecular rather than with mechanical forces. ”

“ Native graphite crystallises in the form of hexagons, mostly tabular; but really well-developed crystals are of extremely rare occurrence, and by far the greatest quantities of this mineral are found in the condition of dense lumps, in which only the crystalline structure, and not any decided crystals, can be discerned. The hardness of the mineral fluctuates within fairly wide limits, ranging from 0·5 to 1·0. ”

“ They show that there are two modes of closely packing equal spheres, which give rise respectively to a cubic and a hexagonal crystal structure, the latter having a specific axial ratio of the vertical to the three equal equatorial horizontal axes; and that the chemical elements which are solids and crystallise, and the structural units of which can naturally be assumed to be equal spheres, being those of the similar atoms of the same chemical element, do practically all crystallise either in the cubic system or in the hexagonal system with the specific axial ratio indicated by them. ”

“ This is the hexad axis or hexagonal axis, for which the angle of rotation is 60°, or one-sixth of 360°. There can be only one hexad axis of symmetry in any crystal (see figs. 77-80) .
Planes of Symmetry.—A regular octahedron can be divided into two equal and similar halves by a plane passing through the corners a1a3ā1ā3 and the centre O (fig. 3) . One-half is the mirror reflection of the other in this plane, which is called a plane of symmetry. Corresponding planes on either side of a plane of symmetry are inclined to it at equal angles. ”