If a hexagon be circumscribed about a conic, then the diagonals forming opposite centres meet in a point.
If a hexagon be inscribed in a conic, then the intersections of opposite sides lie in a line.
This theorem is obviously Pascal's well-known theorem respecting a hexagon inscribed in a conic, for the special case when the conic has degenerated into the two lines l and l'.
If two pairs of adjacent vertices coincide, the hexagon becomes a quadrilateral, with tangents at two vertices.
And always if two consecutive vertices of the hexagon approach nearer and nearer, then the side joining them will ultimately become a tangent.
If a be inscribed in a curve of the hexagon be circumscribed about second order, then the a curve of the second class, then intersectionsof opposite sides the lines joining opposite vertices are three points in a line.
The result is that the side of the regular hexagon inscribed in a circle is equal to the radius of the circle.
It may be noticed that the least prominent angle of the hexagon is opposite the small drop which is farthest away from it, whilst one of the smaller drops has already begun to be swallowed up by the large one.
When the gelatine has set, we put the plate over a diagram, a hexagon for instance, and place a drop of ferrous sulphate solution at each of the six angles.
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 3^3 must mean something in Geometry; what does it mean?
I was rapt in thought, pondering in my mind some words that had casually issued from the mouth of my youngest Grandson, a most promising young Hexagon of unusual brilliancy and perfect angularity.
The body of the cone in its lower part is an hexagon or octagon, broad at first, then gradually and regularly decreasing to two-thirds of its height.
Small dzedis are often disposed on the lower parts of the hexagon or octagon.
On the South side, next the River Seiont, are three Hexagon and three Octagon Towers, and others on the North.
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.
Of the three figures, the hexagon is the most proper for convenience and strength.
It would seem incontestable, therefore, that the hexagonis not merely the result of mechanical necessities, but that it has its true place in the plans, the experience, the intellect and will of the bee.
For a hexagon nut this angle may be shown to be one of 15 deg.
The obtuse angle of this file is five-canted, while the regular cant is hexagon or six-canted, and it is found to be too obtuse for the purposes required of the saw file.
As B is screwed up with a wrench fitting its hexagon exterior, the work can be held against any amount of cut that the lathe will drive.
The nut to hold the wheel is a round one, which is easier to balance than a hexagon nut.
Furthermore a tap bolt may have a hexagon head, which is usually left unfinished unless ordered to be finished, as is also the case with set screws.
The most common form ofhexagon nut is shown in Fig.
A hexagon measures 2 inches across the flats, what is its diameter measured across the corners?
Each socket takes two sizes of square and one of hexagon heads or nuts.
The proof of the construction is shown in the figure, the hexagon and other polygons being marked for clearness of illustration.
The depth or thickness of both the hexagon and square nuts when left rough or unfinished is, according to the above standard, equal to the diameter of the bolt.
As an example of the allowance to be made for this purpose, it may be stated that in brasses of 6 inches bore and 9 inches long, the hexagon of the brass pattern at D, E, Fig.
If the bolt is to have other than a square head it is still called a machine bolt, but the shape of the head or nut is specially designated as "hexagon head machine bolt," this naturally implying that a hexagon nut also is required.
A machine-finished bolt usually has a machine-finished and hexagon nut.
The hexagon may further degenerate into a triangle, giving the theorem (Fig.
If two pairs of lines in Brianchon’s hexagon coalesce, we have a theorem concerning a quadrilateral circumscribed about a conic.
If two points of Pascal’s hexagon approach coincidence, then the line joining them approaches as a limiting position the tangent line at that point.
If the six points be called the vertices of a hexagon inscribed in the curve, then the sides 12 and 45 may be appropriately called a pair of opposite sides.
Pascal’s theorem, then, may be stated as follows: The three pairs of opposite sides of a hexagon inscribed in a point-row of the second order meet in three points on a line.
Finally, three of the vertices of the hexagon may coalesce, giving a triangle inscribed in a conic.
Snowflakes occur in a great variety of forms, which usually present the outline of either a regular hexagon or a six-pointed star.
The hexagon and octagon can be combined so as to make pretty designs which may be used for perforating and embroidery.
It's the name of our girls' club--the Hexagon Club.
As January passed and February came, a new subject came uppermost in the thoughts of the Hexagon Club.
Harold was unusually quiet, and Genevieve was wondering just how and when peace and happiness were to reign once more in the Hexagon Club.
She gave a full account, too, of the midnight session of the Hexagon Club in Cordelia's berth.
We shall write a day apiece--we six Happy Hexagons of the Hexagon Club.
The girls began the "Chronicles of the Hexagon Club" the next morning.
Just how long Genevieve would have practised is doubtful, perhaps, had there not sounded an insistently repeated whistle of theHexagon Club song from the garden.
As Tilly told of the affair in the Chronicles of the Hexagon Club, it was like this: "We saw a broncho busted this afternoon.
In the Chronicles of the Hexagon Club a fortnight later, it was Elsie Martin who wrote the account of Miss Sally's wedding.
A form based on Kekule's formula consists in taking three pairs of tetrahedra, each pair having a side in common, and joining them up along the sides of a regular hexagon by means of their apices.
I); a plane projection gives a hexagon with diagonals (II).
Oil pump is secured to engine gear case cover by onehexagon head bolt and three nuts.
Here it will be seen that the equilateral triangle and the regular hexagon are the basal figures, and a few of the properties of these polygons might be derived from the study of such a design.
For example, the equilateral triangle and its relation to the regular hexagon is shown in the picture of an ancient Roman mosaic floor on page 274.
We may start with a square, for example, but this is not so good as the hexagon because its side is incommensurable with the radius, and its perimeter is not as near the circumference.
The trefoil is closely connected with hexagonal designs, since the regular hexagon is formed from the inscribed equilateral triangle by doubling the number of sides.
That the side of a regular inscribed hexagon equals the radius must have been recognized very early.
The variations in the use of the triangle and hexagonhave been the object of study of many designers of Gothic windows, and some examples of these forms are shown on page 285.
Making six-sided and eight-sided punches and scratchawls out of hexagon and octagon tool steel is interesting work.
Have an assortment, in different sizes, of both the square and the hexagon nuts.
It was of a hexagon form, five stories high, defended by bastions and a moat, and surrounded with raised walks or terraces.
An idea of the minuteness of the work can be formed from the fact that a side of a hexagon would be overcast with some nine or ten buttonhole stitches.
This ground consisted of buttonhole-stitched skeleton hexagons within each of which was worked a small hexagon of toile connected with the outer surrounding hexagon by means of six little ties or brides (Pl.
Regular hexagon, a hexagon in which the angles are all equal, and the sides are also all equal.
Regular hexahedron, a hexagonhaving six equal squares for its sides; a cube.
The above list will hopefully give you a few useful examples demonstrating the appropriate usage of "hexagon" in a variety of sentences. We hope that you will now be able to make sentences using this word.
