Hexagon 3d ((INSTALL))
Fajar Roux
In geometry, a hexagon (from Greek ἕξ, hex, meaning "six", and γωνία, gonía, meaning "corner, angle") is a six-sided polygon.[1] The total of the internal angles of any simple (non-self-intersecting) hexagon is 720.
A regular hexagon is defined as a hexagon that is both equilateral and equiangular. It is bicentric, meaning that it is both cyclic (has a circumscribed circle) and tangential (has an inscribed circle).
The common length of the sides equals the radius of the circumscribed circle or circumcircle, which equals 2 3 \displaystyle \tfrac 2\sqrt 3 times the apothem (radius of the inscribed circle). All internal angles are 120 degrees. A regular hexagon has six rotational symmetries (rotational symmetry of order six) and six reflection symmetries (six lines of symmetry), making up the dihedral group D6. The longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. From this it can be seen that a triangle with a vertex at the center of the regular hexagon and sharing one side with the hexagon is equilateral, and that the regular hexagon can be partitioned into six equilateral triangles.
Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane (three hexagons meeting at every vertex), and so are useful for constructing tessellations. The cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a regular triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral.
The maximal diameter (which corresponds to the long diagonal of the hexagon), D, is twice the maximal radius or circumradius, R, which equals the side length, t. The minimal diameter or the diameter of the inscribed circle (separation of parallel sides, flat-to-flat distance, short diagonal or height when resting on a flat base), d, is twice the minimal radius or inradius, r. The maxima and minima are related by the same factor:
For any regular polygon, the area can also be expressed in terms of the apothem a and the perimeter p. For the regular hexagon these are given by a = r, and p = 6 R = 4 r 3 \displaystyle =6R=4r\sqrt 3 , so
It follows from the ratio of circumradius to inradius that the height-to-width ratio of a regular hexagon is 1:1.1547005; that is, a hexagon with a long diagonal of 1.0000000 will have a distance of 0.8660254 between parallel sides.
For an arbitrary point in the plane of a regular hexagon with circumradius R \displaystyle R , whose distances to the centroid of the regular hexagon and its six vertices are L \displaystyle L and d i \displaystyle d_i respectively, we have[3]
These symmetries express nine distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order.[4] r12 is full symmetry, and a1 is no symmetry. p6, an isogonal hexagon constructed by three mirrors can alternate long and short edges, and d6, an isotoxal hexagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular hexagon. The i4 forms are regular hexagons flattened or stretched along one symmetry direction. It can be seen as an elongated rhombus, while d2 and p2 can be seen as horizontally and vertically elongated kites. g2 hexagons, with opposite sides parallel are also called hexagonal parallelogons.
A truncated hexagon, t6, is a dodecagon, 12, alternating two types (colors) of edges. An alternated hexagon, h6, is an equilateral triangle, 3. A regular hexagon can be stellated with equilateral triangles on its edges, creating a hexagram. A regular hexagon can be dissected into six equilateral triangles by adding a center point. This pattern repeats within the regular triangular tiling.
From bees' honeycombs to the Giant's Causeway, hexagonal patterns are prevalent in nature due to their efficiency. In a hexagonal grid each line is as short as it can possibly be if a large area is to be filled with the fewest hexagons. This means that honeycombs require less wax to construct and gain much strength under compression.
Irregular hexagons with parallel opposite edges are called parallelogons and can also tile the plane by translation. In three dimensions, hexagonal prisms with parallel opposite faces are called parallelohedrons and these can tessellate 3-space by translation.
Pascal's theorem (also known as the "Hexagrammum Mysticum Theorem") states that if an arbitrary hexagon is inscribed in any conic section, and pairs of opposite sides are extended until they meet, the three intersection points will lie on a straight line, the "Pascal line" of that configuration.
The Lemoine hexagon is a cyclic hexagon (one inscribed in a circle) with vertices given by the six intersections of the edges of a triangle and the three lines that are parallel to the edges that pass through its symmedian point.
If the successive sides of a cyclic hexagon are a, b, c, d, e, f, then the three main diagonals intersect in a single point if and only if ace = bdf.[6]
If, for each side of a cyclic hexagon, the adjacent sides are extended to their intersection, forming a triangle exterior to the given side, then the segments connecting the circumcenters of opposite triangles are concurrent.[7]
A skew hexagon is a skew polygon with six vertices and edges but not existing on the same plane. The interior of such a hexagon is not generally defined. A skew zig-zag hexagon has vertices alternating between two parallel planes.
A regular skew hexagon is vertex-transitive with equal edge lengths. In three dimensions it will be a zig-zag skew hexagon and can be seen in the vertices and side edges of a triangular antiprism with the same D3d, [2+,6] symmetry, order 12.
There is no Platonic solid made of only regular hexagons, because the hexagons tessellate, not allowing the result to "fold up". The Archimedean solids with some hexagonal faces are the truncated tetrahedron, truncated octahedron, truncated icosahedron (of soccer ball and fullerene fame), truncated cuboctahedron and the truncated icosidodecahedron. These hexagons can be considered truncated triangles, with Coxeter diagrams of the form and .
I was designing a hexagon-based logo recently and came to horrid realisation: 6-sided polygon in Figma IS NOT a perfect hexagon, where all sides should be equal and angles 120 degrees. Everything is slightly off.
For JIT, you could perhaps initially use the mlir-cpu-runner (which really just calls mlir::JitRunnerMain()), but over time, I would expect this to evolve into something like mlir-hexagon-runner to deal with arch specifics.
Thanks! And I think that's a great idea. The little hexagons work up really quickly and they're perfect for crocheting on the go (in waiting rooms, on the bus, etc). You could also make something smaller like a pillow case or a wall hanging if you want a quicker project. Either way, it'll be awesome :)
I left a long yarn tail at the end of each hexagon and then used those tails to weave the pieces together. So I just used the tail for the next hexagon I was attaching. Since I used the mattress stitch to sew the pieces together, the colors of the yarn tail aren't very visible either way, so it didn't really matter which color tail I was using.
Using Inkscape, I can easily create an hexagon with rounded courners. The result in below in yellow (the right hexagon). It's possible to see that not only the corners are rounded, but the entire shape is (the lines around the hexagon aren't completely straight). I am now trying to acheive the same effect with Illustrator CS6, but I can't find out how. The closest I got is the white hexagon below (on the left). The corners are rounded but the lines are straight.
I would like to find unambiguous terms for the following two orientations for hexagons -- ideally, these would be definitive and succinct (I could refer to these as "Flat on Top" and "Pointed on Top" but would prefer something more technical and authoritative).
Jónapot. I think this problem not only applies to the hexagon but all geometric forms, the hexagon is just a good example. If you draw a hexagon and in- or extrude it it is really hard to resize as you can not see or set the original values of the hexagon. The measurement tool does not show the values of the original geometric form. Not sure if this is a feature request but if shapr recognizes a geometric form in a marked part the measurement tool should display the values of that form.
So if I mark the walls of an intruded hexagon the hexagon size should be displayed. This would help a lot. I am sure many of us remove the intrusion and restart instead of doing a resize over and over by trial and error for a fit or do some other workaround like drawing a second hexagon on a plane. The simple shortcut just does not work. Well it works but without measurement.
So this marking of all walls should display the mm2 surfaces of all rectangles and the hexagons and also width and heights or the hexagons and rectangles. Then resizing to a specific size would be easy. Without measurements resizing from that marking its just a wild guess.
The hexagon's "base" is vertical. (Mainly because I couldn't be bothered to figure out how to rotate the damn thing XD)Now the thing is, I have no idea how to figure out the texture coordinates. I've looked all over the web, but still wasn't succesful.
Texture coordinates work almost like percentages from 0.0 to 1.0 where (0.0, 0.0) is in the lower left. If your texture image is 128 x 128 pixels, then the point (0.25, 0.25) would be 32 pixels in from the left and bottom. Working with what you had there, if you were trying to have the hexagon inscribed exactly inside a square texture graphic, your coordinates should look something like this:
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