7 Hexagon Shape

[Mirtha Shikles]

Aregular 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 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, 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 .

On my legend in the Layout view how can I symbolize the square patch as a hexagon to match the layer shape using Graduated colors? I can change the symbology to Graduated Symbols and change the symbol shape that way, but then of course they are graduated in size. Even when I change them all to be the same point size they change based on the map scale, so I don't want graduated symbols. I need graduated colors, but I can't seem to change the patch shape using that method. I normally wouldn't care if the legend shape matched the data shape, but I will be overlaying some more data that is in a square grid instead of a hex grid, which will make it confusing.

This seems like making the legend patch size should be a simple thing, but I've spent about 3 hours trying to figure this out with settings and just can't. Alternatively I can export this as an SVG and bring it into an image editing program, but I have several of these to produce and simply don't want to have to do that given the time constraints.

It could be that there has now been some progress on aspect ratio, but hard to be sure, because the shapes now become invisible on drag+release - they are converted from Bezier to Rectangle and lose their visible line etc properties

So I figured I could probably create one by drawing 6 equilateral triangles with shift and then moving them in to position. Unfortunately, they don't snap together perfectly, and they are actually 6 separate shapes which means I can't add an outline without them looking weird:

Then I tried making a hexagon using shift that is the same height as my triangular hexagon thing, and then using the yellow handle to adjust it properly so that it matched the internal angles of the triangle. This too did not work perfectly since I was winging it, and while very very close, it wasn't perfect either. Doing a google search didn't help much either.

After I created the approximate shape using the overlay on the close-but-not-quite equilateral triangle, I decided to get programmatic. I used some VBA to check what the position of the handle was (with the hexagon selected):

The value for the close-but-not-quite hexagon was .28002, so I started fiddling around and trying to do math assuming that this value was somehow based on angles. It isn't. I tried setting it to .28 -- that doesn't work either.

So I set it to the furthest left it could go (turning the hexagon into a square) and the value was 0. Then I tried setting it as far right (turning the hexagon into a diamond) and got .57412. Given the starting value of the close-but-not-quite hexagon of .28002, and my many attempts to get it right with none of them working, I tried taking half of .57412, which was .28706, and lo and behold, that was the magic number.

Found an easier method on creating a perfect Hexagon:In PowerPoint, First Create a Perfect Circle: Eg 4cm x 4cmNow create a Hexagon on top of the Circle and resize till all edges "snaps" to the Circle: Perfect Hexagon

Taking Misnomer's answer a bit further...Once I had the six equal length sides configured and grouped, I overlayed the hexagon object from PPTX. I sized it to exactly match the equal-sided hexagon.Using 1" sides, the matching hexagon dimensions are Height 1.73" Width 2.00".So just may a hexagon that is this size, lock the aspect ratio, and resize the hexagon to meet your needs. Doing this gives you a shadeable object as well.

As suggested by jmac I also recommend using vb-editor, not changing widths or heights since it won't correct the false position of the corner point, which you will se when you rotate the hexagon and join with other similar hexagons. Regardless if you draw the hexagon with or without holding shift, your hexagon needs to be adjusted at its corner setting.

Draw a perfectly horizontal line that is as long as your desired side length. Copy that line and rotate it 60 degrees and -60 degrees to create the other sides of the hexagon. The lines should snap together at the points ensuring everything matches up. When complete, hold control and click on each line to select all 6 lines. Then, right click on one of them and group them together. Now you have a regular hexagon that you can copy and paste.

I've just made a regular hexagon in MS Powerpoint. Given that the maths applied is rounded to two decimals, this is not a perfect solution, but it should be sufficient for most presentational purposes. This is how I did it:

We can create right angle triangles in MS Powerpoint readily. Select 'Right Triangle' from the 'Basic Shapes' section of the 'Home' ribbon and draw the triangle. Then right click on it and select 'Format Shape'. Select 'Size & Properties' and set Height and Width. After doing a little maths, I used height of 4.66cm and width of 2.5cm (following Pythagoras theorem: a + b = c, the long side will be 5cm in this case). If you don't want to do any maths, just use these sizes - you can always resize the final shape later.

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