Octagons in Baku

Besides the window in the Divankhana, I saw a lot of geometric design in Baku based around octagons.

For example, consider this pattern in a window in the external courtyard wall at Baku’s Taza Pir mosque.

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This beautiful pattern, with its eight-pointed stars set within octagons, turns up on plate 67 in Jules Bourgoin’s 1879 Les Éléments de L’Art Arabe (which you can download from archive.org).

Bourgoin plate 67 3x3

It’s wallpaper group is the fairly common *442 (p4m) and it is generated by tessellating a square cell.

Bourgoin plate 67 single tileConstruction of this pattern is straightforward. The eight-pointed star in the centre is inscribed in a circle whose radius is one quarter the side of the square. The vertices of the octagon are found by extending the sides of the star. The rest of the construction lines are extensions of the octagon sides, and lines connecting star dimples that are three apart.

 

bourgoin pattern 67 construction lines

But one enters the Taza Pir compound via a stairway from the street. The panels in the stairwell are related, but subtly different from the window design!

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Seen head-on…

taza pit entryway pattern 4-tileWhat did they do here?  There is the same eight-pointed star in the centre, and the same enclosing octagon, but in this case they’ve trimmed back,  to the borders of the octagon, the square that one repeats.

taza pit entryway pattern square2

As a result, the square tile borders stand out strongly as lines, and around each point where four tiles meet we get a big diamond holding four small diamonds.

taza pit entryway pattern 12-tile

(This also belongs to the *442 wallpaper group.)

Now, in the Old City, I came across a piece of octagon-based decoration that illustrates what happens if one doesn’t follow best practices, as explained by Eric Broug.  This pattern involves starting with the same pattern as in the Taza Pir windows (above; a.k.a. Bourgoin’s Plate 67), but then repeating a somewhat random subset of it. In other words, incorrect tessellation.

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It would appear that the manufacturer of these pre-cast concrete blocks selected a piece out of the overall pattern that was not the all-important basic square, but rather a rectangle.

subset taken for pre-cast

Hence each of the concrete blocks looks like this:

single block

When you put them together, lines match up, but the effect of the original design is lost.

tiled pattern

The wallpaper group of this pattern would be*2222 (pmm).

Elsewhere in the Old City, there were pre-cast patterns that did tessellate pleasingly, again with octagons.

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But back at the Taza Pir mosque, I spotted this on an adjacent building, which I believe is Baku Islamic University:

Taza Pir

The grill pattern is octagons packed together, with squares in between; and eight radial lines emanating from the centre of each octagon. It’s basically the central column of this pattern:

Taza Pir end pattern

But look at what they did in the point of the arch. It’s beyond my knowledge to know whether this is best practice or not, but it is definitely creative.

 

 

 

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A Little Geometric Creativity

There’s a nice old geometric pattern…

figure00

…that Eric Broug presents in his book Islamic Geometric Patterns as being from the Great Mosque of Herat.

Its construction is based on a hexagon, and the pattern repeats as additional hexagons are tiled around the first one. The underlying hexagons (which are not drawn in the finished pattern) are shown in red here:

figure00a

This makes a fine star/wedge/triangle pattern that is quite satisfying to look at, and suitable for decorating things in your house.

The construction of one of the hexagon-based units of the pattern begins with drawing a circle, and then a hexagon within that circle. Once the hexagon is drawn, the lines of the pattern can be drawn within in it.

But in the real world, a medium on which we are drawing (or painting) always has edges, and those edges do not go along the edges of the hexagons. For example, let’s say we want to draw one of the pattern units, plus however much extra fits, on a square board or tile. Something like this:

figure00b

Draw in the supporting hexagons and you will see that the one complete pattern is indeed surrounded by only parts of adjacent copies of the pattern.

figure00c

It’s a bit of a construction conundrum. How will we fill in the portion of the pattern that lies outside the one central hexagon that we will have room to draw? How will we extend the pattern to the edges of the square when we cannot place a compass foot at the centre of any of the adjacent circles that would define the basic hexagons?

There is a way.

Let’s review how the pattern is made, and then examine how to extend it without being able to draw more circles. Follow along with your own piece of paper, pencil, compass and straightedge.

There is something magical in the fact that this pattern, with all of its exact proportions, can be constructed solely with a compass and a straightedge. You never need to measure an angle or a line.

To construct a hexagon, one begins by drawing a circle.

figure01

Within the circle inscribe a hexagon using the standard method of walking the legs of your compass (still set to the same radius) around the circle. We’ll call this Hexagon A. It is oriented so it is a point-up hexagon.

figure02

Draw radii through all the vertices of the hexagon. We’ll call these the ribs of hexagon A.

figure03

 

Now we need to construct a second hexagon of the same size, and on the same centre, but rotated 30°. To do this we will connect the midpoints of the six arcs from the hexagon already made.

If we had plenty of room, we would find those midpoints by first drawing six more circles (of the same radius) centred on the six vertices of hexagon A; and then draw lines from the centre of the original circle to the outer points of intersection of adjacent secondary circles. These ribs (purple, below) would show us the midpoints of the arcs.

 

figure22But with the limited space we are working with, we need instead to bisect one side of the hexagon. This can be done with two compass arcs from adjacent vertices of hexagon A; we can then connect their intersection outside the circle with the circle’s centre.

figure04

This line bisects one of the arcs, and we can use that as a starting point for making a second hexagon. We’ll call this Hexagon B, and it is oriented so it is a side-up hexagon.

Draw ribs for hexagon B.

figure05

Note that the ribs of hexagon B (purple) go through the midpoints of the sides of hexagon A (red).

Connect the hexagon A midpoints to make a six-pointed star. We’ll call these star 1 lines (green in the next figure).

figure06

Similarly, we can note that the ribs of hexagon A pass through the midpoints of the sides of hexagon B. Connect the midpoints of hexagon B to make Star 2 — being sure, however, in this case to extend the star lines beyond hexagon B as far as the first edge of hexagon A you encounter. We’ll call these star 2 lines (blue in the next figure).

figure07

Finally, we ink certain portions of the Star 1 and Star 2 lines to make the final pattern. Note that the entire pattern lies within hexagon A.

figure08

The pattern that is inked consists only of portions of star 1 and star 2 lines. In fact we drew hexagons A and B, and their ribs, only to create the star 1 and 2 lines. And we drew the initial circle only to create hexagons A and B.

In order to extend the pattern we need to draw more star 1 and 2 lines outside our original hexagon. On a larger medium we could easily place the centres of new circles outside hexagon A, but on this limited surface where we are working we cannot.

It may help to look at what we need to find.

figure21

From our earlier construction of adjacent hexagonal cells, we know a few things about what should be happening. Star 1 lines leaving the hexagon (A in the diagram above) go a certain distance to a point Y (which we’re not really sure how to locate) and then turn through 120° to follow a line D.

Star 2 lines leaving the hexagon (B in the diagram above) go a certain distance to a point X (which we’re not really sure how to locate) where they cross a star 2 line (C) in the adjacent hexagons. At some farther point Z they also go through 120° turns.

As well, it’s valuable to make a few observations about what happens as these lines enter adjacent hexagons…

1. Hexagon A sides, when extended, become the ribs of other hexagons A, and vice versa.

figure10

2. Star 1 lines become some of the star 1 lines of adjacent hexagons.

figure13

3. Star 2 lines become some of the star 2 lines of adjacent hexagons.

figure14

This means that by extending these hexagon sides, ribs and star lines, we have much (but not all) of the construction information for the pattern outside the original hexagon.

figure16

What’s missing are lines C and D, and points X, Y and Z.

But now we can see that point X is the place where the extended hexagon A sides meet the extended star 2 lines. Drawing a line through them gives us line C.

figure17

Similarly, Point Y is the place where the extended hexagon A sides meet the extended star 1 lines. Drawing a line through pairs of points Y gives us line D.

figure18

At this point we have all the construction lines we need to ink the rest of the pattern outside the original hexagon…

figure19

…and then remove the construction lines.

figure00b

It’s very satisfying to be able to construct this figure in a limited space, and to solve the problems associated. But now, as a bonus, it appear that the pattern presents us with a fascinating geometry problem!

As the pattern is extended, a new, larger hexagon has appeared, a hexagon that is similar to hexagon B, but is formed by star 1 lines that pass on the outside of the six small triangles. We’ll call it hexagon C. In the illustration below, hexagon B is purple, and hexagon C is blue.

figure23

It’s a bit of a puzzler, but I’ll just leave the problem here for the intrigued reader to solve. In terms of the side length of hexagon B, what is the side length of hexagon C?