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When is a Hexagon Not a Hexagon? (And why do we care in temari)

Part 1: Some Spherical Geometry

Much of the geometry of the sphere is the same as the geometry on a flat surface, or plane, that you probably studied at some point in school. There are still points, lines, angles and polygons, but with a few key differences.

Lines on a sphere

Perhaps one of the most basic differences has to do with what a line is. On the plane we think of a line as straight and as the shortest distance between two points. On a sphere the shortest distance between two points is a segment of a great circle, that is, a circle that goes all the way around the circumference of the sphere. So, on a sphere a line is necessarily curved. On the plane it is possible to have two lines going in precisely the same direction so that they will never intersect (parrallel lines). But, on a sphere, any two great circles will intersect in exactly two opposite points. This seemingly small difference in definition makes all the difference in the world when it comes to angles and shapes.

Shapes on a sphere

There is an entirely new shape possible on a sphere that you cannot have on the plane: a lune. It is a shape with only two sides and two angles. It looks like the segment of an orange. In temari, it is the shape of a spindle.

Polygons like triangles, squares, pentagons and hexagons still exist on a sphere, but their sides are made up of curves rather than straight lines. It may appear that the lines are straight when you mark them on a temari but there really is a subtle curvature that is difficult to see until the shapes get large. You can still have regular polygons like an equilateral triangle or a square, but their angles are not same as the ones we are used to on the plane.

Size of the sphere

A flat surface or plane can continue infinitely in any direction but a sphere is confined to a given size dictated by the radius of the sphere. While it seems that it would make a big difference to have different sizes of spheres we can largly discount the difference if we keep our study and comparisons confined to the same sphere. In math we use a unit sphere, that is, one whose radius is considered to be one unit. By confining ourselves to one size sphere we simplify the mathematics. Later, if we have a need to calculate specific distances we can use the actual radius as a factor in our equations.

Angles on a sphere

This is where things really start to get a little wierd. You measure angles in more or less the same way on a sphere and on a plane. There are still 360 degrees around a point. However, when you start creating shapes with angles and great circles on a sphere some remarkable differences emerge.

1. The sum of the angles of a triangle on a sphere is always greater than 180 degrees unlike on a plane where it is always equal to 180 degrees.
2. The amount that the sum is greater than 180 is directly related to the area of the triangle. That means that the bigger the shape is, the larger its angles will be. No way! Show me. (This is an off site link; scroll down to The Area of a Spherical Triangle Part 1:)
3. Because of 2, all mathematically similar shapes on a sphere are congruent.

These details of spherical geometry have the most impact on how we see and stitch shapes on temari. Let's start by looking at the shapes and the interior angles.

Angles of a polygon on the surface of a sphere

Item number 2 above states that on a sphere the sum of the angles of a triangle is related to its area. When you measure the angles with radians rather than degrees the formula is very simple: Area of triangle = sum of the angles - π/2 (Note: π/2 radians = 180 degrees.) So the sum of the angles of a triangle must be more than 180 degrees or the triangle will have no area. And, the larger the triangle is, the larger the angles must be. (The link in item 2 above gives a nice visual representation of this.)

The more area the shape covers, the larger its angles will be. Consider that all polygonal shapes can be split into triangles, like the square to the left is. In this case there are four triangles that meet in the center so the central angle (A) is 360 ÷ 4 = 90 degrees whether the square is on a plane or a sphere. Now look at the interior angle (angle B, the angle between the sides). It is made up of two of the other angles of the triangles we split the square into. We are used to the square having 90 degree interior angles. But on a sphere those angles have to be greater than 180 - the central angle or 180 - 90. That means that for a very small square that takes up barely any area the angle would be close to, but still greater than 90. For a bigger square that takes up a greater amount of area those interior angles will be well over 90 degrees.

While we can't calculate exactly how large those angles will be in all circumstances, sometimes the configuration of them gives us a clue. On a C8 division the squares we typically use have their corners at 6-way intersections. There are three squares meeting at that point so the angles are 360 ÷ 3 or 120 degrees.

Additionally, the sides of those squares are along a longer segment of the great circle so the curvature is more apparent to our eyes than on a smaller segment. That's why even regular shapes can look a little funny to our eye when they are on the sphere.

There are stitching implications to the changes in angle size. They show up when you are stitching a closed shape such as the classic beginners squares design, and also when you are stitching a shape intended to give complete coverage of the ball.

As the shape gets larger, the angles get larger also. In most cases the amount of change is small enough that the thickness of our stitching thread absorbs the differences. But as the shapes get really big it can get to the point where the thread is no longer giving proper coverage between rows. We have to compensate by making the distance between the stitches smaller to get the coverage we want. Many stitchers naturally do this by placing the thread along the row and seeing where it will cross the marking line to take the stitch. It is the process used to stretch the points when we are stitching the spindle element or the bottom of a kiku but it is relevant any time you are stitching a shape that gets substantially larger from when you started. The distance between your stitches changes because the angles are changing. It is one more reason why that beginning squares design is actually harder to execute than it looks.

Part 2: Regular polygons on a sphere

Definition of regular polygon

A regular ploygon is a closed shape with equal sides and equal angles. That seems simple enough. The definition is the same whether the polygon is on a plane or on the surface of a sphere. The following diagram of a regular hexagon demonstrates some of the special characteristics of any regular polygon.

1. All of the central angles are equal. These angles are a constant number based on the number of sides of the polygon. It will be the same on both a plane and a sphere.
2. All of the angles between sides, the interior angles, are equal. On the plane this is a constant number based on the number of sides of the polygon, but on a sphere it increases as the area of the shape increases.
3. As a result of 1 and 2, these smaller angles are also equal and are half the size of the interior angles.
4. The distance from the center of the polygon to each vertex is equal. The distance from the center to the midpoint of each side will also be equal.
5. The lengths of the sides are equal.

If you find that any of the characteristics are missing then you can assume the polygon you are looking at is not regular. But that does not mean that you have to check all five items each time. Just two of them will suffice as long as one is an angle and the other is a length measurement. Next we'll look at how you can check.

Techniques to use to check for a regular shape

When stitching a temari it is nice to know if the shape we are stitching is supposed to be regular. If it is regular then we can do our best to maintain the regular shape as we place our stitching inside or around it. If it is not regular, then we can either adjust our stitching to make it appear more regular, or in most cases, accept that it is not supposed to be regular and enjoy the shape that it is. There are several useful methods to use when analyzing a shape on a temari division.

• First of all, we know that on a sphere all similar shapes are congruent. And congruent shapes have equal lengths as well as equal angles. Conveniently, when we use a basic temari division such a Simple, C6, C8 or C10 we are dividing the ball into congruent triangles. We can use those congruent triangles that have the angles or the sides of the polygon in them as the basis of comparison.
• Secondly the sum of the angles around a point is still 360 degrees on a sphere. And, the sum of the angles that make up a straight line is still 180 degrees. So, instead of looking to the sum of the interior angles of the triangle, we can look at the angles around each point (vertex) of the triangle or along a straight line to discover the measure of an angle we want to know.
• Lastly we can use triangles to help us out. Now, on a plane if you know two of the angles of a triangle, you can subtract from 180 to find the third. But on a sphere if we do that subtraction we will find what the minimum size of that third angle is; it is guaranteed to be larger. While you can't get the exact measure of the angle, you can get enough information to compare it to others that you do know.

How can I use this on a temari?

How does all of this work together? Consider a special temari marking with 16 faces (12 pentagons, 4 hexagons). It starts with a C6 or C8 division and then adds support lines to create the faces. The hexagons are regular because we start with equal central angles at a 6-way intersection and then measure out an equal distance to place the verticies (dotted lines on diagram). But are the pentagons regular? They can appear distorted on a diagram but they look pretty regular on the stitched temari. It turns out that they are not regular, but can be made to look like it with careful stitching.

Look at the diagram closely as we consider the marking.

• - The central angle of the hexagon is 60 degrees (360 ÷ 6). It is labled A on the diagram.
• - The base angle of the small blue triangle (labeled angle B) of the hexagon must be greater than 60 degrees so that the sum of the angles of that triangle will be greater than 180 degrees on the sphere.
• - Notice that one of the angles of the pentagon (angle C) together with angle B make up a straight line or 180 degrees. So angle C must be less than 120 degrees since angle B is greater than 60 degrees and together they add to 180.
• - Now, look at a different angle of the pentagon, labled D. It is on one of the original 6-way intersections for the C8 marking. It takes up 2 sections for exactly 120 degrees.
• - So, angle C is strictly less than 120 degrees but angle D is exactly 120 degrees so the pentagon cannot be regular.

Those pentagons really do look regular when I stitch the marking, but strictly speaking they are not. In order to have a regular look to my design on this marking I will need to pay close attention to my stitching in the pentagons and make sure I am conpensating as needed.

Don't forget that this part about whether a particular region of a temari marking is regular or not is a theoretical discussion. It applies to a perfect sphere with a smooth surface and infinitely thin marking lines. In actual practice our marking thread has a certain thickness, there is a degree of error inherit in our measurements and in the roundness of our base so that sometimes a non-regular shape is close enough to regular so that for all practical purposes it is.

Part Three: Applying the regular polygon ideas to temari designs and stitching elements

Wrapped crossover bands form a hexagonal focal point.
Three intersecting crossover bands create a star shape. This design can be stitched with spindles as well.
An interlocked five point star creates a pentagon center.
An interwoven triwing element has a hexagonal center.

Thus far we've been focused on the underlying shapes of the temari marking and determining if they are regular. But in temari we also encounter regular shapes that occur as the result of specific temari design elements intersecting. In one style of design you have individual elements that intersect to form a focal point for the design where it is intended for the intersecting area to be a regular shape. That shape can be a regular polygon or more of a star shape. In other cases, the actual stitching element intersects itself to create a central area that you might like to have a regular shape.

Individual elements that intersect (eg. spindles)

Intersecting individual elements is particularly common with the spindle stitching element. How do we know that the intersection of the spindles will be a regular shape? We can use the math ideas for regular shapes to deduce that you'll get a regular shape at the intersection as long as the intersecting elements are all equally spaced around the intersection and are intersecting about their central point. The following examples are drawings of three crossed spindle shapes which form a hexagon at their intersection. Notice how the hexagons become distorted if the original lines are not equally spaced or if the spindles are not set to intersect around their centers. This will be true for any number of intersecting spindles. The same basic principle can be applied to other types of intersecting elements such as wide crossover bands. Just be sure that they are intersecting about their center points and the lines they are created on are equally spaced and you will get as regular a shape as possible.

Three spindles intersecting where the lines are not equally spaced.
Three spindles intersecting where the spindles are offset so that they intersect closer to the points.
Three spindles intersecting about their centers with the lines equally spaced.

Single elements that intersect themselves (eg. triwings)

The most common of these elements are the triwing and the five pointed star. Stars are very straight forward; be sure the lines for the element are equally spaced and be sure the stitches are all taken an equal distance from the center. The tri-wing (also know as the mitsubane, three wing, or mitsubishi) is more problematic. In this case the intersection at the middle may not be a regular hexagon even if you stitched it correctly. Triwings are unique because there are two different angles in the resulting hexagon: those formed by the stitches and those formed by the intersecting lines. The distance we use to establish the outer points of the triwing has a tremendous effect on the angles where the threads cross. Yikes! What is a stitcher to do? Fall back on the definition of a regular shape. It needs to have equal central angles and the distance from the center to a vertex needs to be constant. First, be sure the lines you are placing the tri-wing on are evenly spaced to ensure equal central angles. Next, you will want to get the optimum distance for stitch placement of the outer points so that your lines cross at a distance from the center that is equal to the stitch placement for the inner stitches. There are three methods you can use to make sure you have nice centers to your tri-wing elements.

It is easier to get a regular looking center to your triwing if you start by stitching a small hexagon in the center. Once the regular hexagon is established you can use the side of it to establish the line for the wing and thus the placement for the outer point.

Depending on your thread, you can also use the friction of the thread to groom the stitches around the hexagon thus preserving the regular shape when you need to cheat and shorten the points a bit for other design considerations. (Not a mathematical solution, but eminently practical.) The pictured example was not stitched with a triwing element, but the structure is similar and makes it a good example for this technique.

You can also use the method of laying the thread in place to determine the optimum placement for the outer stitches when you start the triwing in the first place. This is less precise but basically you would eyeball the distances out from the center to make them as equal as possible. Placing pins and wrapping the thread around them first can help you to see where to make adjustments. Place the first stitch, then lay the thread toward the next stitch, making sure that it crosses the intermediate line at the same distance from the center as the inner stitch. The first picture shows a placement that will not give a regular hexagon center since the distances from the center point to the thread crossing the marking lines are not equal. The second picture shows a placement that will give a regular hexagon center.

In any case you will need to continue to pay attention to the hexagonal center of the triwing as you stitch it. It is very easy for the lines to be off just a little in the beginning and have the errors grow as you do more rows. Use the techniques above to make adjustments as you stitch and keep the hexagon regular. Also, consider carefully if you really need to have a regular hexagon for the center. If it is not a focal point for the design then it may be just fine to allow it to take its own shape without stressing about it being regular.

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