Drawing Islamic Geometric Ornament. Basics: Dividing the Circle

Notes, References and GlossaryNotes and curiosities

1) A curiosity in the naming of degrees and divisions. One of the very useful results of geometry / trigonometry is that the sum of the internal angles of a triangle is 180º. In the older nomenclature, that would be three sixths. For a square the internal angles sum to four fourths. For a pentagon they sum to 15 tenths. For a hexagon they sum to 12 sixths. For a heptagon they sum to 35 fourteenths. For an octagon they sum to 24 eighths. The sum of the angles of the polygons by reference to the whole circle is therefore; triangle one half, square 1 whole, pentagon 1.5, hexagon 2, heptagon 2.5, octagon 3…..

2) Notes on Compass Use There is an understandable tendency to think that a six inch compass (150mm) will draw a 6 inch radius circle since you can open it that far. This is not good practice as detailed below.

Opened as in (1) to the absolute maximum radius, you will have two problems. It will dig a hole in the center of your drawing. The steel needle point is leaning far to the outside of the circle. The needle will tend to “skate” and you will need to press quite firmly to keep it on the center. A hole results. That same firm pressure to keep it from skating will tend to flex a poorly designed compass, creating an inaccurate circle. Most modern compasses are not well designed to handle flexing pressures. The setting at (2) is better. It will not dig a hole in your center but it will still put some flexing stress on your compass. The setting at (3) is best with a vertical steel point and no flexing stress on the compass. You give up almost half of the opening to use your compass correctly. The less expensive your compass is, the more important these points are. To put it another way, your compass was not designed to do what you thought. Those joints are there for a reason. If they ARE NOT there, you are even more limited.

1) Absolute max 150 mm Rad 2) Design max ~115 mm 3) Best practice ~85 mm

The side of a Pentagon, and Why √5 Appears. If you don’t like math, skip this section.Since √5 appears in the algebraic solution of a side of a pentagon, we needed to constructed it here graphically. If the original circle radius is 2, line [oG] is 1. This makes line [GC] in Fig 16 exactly √5.

One formula for the side of a pentagon as drawn here is (10 x [oA]) / (√(50 + 10 x √5)) =[CJ] (I have no idea where they got that one)

From both trigonometry and golden ratio lore; cos 36º = [(√5)+1] / 4 , one half of the golden ratio. And I will send you to wikipedia for the proof of that one. The √5 always pops up.We can check that our triangle [oCH] agrees with these and it does. When the arc [GC] is drawn down to intersect [AB] and identify [H], the line [GH] is √5, [BH] is (√5)+1, and [oH] is (√5)-1. We have a triangle with sides (√5)-1 and 2. The hypotenuse [CH] is the solution of [(√5)-1]2 + 22 = [CH]2. The solution to both equations is 2.351141. Of course we don’t care because we have already solved the length graphically as [CH].As a bonus [oH] is the side of the inscribed decagon of this circle, for checking our later work.

Glossary. Wikipedia is a good source in general for definitions of geometry terms.

Arc: An arc is simply a part of a circle, an incomplete circle.

Compass Constructible Polygons: Only some of the regular polygons can be constructed- theoretically- exactly with compass and straight edge methods. The common set are; triangle, square, pentagon, hexagon, octagon and decagon. A 19th century method was demonstrated for the 17 sided polygon. This common set can be multiplied by the factors of the powers of two to give an infinite number. There are approximate compass construction methods for many of the remaining polygons. Some of these are extremely old.

Divided CircleDivisions of the Circle: This has the obvious meaning, plus some qualifications. The circle is divided into two or more parts by lines which extend across the circle. In this document, 1) those lines pass through the center of the circle i.e. the lines are diameters or radii and 2) the parts are equal parts. This is a subset of the general definition.

Equilateral triangle: This simply means that the three sides of the triangle are of equal length. It also means, as a result, that the three corner angles are equal.

Inscribed polygon: An inscribed polygon has it vertices located on the circle we are using here, circle [oA]. The circle is now called the polygon’s circumcircle. We won’t use it here, but the circumscribed polygon is the polygon drawn outside, around, the circle. A regular circumscribed polygon is tangent to the circle at the midpoints of its sides. The circle used for construction is now called the inscribed- or in-circle.

Inter-radius: Plural Inter-radii. This term is used in a sense introduced to the rosette discussion by AJ Lee. This is not its dictionary definition.

In the divided circles used here for the construction of rosettes, it is useful to refer to two types of radii. Radii pass through the points of the tips of the rosettes. Inter-radii are the radii defining the half divisions between these radii. A 12 fold rosette has 12 radii passing through the tips of the rosette and 12 inter-radii passing between.

Isometric grid: The isometric grid is simply the infinite field of equilateral triangles. Each meeting “tip to tip” or properly speaking sharing vertices with three other triangles. All sides and all angles are the same- iso- same -metric measure.

Nodes: Node in this paper refers to points where lines cross, change direction or end.

perpendicular bisector: A bisector is simply a line segment which divides something exactly in halves. You can have a bisector of a line segment, an angle, a polygon, a given space, etc. Perpendicular simple means that the 4 angles which the bisector form with the a line segment (in this case), are 90 degrees. Bisectors can also be planes but we won’t use it in that sense here.The segment appears here since a line, in geometry, may proceed infinitely in both directions. We use some segment cut out of that line.

Polygon: Wikipedia is a good source for definitions. Polygon is usually defined as “many sides.” Actually it means “many angles.” In this paper it will refer to regular polygons unless specified as irregular. See; Regular polygons, compass constructible polygons.

Radial Structure: The layouts we will create rely on circles interacting with the underlying divided circle. They are sometimes drawn or discussed as polygons, but all rely on circles. Rosette layouts usually possess radial symmetry.

Radial symmetry: Radial symmetry simply means that the pattern can be rotated around its center point and it will overlap perfectly with a copy of itself at a specific fractions of one rotation.

Radius: Plural is Radii. This is the straight line segment from the center of a circle to any point on the circle. In many cases in this paper, radius is used to refer to a line extending that radius out beyond the circle. Technically this meaning is a radial line rather than a radius proper.See inter-radius.

Regular polygons have all sides and corner, vertex, angles equal.Polygons can be convex or concave. All of the regular polygons discussed in this paper are convex. Many of the polygons used in the definition of the rosette are star polygons, concave polygons.

Sector: Defined by Euclid; Book III Definition 10. This has the common English meaning. This is usually the part of a circle cut by two radii which define a central angle- the sector of an angle. It can also be the part of a sector cut off by any Line which cuts the circle circumference twice. In this case it is the sector of a chord.

Sine qua non: That which is absolutely necessary. The base without which geometric design cannot exist.

Tangent Circles: Two circles which touch at only one point, share one point, are tangent

Vertices: Plural of vertex.

Vertex: For our purposes in this paper this is used to refer to corner, usually of a polygon. It is the node at the corner of a polygon. This is a small subset of the general definition.

Alan D Adams, Holland, New York, Mar 2021. License: Creative Commons -Attribution 3.0 Unported (CC BY 3.0) Text, photos and drawings.