In earlier civilizations, science (math in particular), religion, and art were not separate. We don't even have a word for the broad and fluid field they formed together, but we can get a feel for it by gazing at any of the many astounding works of geometric art which have survived till now, and are usually part of a sacred structure.
Geometry is nothing but numbers made visible. In fact it is the very
first manifestation of numbers, well before shorthand symbols—1,2,3—were
created for them. Early geometers understood the relationships between
numbers by looking at the way geometric shapes related to each other,
and as numbers were deeply meaningful, so were the patterns emerging
from them charged with meaning. The two-dimensional, abstract nature of
geometry was understood as being one step closer to the zero-dimensional, unknowable Divine than our physical world, and its beauty was quite
literally out of this world.
The fascination with geometry and mathematical patterns is reemerging today: we can see it in the growing popularity of fractal art. There is no need, however, for special software to create highly complex geometric designs, and it is in fact deeply satisfying, even meditative, to slowly draw them out of the white nothingness of a sheet of paper, as we are going to do in these tutorials.
We will start with the building blocks of geometry, mastering simple constructions during the first few lessons. Then we will move on to patterns and more elaborate constructions, and the last few lessons will tackle truly complex, but rewarding, works of geometry.
To begin with, let us define a few terms that will come up regularly in these lessons. You are probably already familiar with several of them.
- A circle is the simplest geometric shape, a closed curve where all points are the same distance from the center.
- A diameter is any line that connects two points on a circle and passes through the center.
- A radius is any line that connects the center of a circle to its circumference (practically speaking, this is our compass opening when drawing a circle).
- A chord is any line that connects two points on a circle, without passing through the center.
- A semicircle is exactly half a circle.
- An arc is any segment of circle that is not a semicircle.
- A tangent is a line that just touches a circle in a single point.
- An acute angle is smaller than 90º.
- A right angle is exactly 90º, and the little square marked inside it is the conventional way of indicating a right angle on a diagram.
- An obtuse angle is bigger than 90º.
- A triangle is any closed shape with three straight sides. A random triangle, as opposed to the next three, is also called a scalene triangle. The sum of the angles in any triangle is always 180º.
- A right triangle has one right angle. The other two angles don't have to be equal, and the sides vary.
- An isosceles triangle has two equal sides (equal lengths are indicated by dashes on a diagram).
- An equilateral triangle has three equal sides, and its three angles are also equal (60º).
- A quadrilateral is any closed shape with four straight sides. The sum of angles in a quadrilateral is always 360º.
- A rectangle is a quadrilateral with four right angles. By necessity, the two sides opposite each other are parallel and the same length.
- A square is a specialized rectangle where all four sides are equal.
- A rhombus also has four equal sides, with the two opposite each other parallel, but no right angles.
- The next eight shapes are polygons (closed shapes with more than four sides) with five, six and up to 12 sides. All their sides and angles are equal.
Geometry was originally practiced with nothing but a rope and pegs, so it really doesn't require fancy tools, only accurate ones, when working on paper. You only need three things: a pencil, a straight edge, and a compass.
A basic lead pencil is perfectly adequate for the job, but don't just grab the first one you find: it needs to be the right hardness. In the picture below you'll notice the label HB on the orange pencil, and 6H on the grey one. These are indicators of hardness. B indicates a soft lead, and the higher the number (4B, 5B), the softer.
A soft lead will leave a darker mark that does not score the paper, but smudges easily. H indicates a hard lead, similarly graded, that will only leave a light mark, and not smudge, but will score the paper if pressed too hard. HB, obviously, is a happy middle.
When constructing geometric patterns, you don't want soft pencils! The reason is that dark construction lines quickly get confusing, and smudging is inevitable. Soft leads also lose sharpness very quickly, resulting in either constant sharpening, or loss of accuracy when drawing.
What we want, instead, is to build up the drawing with light construction lines, and use a softer pencil to pick out the final lines of the patterns. This is what these two pencils are for: the 6H remains sharp a long time and makes a very light line, over which final lines made by the HB really stand out.
For very complicated patterns, an intermediate darkness of line can be added in‑between, for instance a 3H or 2H. It is important, however, to learn to draw lightly with H pencils, because they do score the paper and that's a mark that can't be rubbed out. When penciling is complete, the pattern can either be inked, and the pencil then rubbed out, or painted, which will cover the pencil, or transferred to a completely clean sheet of paper using tracing paper if so desired.
The advantage of traditional lead pencils is their affordability, but the downside is how frequently they need sharpening, and their impact on the environment. An alternative which I personally prefer is a good 2mm mechanical pencil (aka clutch pencil or leadholder), such as the one pictured below, with a special sharpener and boxes of leads. You can have just one such pencil and interchange leads as needed. Avoid those with thinner leads, such as 0.5 mm, because they can't be sharpened to a real point (0.5 mm is pretty blunt for our purposes!) and because you won't have a choice of hardness or softness.
Straight Edge or Ruler
Strictly speaking, measurements are never used in geometry as they are not as accurate as proper constructions, and we are never going to use them in this course. We would have to go out of our way, though, to find a straight edge without measurement markings, so we might as well pick a good ruler.
For precision tools, you can't go wrong with brands that cater to architects, and every art shop will have at least one of those. You might wonder, wouldn't any ruler be good enough? Well, no: markings may not matter so much, but straightness is very important!
Here is how to test a ruler's straight edge: draw a line along the edge of the ruler, then turn the ruler around and draw a line on top of the first one along the same edge. I have tested this below with a trusty ruler I have been using since 1997:
Let's have a close-up look: see how the line is still definitely a single line? This means the edge is perfectly straight.
Next I tested it with a metal edge, and you'll see why such edges are okay for cutting but should never be used for precision work.
In the close-up above, see how the line splits towards the right? If the whole image could fit on this screen, you would see that even though the two lines are drawn along the same edge, they enclose a narrow space, indicating that the edge is slightly curved. Just what we want to avoid!
Our most interesting and most important tool is also the most costly one, but a good compass is worth its weight in gold and will last a lifetime. It's always fine to use a cheaper school-type compass for learning, of course, and upgrade when moving on to serious work (or when you get too frustrated with the lack of precision).
A compass basically has two legs connected by a hinge: one leg ends in a needle point, the other in a pencil point. The pencil leg can be adjusted to obtain different openings, and turned while the needle point is held steady on the paper, which creates a circle. It is entirely possible to create complex geometric figures with nothing but a compass, replicating the rope-and-pegs methods used in architecture a long time ago.
What to Look for in a Compass
- A screw mechanism to change the compass opening (or at least a screw to tighten the hinge so that it stays put once you've set the desired opening). You do not want a compass that opens and closes easily, because the opening will inevitably change while you work.
- An interchangeable pencil end. The end of the right arm of the compass above can be taken off and replaced with the little gadget on the right, in which any drawing tool can be inserted: pencil, pen, ruling pen, or even brush. This is incredibly useful, as the alternative would be to ink or color the lines freehand, endangering the perfection of the curve.
extension arm: This is the longer accessory at the bottom. It makes it
possible to draw much larger circles. For instance, this compass can
manage a circle with a radius of around 25 cm, but the extension arm
stretches this to 35 cm.
Some compasses don't have a lead like this one does, but are designed to be fitted with a pencil. That's fine: it is then a matter of personal preference, subject to the same pros as cons as I have explained when comparing traditional pencils to mechanical pencils.
Tips for Using a Compass
- Cover your work surface with a large piece of card or mount board (at least the size of your paper), both to protect it from the needle point, and also so that the point can penetrate enough to stay in its place. Otherwise it can get very frustrating, as it'll keep slipping out.
- Place the needle or dry point, with great precision, where you want it to be, and then hold the handle (at the top) between thumb and forefinger to rotate it and create a circle. Getting a nice, even circle this way may take some practice at first—that's normal. Try to keep the compass reasonably upright while you draw. Never hold the compass with one leg in each hand, as that alters the opening.
- I must stress this: take great care to place the needle point accurately, and to keep the pencil end sharp. The reason some people are good at geometry and some aren't is all down to precision.
That's enough theory, let's start drawing! Gather your tools and some cartridge paper, and let's get started.
In the construction diagrams throughout the course, I use the following types and colours of lines. Here is what they mean:
Triangle (on a Given Side)
This is how to proceed if you're starting from a line segment, which means you already have one of the sides of the triangle.
Dry point on A, draw an arc from B.
Dry point on B, draw an arc from A to find the third point C.
Join. If your compass opening is larger or smaller than AB, the triangle is isosceles.
Triangle (in a Circle)
If you have a given circle and you need to inscribe an equilateral triangle in it (meaning its three points will be on the circle), follow these steps:
Draw a line through the center, cutting the circle at A and B.
With the same compass opening, draw an arc that cuts the circle at points C and D.
This technical-sounding term refers to a line that does two things: it divides a segment (or an angle) in two equal lengths (or angles), and it is at a right angle to the segment it divides. This is a rather important device and is frequently used in the process of constructing other figures.
With the point on A and the compass opening equal to AB, draw an arc.
Repeat with the point on B. The two arcs intersect above and below.
Join the two intersection points. The segment is now bisected and O is the mid-point between A and B.
Tangent Through a Point on a Circle
If you have a given point (P) on a circle and need to draw the tangent through this particular point:
Start by drawing the diameter that passes through P and the center O, and cuts the circle at another point A.
Set your compass opening to the distance AP, place the point on O and draw a large arc, almost a semicircle. It cuts the line AP at B.
Without changing the compass opening, place the point on B and cut the arc at C and D.
The line CD is your tangent at P.
Tangent to a Circle From an Outside Point
Suppose now that P is a point outside the circle and you need to draw the tangent that passes through it:
Join the segment PO.
Bisect PO at point A.
With the dry point on A and the opening set to AO, cut the circle at points B and C.
PB and PC are the two possible tangents from point P.
Parallel (Through a Given Point)
Parallel lines are lines that never touch, so they travel in exactly the same direction. If your schooling was anything like mine, you were taught a vague shortcut to drawing them, but always ended up just relying on the grid printed inside your copybook. This, however, is the right and proper way of getting true parallels!
Let us start with a given line and suppose we have an outside point P through which the parallel needs to pass.
With P as the center, draw any arc to cut the line at A.
With the same compass opening, put the point on A and mark point B.
Now place the point on B to draw an arc that passes through A and cuts the first arc at C.
The line PC is your parallel.
Parallels (at a Given Distance)
Slightly more tricky is to draw a parallel at a specific distance from the original line.
Start by marking two pairs of points on the line. Distances are not specific, but the further the pairs are from each other, the more accurate the result.
Find the bisector for each pair of points.
Open your compass to the desired distance and mark that distance on each of the two bisectors.
Dividing a Segment
We'll finish this first lesson with a very nifty method to divide a segment into a number of equal parts. This is useful of course if you don't have a ruler with markings at hand, but even a ruler is no help if you have a segment measuring 5.63 cm which you need to divide into seven sections. This method is completely accurate and will spare you awkward calculations.
In the following example, we want to cut a segment AB into seven.
Draw two arcs with the point on A and B respectively. Their radius doesn't matter as long as they intersect.
Join A with one of the intersections and B with the other. This results in two parallel lines.
What we're going to do now is mark evenly-spaced points on each parallel, using the compass. The opening doesn't matter but keep it small so all the points fit on the line. Their number is [number of segment portions minus 1], which in the case of our example, is 7–1 = 6 points. Here the first point is marked from A.
Move the compass point to the point just marked, and mark another, then repeat till six points are marked, then do the same starting from B.
Connect the points, and the lines cut the segment into seven equal parts.
So, we have taken our first steps into geometry as an art, with basic operations that will come in handy in future lessons or in your own explorations. Next time we will be jumping right into actual shapes and patterns, working with the numbers 4 and 8...