## Introduction: Fun With Ruler and Compasses - Basic Geometric Constructions.

With the prevalence of drawing software, I have noticed that certain skills seem to be fading away.

This Instructable is the result of a request* for an outline of some of those skills. If you can already use a ruler and compass, this is not the project for you.

This Instructable is the result of a request* for an outline of some of those skills. If you can already use a ruler and compass, this is not the project for you.

_{*Sometimes I do take requests. Keep them polite, though.}## Step 1: The (very) Basics - Equipment.

For these simple projects, you will need:

A pencil. Nice and sharp, not too hard. In most of the world, these are referred to as "HB". In the US, they are called "#2".

A ruler or straight edge. Any rigid material will suffice. I prefer steel rulers, but as long as you avoid soft bendy rulers, you should be OK. Avoid notched edges.

A compass. Not the magnetic kind, but the drawing kind. Any decent stationers will have a wide range to choose from. You need one that will hold your pencil firmly, does not have a loose hinge, and has a point - the "safe" compasses slip too easily. I prefer those compasses that have a knurled knob at the hinge, as they allow for a smooth, one-handed action. The best compasses are adjusted by means of a wheel on a screw-thread near the hinge, because these are least likely to slip.

If you invest in a compass that does not hold a pencil, but instead holds a small piece of graphite, you will find a small piece of sandpaper useful for maintaining the bevelled point (and far cheaper than dedicated sharpeners).

Paper. Plain printer paper is fine.

You will also find a sharpener and a rubber useful, even if you are competent with your tools.

A pencil. Nice and sharp, not too hard. In most of the world, these are referred to as "HB". In the US, they are called "#2".

A ruler or straight edge. Any rigid material will suffice. I prefer steel rulers, but as long as you avoid soft bendy rulers, you should be OK. Avoid notched edges.

A compass. Not the magnetic kind, but the drawing kind. Any decent stationers will have a wide range to choose from. You need one that will hold your pencil firmly, does not have a loose hinge, and has a point - the "safe" compasses slip too easily. I prefer those compasses that have a knurled knob at the hinge, as they allow for a smooth, one-handed action. The best compasses are adjusted by means of a wheel on a screw-thread near the hinge, because these are least likely to slip.

If you invest in a compass that does not hold a pencil, but instead holds a small piece of graphite, you will find a small piece of sandpaper useful for maintaining the bevelled point (and far cheaper than dedicated sharpeners).

Paper. Plain printer paper is fine.

You will also find a sharpener and a rubber useful, even if you are competent with your tools.

*You can spend tens of pounds on decent equipment. My local stationer stocks a "pretty good" compass for 6.99GBP. However, to show that a little care can produce useful results with any equipment, I invested all of 79p in the kit I used for this Instructable.*_{I should also point out that some of the lines drawn in the photos look faint, curved or both. Pencil lines on white paper are a pig to photograph, and the wide-angle setting on my new camera bends lines that do not cross directly through the centre of the image.}## Step 2: The (very) Basics - Handling Your Equipment.

The Ruler

Believe it of not, I often hear the complaint "this ruler won't draw straight".

Appalling, but true.

To draw a reliably-straight line, where you want it, position your ruler carefully, then spread your fingertips along it, keeping them away from the edge. Press firmly, but not so hard to turn your knuckles white. If you only press at one point, it is all too easy to pivot the ruler and draw a wonky line.

Hold the pencil in the traditional manner, but lightly, and further away from the point than usual.

Draw* the pencil towards you. To get the drawn line as close to the ruler as possible, the pencil should be held at 45

^{o}to everything - to the ruler

*and*the page. Press lightly, or you will splinter or break the pencil's point, and carve an unwanted groove in the paper (which makes it hard to erase mistakes).

Lift the ruler, and admire your handiwork.

The compass

Close the compass, and clamp in the pencil so that the point of the pencil meets the point of the compass.

Open the compass to the desired width. Don't forget, the distance point-to-point will determine the

*radius*of the circle, the distance from centre to perimeter ("edge"). The diameter of the circle is twice the radius.

Place the point of the compass where you want the circle's centre to be and press in gently - you want the point to "bite" into the paper, without punching right through.

Place the point of the pencil on the paper, lean the compass over and then smoothly turn the compass in the direction of the lean. With just a little practice, your should be able to draw a smooth, continuous circle with just one hand. It does not matter which way you turn the compass (in fact, I often turn the compass both ways for one circle), as long as you turn it in the direction you leaned - remember,

*draw*the line.

_{*"Draw" means pull. You do not push a line, you draw one.}

## Step 3: Construction #1 - a Perpendicular Line.

"Perpendicular" means "at right angles to the line".

Draw a straight line, then mark the point where you want your perpendicular line to meet the line you have just drawn. It is usual to mark points like this with a small line crossing the long line.

Place the compass on the point you have just marked.

Draw small arcs (fractions of a circle) crossing the line on each side of the starting point, marking two new points.

Reposition the compass on one of the new points and draw an arc above the line, roughly where your perpendicular line will go.

Reposition the compass onto the second point, and draw a second arc to cross the first. This one can be much shorter.

Draw a ruled line to join the starting point on the first line to the point marked by the crossed arcs.

Congratulations!

Draw a straight line, then mark the point where you want your perpendicular line to meet the line you have just drawn. It is usual to mark points like this with a small line crossing the long line.

Place the compass on the point you have just marked.

Draw small arcs (fractions of a circle) crossing the line on each side of the starting point, marking two new points.

Reposition the compass on one of the new points and draw an arc above the line, roughly where your perpendicular line will go.

Reposition the compass onto the second point, and draw a second arc to cross the first. This one can be much shorter.

Draw a ruled line to join the starting point on the first line to the point marked by the crossed arcs.

Congratulations!

## Step 4: Construction #2 - Bisect a Line.

"Bisect" means "cut into two equal parts" (not the same as "dissect", which is simply general cutting apart).

Open the compasses so that they are longer than half the line - this is easily checked by eye.

Position the compasses on one end of the line, and draw arcs above and below the line.

Reposition the compass on the other end of the line, and draw two more arcs, crossing the first two you drew.

Position the ruler as if you were going to draw a line between the two points created by the crossed arcs, but only draw enough of a line to mark a point on the line you were bisecting.

Congratulations!

Open the compasses so that they are longer than half the line - this is easily checked by eye.

Position the compasses on one end of the line, and draw arcs above and below the line.

Reposition the compass on the other end of the line, and draw two more arcs, crossing the first two you drew.

Position the ruler as if you were going to draw a line between the two points created by the crossed arcs, but only draw enough of a line to mark a point on the line you were bisecting.

Congratulations!

## Step 5: Construction #3 - Bisect an Angle.

Place the compass on the point formed by the angle.

Draw an arc that crosses both sides of the angle, marking points on each line.

Do not change the angle of the compass (there is no need), place it on one of the newly-marked points. Draw an arc between the lines of the angle.

Position the compass on the other side of the angle, and draw another arc to cross the last one you drew.

Use your ruler and pencil to draw a line from the point of the angle through the point marked by two arcs.

Congratulations - the angles either side of the line you have just drawn are exactly half of the angle you started with.

Draw an arc that crosses both sides of the angle, marking points on each line.

Do not change the angle of the compass (there is no need), place it on one of the newly-marked points. Draw an arc between the lines of the angle.

Position the compass on the other side of the angle, and draw another arc to cross the last one you drew.

Use your ruler and pencil to draw a line from the point of the angle through the point marked by two arcs.

Congratulations - the angles either side of the line you have just drawn are exactly half of the angle you started with.

## Step 6: Construction #4 - Copy an Angle.

- Angles that are the same are said to be "congruent".

Set the compass to an arbitrary size, and draw an arc across both sides of the angle.

Without changing the setting of the compass, draw a similar arc on the line you are copying the angle to.

Set the compass so that the point and the pencil land on the crossing points of the arc on the original angle.

Keep the compass set the same, and draw a small arc to mark a point on the larger arc you drew on the line you are copying the angle

*to*.

Draw a ruled line from the point of the new angle, through the point marked by the crossing arcs.

Congratulations!

_{The angles look odd because of the camera angle. The right-hand angle is further from the lens.}

## Step 7: Construction #5 - Draw Parallel Lines.

- Parallel lines are always the same distance apart - they neither meet nor diverge.

Start by drawing a diagonal line through the point that crosses the existing line.

This creates an angle between the original line and the diagonal line.

Using the directions from step 6, copy that angle onto the diagonal line, at the point where the new parallel line is supposed to go through.

Congratulations!

## Step 8: Construction #6 - an Equilateral Triangle (the Easy Way)

- Equilateral triangles have all three sides the same length. This means that all three angles are also identical.

Set your compasses to the length of the first side, position them at one end of the line, and draw an arc.

Move your compasses to the other end of the line, and draw an arc that crosses the first.

The crossed arcs mark the final point of the triangle. Draw ruled lines from each end of the original line to meet at that point, and you have created an equilateral triangle.

Congratulations!

## Step 9: Construction #7 - the Regular Hexagon

- Regular hexagons have all six sides the same length, and all six angles the same size.

Without changing the compass-setting, place the compass on the edge of the circle and draw arcs that also cross the circumference.

Place the compass on one of these two newly-marked points, and draw two more arcs on the circumference. One will pass through the point where you originally placed the compass.

For a third and final time, move the compass to one of the newly-marked points, and draw two more arcs.

The circle will now have six points marked by small arcs on the circumference. Join them with straight lines, and you will have constructed a regular hexagon.

Congratulations!

If, when you construct the hexagon, you draw continuous arcs, your hexagon will be filled with a six-petalled flower. Extending this pattern, always with the same setting on the compass, can create extensive, decorative patterns.

## Step 10: The Ultimate Point of This Thing - Constructing a Strip of Triangles for a Hexaflexagon.

You see? This isn't just a set of skills for drawing abstract forms.

Draw a line, as long as you desire. Construct an equilateral triangle at one end of the line, then another, and another and another....

(You will find it useful to check the setting of the compass occasionally, as the pencil will wear down and change the size of your triangles.)

When you get far enough, draw a line joining the tops of the triangles you have just constructed. This will create a whole set of triangles between the first row you drew, and make a strip that you can cut out and use to fold a hexaflexagon.

You can, of course, use this basic set of constructions to create more complex or subtle forms. For instance, if you bisect all the angles of a triangle, the new lines will all cross at the centre of the triangle and allow you to circumscribe the triangle.

Draw a line, as long as you desire. Construct an equilateral triangle at one end of the line, then another, and another and another....

(You will find it useful to check the setting of the compass occasionally, as the pencil will wear down and change the size of your triangles.)

When you get far enough, draw a line joining the tops of the triangles you have just constructed. This will create a whole set of triangles between the first row you drew, and make a strip that you can cut out and use to fold a hexaflexagon.

You can, of course, use this basic set of constructions to create more complex or subtle forms. For instance, if you bisect all the angles of a triangle, the new lines will all cross at the centre of the triangle and allow you to circumscribe the triangle.