Most people have some experience using a compass and straightedge to construct various shapes and arcs from either junior high or high school. Much of this information is forgotten because there is usually no utility for such things besides passing geometry. But you may not know that you can use these techniques to draw. Architects and masons as well as painters have used Euclidean geometry for a long time to construct anything from portraits to buildings. There is no prior experience in drawing or geometry required for this instructable.

By the end, you should know:

Basic construction techniques
Some Euclidean terminology
How to construct the egg

Step 1: Tools

You will need some tool for this, which are quite easy to come by.

Tools:

Compass
Pencil
Marker(s)
Straightedge (This means any edge, marked or otherwise. A ruler will do well, if not, even the spine of a book, etc.)

If you do not have these tools, don't worry. You can use a program called Geogebra online for free. But it will take some getting used to. https://www.geogebra.org/classic

Step 2: Basics (Lines)

Lines, or line segments, will be very common in all Euclidean constructions. You need two points to draw a line. Any two points, then, can be used to create a line segment, which is denoted by those two points (AB for example). You can also extend the lines beyond the points as pictured. This information will also be key.

Step 3: Basics (Perpendicular Bisector)

The first technique to get acquainted with is called the perpendicular bisector. "Perpendicular" because it is at a right angle, and "bisector" because it cuts a line segment into two equal parts.

You can draw a line with your straightedge and arbitrarily mark the points X and Y on it. This is line XY. If on Geogebra, all you need to do is select the line segment tool and mark any two points you want.

Starting at X, draw a circle with radius XY. That is to say, draw a circle with the stationary end of the compass at X and the pencil end at Y. Keeping this radius, do the same at point Y. You will have two circles which intersect. at two points. I have labeled them V and W. Drawing a line segment between them (line VW) will complete the bisection.

Step 4: Beginning the Egg

Make a point where you want the center of the first circle to be (A). You can choose any radius length by adjusting the compass. Now draw a full circle. Draw a line (BC) such that A is also on the line.

Step 5: Perpendicular Bisector

Just like we learned before. Construct a perpendicular bisector between B and C. D and E are where the bigger circles intersect. You don't need to draw the whole circle, either.

Step 6: More Lines

Use your straightedge to draw a line between DE. You see where DE intersects our first circle with radius AB? Label that point F. Using either B or C as a starting point, draw a line between it and F. Now we need to extend that line to one of the bigger circles. This gives us point G.

Step 7: The Final Circle

Putting the stationary arm of the compass on point F, construct circle FG. This will make the top of our egg.

Step 8: Finishing Touches

Now you can fill the egg in, erase the extra stuff, or whatever you want to do. To make the egg a bit more clear, I have added some extra points. The egg is between IH to BH to BC to CG back to GI.