This projection is not so pretty, but it has about as little distortion as you can get on a flat map. This is because the individual spears (the triangles representing a swath of longitudes) are not distorted into rectangles, but retain their triangular shape.
Materials: You will need pencil and paper and/or a computer; a calculator; a good idea of where your continents are; and an orange.
The basic math of the projection is simple enough. Let's look at one half-spear, 45o in width. (You can use any width, of course, up to the entire circumference of the planet.) The entire globe will be made of eight of these spears.
Choose a drawing width for your spear, based on the size of the paper or image. If you're using an 8 1/2 x 11" sheet of paper, these 45o spears can be up to 11 / 8 = 1 3/8". (1.25" is easier to measure. 3.4 cm also works, though if you have A4 paper you can use 3.5 instead.)
(In case it's not clear, the long axis of the map is the planet's equator; since this is 360o, there are 360/45 = 8 spears total, and I am making the spears as wide as possible to fit within the paper. Make adjustments as necessary for different sized images.)
A spear goes from pole to pole, that is, half the circumference of the planet. If the spears are 1.25" wide, 10" total, then the spears are 5" tall, and the half-spears are 2.5" tall. At left is the skeleton for one spear, with these measurements. (I'm using Photoshop Elements, which allows you to draw in inches; you can of course measure everything in pixels instead.)
If you chose a different degree width, the spears would be narrower, but about the same height. E.g. if you choose 30o spears, you'd have 12 spears, each 7/8" wide, for a total width of 10.5"; so each spear would be 5.25" tall.
Now comes the clever bit. The sides of the spears are not straight; they're curved. And the curve obeys a simple rule: the width at any degree of latitude x is cos x times the width at the equator. Let's work that out for increments of 10o and a spear width of 1.25":
|x||cos x||Spear width|
Note that when combining an even number of spears, the longitude in the middle is straight. This is true of any spear, actually; in the first map I simply didn't draw this longitude. A straight longitude is free of distortion, so it's best if you can cut the orange such that a straight longitude goes through the center of each continent. In the sample map I chose to have the straight longitude go throught the 0o line (not marked, but it's the one that passes through Eretald).
I've talked about drawing the spears and then combining them for ease of exposition. In practice, of course, it saves time to decide which spears you are combining ahead of time
You can, if you like, combine all the spears. Here, the length of each latitude line x is just the equator times cos x. The resulting map gets rid of the orange-slice appearance, but the left and right edges do end up pretty stretched out.