All regular polygons (equilateral triangles, squares, octagons, etc.) and all of the figures on this page can be constructed with a compass and straightedge.
To construct a vesica piscis, draw a circle, then move the needle point of the compass to a point on the circle's perimeter and draw another circle with the same radius. The mandorla is the perimeter of the overlap. Equilateral triangles can be inscribed in the upper and lower halves of a mandorla.
Vesica piscis and mandorla construction, with equilateral triangles.
A golden rectangle can be divided into a golden rectangle and a square, to infinity. To draw a golden rectangle, first construct a square. Place the needle point of the compass on the midpoint of one of the sides (A). Place the pencil lead on an opposite corner (B). Draw an arc. It defines the height of the golden rectangle (C).
Golden rectangle construction.
A golden spiral is a line that curves exponentially at the rate of 1:φ (1:1.618..., the golden ratio). To draw it, divide a golden rectangle into a square and a golden rectangle. Then divide that golden rectangle into a square and a golden rectangle. Repeat for as many times you desire. Then draw a line through this spiraling construction that becomes smaller (or larger) at a constant rate. This is done by placing the needle point of the compass at the inside corner of each square and drawing an arc through it. (In other words, you have to keep moving the needle point.)
Golden spiral construction.
To construct a golden triangle, first construct a golden rectangle. Place the needle point of the compass at one corner of one of the rectangle's longer sides. Place the pencil lead on the other corner of that side and draw an arc. Repeat on the other longer side. Then draw lines from the point where the two arcs intersect to the corners of the opposite short side. (You are essentially removing one of the rectangle's short sides and turning it into a three-sided figure. The lengths of the sides remain the same.)
Golden triangle construction.
Because a golden triangle's base-to-leg ratio is 1:φ, it can be divided ad infinitum into a golden triangle and another geometric figure — a triangle called the golden gnomon. A golden triangle can be used to construct a golden spiral.
Golden spiral construction using a golden triangle.