Sacred Geometry exercise no. 1

Constructing a Pentagram

This method was devised by 16th Century artist and geometer Albrecht Dürer, and is by far the easiest and most elegant method of drawing a pentagram. This is a classic construction uniting the pentagon and hexagon, and is accomplished in the same way as the Creator traditionally designed the Universe, “with the opening of the compasses unchanged”. Dürer used this geometry as the basis for several of his pictures, most notably Melancholia (left).

Requires only compasses and straightedge.

Step 1: Draw a horizontal line. This is not strictly necessary, but it helps to keep things square to the page. Step 2: Set your compasses to the length you want the base line of the pentagon to be; a-b in this case. Step 3: Place point of compasses at point a and point b in turn and draw two overlapping circles, the centre of each on the circumference of the other. This is the essential sacred geometric form called the Vesica Piscis. Step 4: Bisect the vesica by drawing a vertical line c-d connecting the points where the two circles cross. Step 5: Without changing the setting of your compasses, put the point at the bottom of the vesica where the two circles cross at point c, and draw another complete circle. You now have three overlapping circles, two side-by-side and one lower.

Step 6: With your straightedge, draw two lines e-f and g-h diagonally upwards from the points where the lower circle cuts the two circles of the vesica, passing through the point where the lower circle cuts the vertical centre line (point o). Extend these lines until they touch the upper edges of the two vesica circles at f and h. These two points will be the side 'corners' of your pentagon.

Step 7: Still without altering your compass setting, place the point in turn on each of these two points f and h that we've just established. Draw overlapping short arcs above and between these points to define our top corner of the pentagon at point i.

Step 8: Connect this last point with the previous two and then connect those to the original base points to make the pentagon a-b-f-i-h .

Step 9: Finally, draw in the internal diagonals of the pentagon to get your pentagram! You can erase the construction lines if you like, but it is traditional in sacred geometry that you leave them visible so that others can see the method of construction and be impressed with your geometric skills.

For an encore, you can continue with the drawing to produce the hexagon/hexagram. This is very easily done; still with your compasses setting unchanged, "step off" short arcs from points e and g to give points j and k. Do the same from these points to check that you have everything accurate. This "stepping off" technique is well-known to most people as a method of making a six-petalled flower shape.

Now draw in your hexagon a-b-g-j-k-e, and then its a simple matter to connect the diagonals to make the hexagram star. Congratulations! This little exercise has profound significance in sacred geometry as it encapsulates the union of five and six, life and regeneration unfolding into order and structure. This basic form can be expanded in many interesting ways to produce area-filling tiling patterns such as those found is Islamic art for instance. It can also be used to enclose three-dimensional space in the form of a geodesic dome, which is constructed from these basic hexagons and pentagons.

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© Grahame Gardner 2005