In the column in which I described the approximate construction of the regular heptagon from an equilateral triangle and a regular pentagon (DRAWING 3) and demonstrated the level of accuracy of the approximation (DRAWING 4), I failed to reiterate the fact that one should begin with an exact regular pentagon. Such an explicit statement would have been helpful since earlier in the column I described an approximate construction of the regular pentagon (DRAWING 1). Moreover, I failed to state explicitly that the construction of DRAWING 1 is, in fact, approximate. Finally, rather than stating that this approximate construction "is credited to Albrecht Dürer", I should have written that it "is often associated with Albrecht Dürer", but in fact, no one knows for certain where the pentagon/vesica construction originally comes from.

The following three points deserve mention:

1. Dürer didn't originate the vesica/pentagon construction. In the book Geometrica Deutsch, probably published by Roriczer ca. 1472-1484, we find the construction, placing its existence about 50 years prior to Dürer's 1525 publication of Underweysung der Messing mit Zirkel und Richtscheit (Treatise on Mensuration with Compass and Ruler), in which this drawing can be found (I am indebted to Professor John Sharp for this observation);

2. To reiterate, the construction often credited to Dürer is not 100% accurate, and in order to assure success when doing the complete series of drawings, it is advisable to begin with the best possible pentagon. In practice, of course, no drawing is 100% accurate in as much as humans are not perfect. As a matter of fact, the approximate construction of the pentagon is so close that its difference from an exact pentagon is imperceptible in practice. Perhaps that is why Renaissance greats like Dürer and Daniele Barbaro cite this approximate construction in their writings (it appears in Daniele Barbaro's La pratica della perspettiva of 1569).

No one knows for certain from the source of the approximate construction, or when it was first done, although we are in agreement that it was probably a Masonic procedure that was not in the public domain. Certainly we have earlier references to the pentagon and its uses. R. Merlet, for example, in Dignitaires de l'englise Notre-Dame de Chartres, cites the star-pentagon in the seal of Simon de Beron, canon of Chartres, in 1209, but there is no document we know of to say, unequivocally, where Simon received knowledge of its construction. It would appear that any procedural text on pentagon construction is lacking conclusive scholarly proof. Euclid mentions it, especially in detail, in Book XIII of the Elements, with the dodecahedron, but as far as I can find, he lays out no true steps for construction. The impression is that its construction was commonly known at least to scholars and geometers. In Euclid IV:10, reference to the triangles within the pentagon are discussed, yet Heath refers the reader to Pythagorean sources, and refers also to Plato by referencing Proclus, for deeper discussions on the pentagon itself. If we then go back to Pythagoras, then the secrecy level becomes extreme. Either that, or the censorship and destruction of Pythagorean thought after his death were paramount to eradicating his philosophy. I guess all we can surmise with relative comfort (?) is that there may have been an initial link with the pentagon and division in extreme and mean ratio. As we know, there may be a long standing problem regarding a proof for when the golden number first issued from civilization.

Please note that the figures given in the proof are precise, and the proof is correct as it stands.

The approximate construction of the regular pentagon was shown for historical interest—and to show the efforts put forth by artists of the Renaissance to verse themselves in the methods of geometric constructions. In my column I also mention that other pentagon constructions as well as the one shown are available for use. The rule of thumb when doing any geometric construction is to follow the proper tradition of working from Euclid's procedures and constructions whenever possible. At times, other masters like Archimedes, Apollonius, Theon, Ptolemy, and Kepler can be studied for constructions, some of which were unknown to Euclid. (Karl Friedrich Gauss's regular 17-gon is one splendid example.) Most fundamental constructions do come from Euclidean and other classical traditions, and they can include Greek and Arabic sources. (It is advisable to have Euclid's Elements in your library, and to have Sir Heath's volumes on Greek Mathematics accessible as well. Arabic references should include Alhazen and Ibn-Sina, known to us as Avicenna.)

In the classical tradition, only compasses and straightedge with no measures could be used. For our purposes in doing a study or practice drawing involving regular polygons, it is acceptable to lay off vertical and horizontal axes that cross at 90° precisely, and to place the calibrated protractor onto these axes to mark off the required angles. In doing my studies for the heptagon generation, I employed five different methods for the initial master pentagon construction to add validity of foundation to the accuracy of the final drawing. In this technological age, I would imagine that AutoCAD constructions can be acceptable as well, but note that I actually saw an error last year in an AutoCAD rendering of a hexagonal floral tessellation where lines did not line up the end of the design with the beginning because of one pixel used by the program instead of another. The error was very small, being only one pixel off, but the line did have a blip, and it could be seen.

In all cases, it is necessary to check each and every construction for accuracy by reviewing proper procedures, and by the use of the dividers and the compasses. Draw very lightly (and usually with a 4H lead) at the outset, and only after verifying the precision of the construction should a sharpened F or HB lead be applied firmly- not with force or pressure - to "tighten up" the work.