Book II — Plane Figures, Regular Polygons, Parquetries & Quadrature¶
Plane Enclosure, Exterior Corners vs. Interior Angles & Quadrilaterals (Figs. 1–8)¶
Having shown in the first book how various lines are to be drawn, I now proceed—as promised at the outset—to planes (planos oder ebne felder), and teach how to construct several of their many figures. To understand what a plane figure is: it is a magnitude bounded and set apart by lines that does not yet enclose a solid body (Corpus). Just as lines enclose a plane, so planes enclose solid bodies.
Now it is known through Euclid (euclid:elements) that two straight lines cannot enclose a plane: if drawn parallel (Barlinien), they remain open at both ends; if inclined toward one another, they meet in a point at one end yet gape open behind (Fig. 1). Two curved lines, by contrast—whether turned with their concavities facing one another to form a lens, or convex against concave like the crescent of the new moon—do enclose a region, as does a curved arc spanning a straight chord (Fig. 2).
Before constructing polygons, one must note the craftsman’s distinction between a corner (Eck) and an angle (Winckel, Fig. 3):
When you look from outside upon the sharp salient point, it is called an exterior corner (eyn eck, marked
.a.).When you look inward into the recessed cavity, it is called an interior angle (ein winckel, marked
.b.).
Angles and corners are of three species—right (recht), obtuse (weyt), and acute
(eng)—constructed immediately by describing a semicircle around the intersection .a. of two
crossed lines (Thales’s theorem, Fig. 4): joining three concyclic points .b.c.d. yields a
true carpenter’s square (ein rechter winckelhacken), while tilting ray .c.b. toward .d.
diminishes one side into an acute angle by the exact amount added to the adjacent obtuse angle.
Two straight lines crossing obliquely always form equal opposite vertical angles (Fig. 5), while
circular arcs and serpentine curves generate curvilinear angles shaped like
boar’s tusks (wie die eber zeen, Fig. 6). At least three straight lines are required to bound a
rectilinear plane, giving equilateral, isosceles, and scalene triangles (Fig. 7), followed by
eight species of four-sided figures (a–h, Fig. 8) headed by the true right square
(rechte Fierung, .a.), oblong rectangle (.b.), rhombus (.c.),
rhomboid (.d.), and four trapezia (.e.–.h.).
Demonstrative vs. Mechanice Regular Polygons & the Rusty Compass (Figs. 9–20)¶
Throughout his polygon constructions, Dürer enforces a remarkably modern epistemological boundary between demonstrative constructions proved with exact Euclidean rigor and rapid mechanice workshop approximations (von behendigkeyt wegen in der arbeyt):
Hexagon, Equilateral Triangle, Square, 8-gon & 16-gon (Figs. 9–10, 13–14 — Demonstrative): Stepping an unchanged compass (unverruckter Zirckel) around a circle of center a marks six exact vertices (1..6), since radius equals chord (darum das die der zirkell vnuerruckt selbs gibt, Euclid IV.15). Joining alternate vertices 1, 3, 5 inscribes the equilateral triangle; perpendicular diameters
b.c.andd.e.with repeated arc bisections yield the square, octagon, and 16-gon.Ptolemy’s Exact Pentagon & Decagon vs. the Workshop Heptagon (Figs. 11 & 15): In Figure 15, Dürer presents Ptolemy's Almagest I.10 construction inside circle a with horizontal diameter b–c and upper pole d: bisecting radius a–c at e (explicitly corrected by Dürer in his 1525 Errata, p. 182!) and swinging arc
e.d.down onto diameter b–c atfproduces both the exact regular decagon sidef.a.and the exact regular pentagon sidef.d.over $\mathbb{Q}(\phi)$. On the very same diagram (and earlier in Fig. 11), Dürer notes that the vertical half-chord erected from midpoint e up to the circle—equivalently half the side of the inscribed equilateral triangle, $s_7 = \frac{\sqrt{3}}{2}R$—steps around the circle seven times (W0, W1, W2, …, W6) mechanice (so hast du ein sibenteil des zirckels Mechanice), generating the 14-gon and 28-gon by arc bisection (Fig. 12).Single-Aperture (“Rusty Compass”) Pentagon (Fig. 16): Immediately after the demonstrative pentagon, Dürer teaches how to construct an equilateral pentagon with an unchanged compass opening (ein fünfeck auß vnuerruckten zirckel zuᵒ machenn, inherited from Abū al-Wafāʾ abulwafa:craftsmen and Matthäus Roriczer's Geometria Deutsch roritzer:1487):
Draw two unit circles through each other’s centers a and b; segment a–b is the base of the pentagon, and their vesica intersections c (top) and d (bottom) define perpendicular axis c–d.
Keeping the compass unchanged, place one foot at d and sweep an arc through a and b, cutting the two circles at e and f and cutting vertical axis c–d at g.
Project straight rays from e through g to cut the right circle at h, and from f through g to cut the left circle at i; finally, step the same unchanged unit compass from i and h to meet at apex k (contrasted with true Euclidean vertices Hphi, Iphi, Kphi governed by golden point phi).
Exact 15-gon, Workshop 9-, 11-, and 13-gons & Arc Trisection (Figs. 17–20):
15-gon (Fig. 17 — Demonstrative): Subtracting the pentagon arc ($72^\circ$) from the equilateral triangle arc ($120^\circ$) leaves $48^\circ$, whose bisection yields $24^circ = frac{360^circ}{15}$ (Euclid IV.16).
Nonagon (Fig. 18 — Mechanice): Three radial Fischblosen (fish bladders) are inscribed in a circle; trisecting the axis
b.a.of the upper lens and drawing a perpendicular cross-chorde.f.through the inner division gives a 9-gon chord accurate to $+0.16\%$.Hendecagon & Tridecagon (Fig. 19 — Explicitly Mechanice): For an 11-gon, take a quarter-diameter ($\frac{1}{2}R$) and lengthen it by one-eighth of itself ($s_{11} = frac{9}{16}R$, “also das es sich Mechanice / aber nit demonstratiue findet”); for a 13-gon, take half the radius ($s_{13} = \frac{1}{2}R$, “ist aber auch mechanice vnd nit demonstratiue”).
Circular Arc Trisection (Fig. 20): Trisecting chord
a.b.combined with an iterative compass refinement cuts any circular arc into three nearly indistinguishable parts (“wer es will geneuer haben / der such es demonstratiue”).
Gothic Tracery, Planar Tilings & Pentagonal Parquetries (Figs. 21–27)¶
Moving from isolated polygons to ornament and architectural pavements (estrichen vnd stuben tillen), Dürer investigates how congruent circles and regular polygons fill the plane or leave structured interstitial stars:
Square vs. Hexagonal Circle Packings & Tracery Stars (Fig. 21): Circles arranged in a $3 \times 3$ square grid touch four neighbors leaving concave four-cusped voids, whereas circles centered on a triangular rhombic lattice pack so that six circles always touch the seventh (so rüren alweg .6. den sybetten an). Intersecting 12-fold circles and 5- and 7-pointed arc rosettes generate Gothic window tracery (seltzams maswerch).
Regular, Archimedean & Pentagonal Parquetries (Figs. 22–26): After showing how equilateral triangles, squares (both orthogonal and staggered like ashlar masonry), rhombi (forming illusory isometric cubes and 5-, 6-, and 8-pointed stars), and regular hexagons tile the plane gaplessly, Dürer devises four extraordinary planar assemblies of regular pentagons (Fig. 24, p. 066)— pairing antiparallel rows of pentagons and fivefold pentagonal rosettes (fünf rosen van den fünf ecken) separated by narrow and wide $\mathbb{Q}(\phi)$ golden rhombi! He then tests heptagon lattices with 4-pointed star interstices (observing explicitly on p. 068 that closed rings of edge-joined 7-gons or 5-gons overlap or fail to close!) and constructs the semiregular 4.8.8 octagon-and-square pavement (Fig. 26)—the direct fountainhead of Johannes Kepler's <kepler> Harmonices Mundi Book II (kepler:1619).
Euclidean Area Transformations & Circle Quadrature (Figs. 28–36)¶
The closing movement of Book II treats comparison and transformation of plane areas (vergleichung der ebnen):
Squaring Triangles, Rectangles & Regular Polygons (Figs. 28, 32–33 — Demonstrative): Bisecting any triangle converts it into an equal-area rectangle; placing the rectangle’s long and short sides end-to-end as diameter
e.f.g.of a semicircle and erecting perpendicularf.i.constructs the exact geometric mean $f i = \sqrt{e f \cdot f g}$ (Euclid II.14 & VI.13), yielding a square equal in area to the triangle or to any $n$-gon decomposed into radial triangles (Fig. 33).Homothetic Scaling & Area Multiplication (Figs. 29–31, 35): Extending the diagonal (Ortstrich) of a rectangle preserves aspect ratio under arbitrary rescaling (Euclid VI.24, Fig. 29); combining semicircle mean proportionals with the diagonal multiplies the area of any rectangle or square twofold, threefold, or sevenfold while preserving shape (Figs. 30–31), and generalizes the Pythagorean theorem from squares to equilateral triangles or any similar figures erected on a right triangle’s sides (Euclid VI.31, Fig. 35).
Workshop Quadrature of the Circle (Fig. 34, p. 075): On squaring the circle (Quadratura circuli), Dürer states with exemplary scientific candor: “One ought also to know quadratura circuli—that is, equating a circle and a square so that one contains as much area as the other—yet this has not yet been demonstrated by scholars (*noch nit von den gelerten demonstrirt). Mechanice, however—that is, approximately, so that in practical work it misses not at all or only by very little—this equality may be made thus: draw a square, divide its diagonal (ortstrich) into ten equal parts, and draw a circle whose diameter takes eight of those ten parts”* (centered at Oq alongside the heptagon in the figure below).