Book IV — Regular & Archimedean Nets, Cube Duplication & Perspective

[Folio N1525 p. 142–157]

First Printed Planar Fold-Out Nets for Platonic & Archimedean Solids (Figs. 29–43)

In the opening of Book IV, Dürer classifies three families of solid bodies that may be fashioned with compass and straightedge: prismatic columns of uniform cross-section, tapering cones and pyramids, and finally bodies equal on every side in their faces, solid corners, and edges, “which :person:`Euclid <euclid>` calls :term:`corpora regularia <corpora-regularia>`, describing five of them because there can be no more that may be inscribed in a sphere touching it at every vertex” (euclid:elements, plato:timaeus).

Whereas Luca Pacioli and Leonardo da Vinci had depicted spatial perspective wireframes of polyhedra in Divina Proportione (pacioli:1509), Dürer invents a transformative medium in the history of mathematics: for every solid, alongside its closed ground plan and elevation, he prints the exact two-dimensional unfolded planar net (gantz aufgethan / offen) with explicit craft instructions on page 148:

“Whoever wishes to make them should draw them larger upon two sheets of paper pasted together (zwifach gepabt papier), and with a sharp knife score along all the drawn lines through one ply of the paper on one side; and once the whole figure is cut free from the remaining paper, fold the body together—it bends willingly along the scored lines.”

1. The Five Platonic Solids & the Gore Sphere (pp. 142–147, Figs. 29–34):

   • Tetrahedron (Fig. 29): $4$ equilateral triangular faces, $4$ three-valent solid corners, and $6$ sharp edges.

   • Octahedron (Fig. 30): “shaped like a diamond point” (wie ein diamant punct)—$8$ equilateral triangles, $6$ four-valent corners, and $12$ edges.

   • Icosahedron (Fig. 31): $20$ equilateral triangles, $12$ five-valent corners, and $30$ edges.

   • Cube (Fig. 32): “like a four-cornered die” (wie ein vierecketer würffel)—$6$ right squares, $8$ three-valent corners, and $12$ edges.

   • Dodecahedron (Fig. 33): $12$ regular pentagonal faces unfolded into two joined six-pentagon rosettes over $\mathbb{Q}(\phi)$, with $20$ three-valent corners and $30$ edges.

   • Meridian Gore Sphere (Fig. 34): A sphere sliced along its meridians and laid flat “acquires the form of a comb” (ein gestal eins kams)—the mathematical template of Renaissance printed terrestrial and celestial globes.

Exact unit-quadrance ($Q = 1$) planar fold-out nets for the Tetrahedron (Fig. 29), Octahedron (Fig. 30), and Cube (Fig. 32), distinguishing interior scored fold hinges (dashed) from outer cut seams (solid).

Dürer's double six-pentagon rosette net for the regular Dodecahedron (Book IV, Fig. 33) constructed with 100% exact symbolic coordinates in $\mathbb{Q}(\sqrt{5}, \sin 36^\circ)$.

2. Seven Archimedean Semi-Regular Solids & Composite Nets (pp. 148–157, Figs. 35–43): Proceeding to solids bounded by two or three species of regular polygons whose vertices all touch a circumscribed hollow sphere (Archimedes, Pappus pappus:synagoge), Dürer publishes planar nets for seven Archimedean solids:

   • Truncated Tetrahedron (Fig. 35): $4$ hexagons and $4$ triangles ($12$ vertices, $18$ edges).

   • Truncated Cube (Fig. 36): $6$ octagons and $8$ triangles ($24$ vertices, $36$ edges).

   • Cuboctahedron (Fig. 37): $6$ squares and $8$ triangles ($12$ vertices, $24$ edges).

   • Truncated Octahedron (Fig. 38): $8$ hexagons and $6$ squares ($24$ vertices, $36$ edges).

   • Rhombicuboctahedron (Fig. 39): $18$ squares and $8$ triangles ($24$ vertices, $48$ edges).

   • Snub Cube (Fig. 40): $6$ squares and $32$ equilateral triangles ($24$ vertices, $60$ edges!).

   • Truncated Cuboctahedron (Fig. 41): $6$ octagons, $8$ hexagons, and $12$ squares ($48$ vertices, $72$ edges!).

Exact unit-quadrance planar nets over $\mathbb{Q}(\sqrt{3})$ for the Platonic Icosahedron (Fig. 31, left, $20$ triangles) and the Archimedean Cuboctahedron (Fig. 37, right, $6$ squares + $8$ triangles).

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[Folio N1525 p. 157–166]

The Delian Altar: Doubling the Cube & Cannonball Scaling (Figs. 44–51)

Recounting how the citizens of plague-stricken Athens consulted Apollo’s oracle and were commanded to double his cubical altar, Dürer explains how Plato reduced cube duplication to finding two mean proportionals ($g : c = c : d = d : e \implies c^3 = g^2 e$) between two unequal segments. Declaring that “because this is a most useful art serving all craftsmen, yet kept in great secrecy and concealment by scholars, I wish to bring it into the light of day—for as I believe, it has never until this day been described in the German tongue,” Dürer presents three classical constructions and applies them directly to casting proportional bells and iron cannonballs (püxsen kugeln) of $1, 2, 3, \dots, 100$ pounds:

1. Sporus / Pappus Movable-Scale Neusis (pp. 157–163, Figs. 44–49): Pivoting a symmetrically graduated straightedge at d until segments g.h. and h.i. across vertical axis a.b.c. become equal constructs $\sqrt[3]{2}$, $\sqrt[3]{3}$, and $\sqrt[3]{4}$, paired with linear scaling gauges for iterative doubling and halving of any given cube.

2. Plato’s Mechanical Carpenter’s Square (pp. 163–164, Fig. 50): Fitting a grooved Winckelmeß r.p.q. with a sliding perpendicular cross-bar t.s. realizes two coupled right triangles g.c.d. and c.d.e., proved rigorously via Euclid VI.8 ($gb : cb = cb : db = db : eb$).

3. Hero of Alexandria’s Rectangular Neusis (pp. 164–165, Fig. 51): Pivoting a straightedge through corner b of rectangle a.b.g.d. until distances e.h. and e.z. from center e are equal yields two mean proportionals a.h. and g.z., proved step-by-step via Euclid VI.2, VI.15, and VI.4.

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[Folio N1525 p. 166–182]

Cube Sciagraphy & Central Projection Perspective Apparatuses (Figs. 52–63)

Book IV culminates in the geometric synthesis of light, shadow, and central perspective (Perspectiva = Durchsehung):

1. Two-View Cube Shadow Sciagraphy (pp. 166–169, Figs. 52–53): Placing a cube on square ground e.f.g.h. in both horizontal ground plan (Cubus im Grund, illuminated by plan light p) and upright elevation (Cubus aufrecht, illuminated by elevation light o), Dürer intersects elevation shadow-range verticals i, k with plan shadow rays through corners b2, c3, d4 to determine the exact six-sided shadow polygon b2–l–m–n–d4—reminding the reader that “both plans and both lights are one single thing, split apart solely for ease of execution.”

2. Transparent Picture Plane & Two Perspective Constructions (pp. 169–178, Figs. 54–62): Sight proceeds exclusively along rectilinear rays (Streimlinien) from the eye point; inserting a transparent planar section (ebne durchsichtige Abschneydung, Alberti's intersegazione alberti:1435) across the cone of rays captures the exact perspective image. Dürer demonstrates two rigorous constructions of the illuminated cube:

   • First Method (pp. 170–175, Figs. 56–58): Coordinate transfer (Abstelen) with two compasses from split elevation and ground-plan eye rays onto orthogonal crosshairs centered at the principal vanishing point.

   • Second Shorter Method (pp. 175–178, Figs. 59–62): Direct planar construction on the picture sheet using the principal vanishing point (Vx), a lateral distance point at equal height, and the $45^\circ$ ground diagonal e.g.—projecting front face A'–B'–C'–D', receding back face E'–F'–G'–H', and foreshortened depth midpoint M'.

Exact central projection of Dürer's cube onto the glass picture frame (Book IV, Figs. 56–63): receding depth edges (:g:`A' `–:g:`E' `, :g:`B' `–:g:`F' `, :g:`C' `–:g:`G' `, :g:`D' `–:g:`H' `) converge at principal vanishing point :g:`V_x` $= (3, 5)$, preserving the harmonic cross-ratio $\operatorname{CR}(A', E'; M', V_x) = -1$.

3. The Glass Picture Frame & Three-Thread Lute Instruments (pp. 178–181, Figs. 62–63): To make central projection tangible to every painter, Dürer concludes with two celebrated mechanical instruments: (a) a hinged glass picture frame paired with a lead-screw and notched peep-sight rod (gedreter stab) holding the artist’s eye fixed while tracing a portrait directly onto glass; and (b) the three-thread sighting apparatus (durch drey feden), in which an assistant stretches a weighted string from a wall needle (“the eye”) through an open frame to successive points on a foreshortened lute, while the master marks each ray’s intersection with wax-affixed cross-threads and swings shut a hinged panel door to prick the exact perspective coordinates!