One Turn Is a Gnomon: the Golden Cycle Step by Step, and the Polar of the Keyed Pair

In short. Take a square of 3 × 3 dots and add an L-shaped strip of 3 + 4 dots along two sides: you get the next square, 4 × 4. The strip is a gnomon: in the classical definition (Euclid, and Heron in general), the piece which, added to a figure, makes a whole of the same shape as the figure, only larger. The gnomon method as reported for Júlia Pimentel works with exactly this strip, n + (n + 1). DICTO's idea was to carry that structure into three dimensions: walk through his golden cycle of twelve nested solids, from an icosahedron out to the next icosahedron φ³ times larger, and check at every step what it adds and whether the new whole has the shape of something before it. We built every step exactly. The answer is clean. The whole turn is a classical gnomon: the next icosahedron has the same shape as the first. No single step and no shorter run of steps is: no station has the shape of any other station, except the two icosahedra. So it is the last step, from M to the next icosahedron, that closes the gnomon. Step by step, the cycle adds spikes, pyramids, Shields and other pieces that never repeat their shapes. By volume, four steps still add a clean golden amount. DICTO's second idea concerned his keyed pair of space-filling pieces, the Lotus Jewel Axes (eight dents) and the Dodecaxes (eight keys). A dented or keyed solid has no ordinary polar, so DICTO's method is to fill the dents in (or take the keys off), take the polar of the smooth solid, and keep what was removed as the remainder. The smooth solids are the golden pair of paper #48; their polars are cubes with a low pyramid on every face, of heights 1/φ and 1/φ², which add up to exactly 1, the height that makes Kepler's rhombic dodecahedron; and the remainder is the same eight Tricaps for both pieces. Both parts read a solid by the piece that lies between it and its neighbour. Theorem A (the classical gnomon of the golden cycle). In the cycle I ⊂ Cleo star ⊂ D′ ⊂ ID ⊂ OG ⊂ rTI ⊂ TI ⊂ H ⊂ jtI ⊂ RT ⊂ M ⊂ I·φ³ the only two similar stations are I and I·φ³ (ratio φ³). Hence one whole turn, I·φ³ minus I, is a classical gnomon of I; no single step and no shorter run of steps is a classical gnomon; and, within the turn from I, the last step, M → I·φ³, is the only step whose new whole is similar to an earlier station. Table 4 lists what each step adds. Corollary (volumes). Every station is an exact multiple a + b√5 of I; Cleo star = 2·I, RT = 81·I/φ², M = φ⁸·I, I·φ³ = φ·M; exactly four steps (1, 6, 7, 11) add a rational multiple of φ k ·I. Theorem B (the keyed pair, hull and remainder). The Lotus Jewel Axes and the Dodecaxes are the Icosaxes and the Unborn Axes minus and plus the same eight Tricaps, each of volume 1/(12φ²). The polar of the Icosaxes is a cube with six face pyramids of height 1/φ (in half cube edges), that of the Unborn Axes one with height 1/φ²; along the whole factor-x family of the two heights are x/(1 + x) and 1/(1 + x), so they always add up to 1, the rhombic dodecahedron's height. A polar of the Lotus Jewel Axes taken face by face about one centre would put spikes over the eight dents. All results are exact (119 checks, none failed). The ideas are DICTO's: carrying the "add the next strip" structure of the gnomon method as reported for Júlia Pimentel, n² + n + (n + 1) = (n + 1)², into his three-dimensional golden cycle, and judging each step by the shape it adds as well as by its volume; the names Julias and Pimentels are DICTO's; the hull-and-remainder polar of the keyed pair, "fill them in, maybe invert, then work out the remainder" (DICTO, 24 October 2026); and, from his earlier papers, the twelve-station golden cycle, the Cleo star, the DICTONIC SOLIDS and their names. Computation, exact proofs and drafting with Claude (Anthropic). In short. Take a square of 3 × 3 dots and add an L-shaped strip of 3 + 4 dots along two sides: you get the next square, 4 × 4. The strip is a gnomon: in the classical definition (Euclid, and Heron in general), the piece which, added to a figure, makes a whole of the same shape as the figure, only larger. The gnomon method as reported for Júlia Pimentel works with exactly this strip, n + (n + 1). DICTO's idea was to carry that structure into three dimensions: walk through his golden cycle of twelve nested solids, from an icosahedron out to the next icosahedron φ³ times larger, and check at every step what it adds and whether the new whole has the shape of something before it. We built every step exactly. The answer is clean. The whole turn is a classical gnomon: the next icosahedron has the same shape as the first. No single step and no shorter run of steps is: no station has the shape of any other station, except the two icosahedra. So it is the last step, from M to the next icosahedron, that closes the gnomon. Step by step, the cycle adds spikes, pyramids, Shields and other pieces that never repeat their shapes. By volume, four steps still add a clean golden amount. DICTO's second idea concerned his keyed pair of space-filling pieces, the Lotus Jewel Axes (eight dents) and the Dodecaxes (eight keys). A dented or keyed solid has no ordinary polar, so DICTO's method is to fill the dents in (or take the keys off), take the polar of the smooth solid, and keep what was removed as the remainder. The smooth solids are the golden pair of paper #48; their polars are cubes with a low pyramid on every face, of heights 1/φ and 1/φ², which add up to exactly 1, the height that makes Kepler's rhombic dodecahedron; and the remainder is the same eight Tricaps for both pieces. Both parts read a solid by the piece that lies between it and its neighbour. Theorem A (the classical gnomon of the golden cycle). In the cycle I ⊂ Cleo star ⊂ D′ ⊂ ID ⊂ OG ⊂ rTI ⊂ TI ⊂ H ⊂ jtI ⊂ RT ⊂ M ⊂ I·φ³ the only two similar stations are I and I·φ³ (ratio φ³). Hence one whole turn, I·φ³ minus I, is a classical gnomon of I; no single step and no shorter run of steps is a classical gnomon; and, within the turn from I, the last step, M → I·φ³, is the only step whose new whole is similar to an earlier station. Table 4 lists what each step adds. Corollary (volumes). Every station is an exact multiple a + b√5 of I; Cleo star = 2·I, RT = 81·I/φ², M = φ⁸·I, I·φ³ = φ·M; exactly four steps (1, 6, 7, 11) add a rational multiple of φ k ·I. Theorem B (the keyed pair, hull and remainder). The Lotus Jewel Axes and the Dodecaxes are the Icosaxes and the Unborn Axes minus and plus the same eight Tricaps, each of volume 1/(12φ²). The polar of the Icosaxes is a cube with six face pyramids of height 1/φ (in half cube edges), that of the Unborn Axes one with height 1/φ²; along the whole factor-x family of the two heights are x/(1 + x) and 1/(1 + x), so they always add up to 1, the rhombic dodecahedron's height. A polar of the Lotus Jewel Axes taken face by face about one centre would put spikes over the eight dents. All results are exact (119 checks, none failed). 1. Where it started, and why one paper Both ideas came on 24 and 25 October 2026, and both look at the piece in between: what a step of the cycle adds, and the eight small pieces that a key or a dent moves from one solid to the other. Exact studies and checks attached (commit ad13a42). Licence CC BY-SA 4.0. More at dictospheres.com.