Independent Dance

Vajrayana

Progenitors (DICTO DOI: 10.5281/zenodo.23413184) set the DICTO Ogdoecontahedron O into the Spheres of Dharma in the dodecahedron's place and found an exact nine-station golden cycle, I ⊂ ID ⊂ O ⊂ rTI ⊂ TI ⊂ H ⊂ jtI ⊂ RT ⊂ I·φ³, closing on φ⁹ in volume. This companion asks where t...

The DICTO Metaxyhexecontahedron

The DICTO Metaxyhexecontahedron is the convex hull of a regular icosahedron and a dodecahedron placed over its faces, at the one size where the icosahedron edge is exactly 9/φ⁴ times the dodecahedron edge: in closed form the hull of (0, ±1, ±φ) and (φ⁵/9)·{(±1, ±1, ±1), (0, ±φ, ±...

The DICTO Palintriacontahedron

The DICTO Palintriacontahedron is the hull of the icosidodecahedron's central pyramids flipped out at full height: 32 corners and 60 triangles, each half a rhombic-dodecahedron rhombus. It is the member of the dodecahedron–icosahedron hull family with edge ratio exactly 3/√5, in...

Double Vision – Both Cycles in One Space

The Spheres of Dharma nest six icosahedral solids, with the Hecatodihedron as an in-between stage; their polar reciprocal is a second exact cycle in which every nesting is reversed (Inverted Shells). We put both cycles in one space. The sphere of reciprocation is then the one fre...

The DICTO Ogdoecontahedron

DICTO asked what the Spheres of Dharma, an exact endless nesting of six icosahedral solids, look like turned inside out. We include the cycle's in-between stage, the DICTO Truncated-Icosahedral Hecatodihedron, as a seventh station and take every station's polar reciprocal: the ne...

Inverted Shells: The Spheres of Dharma Turned Inside Out

The Spheres of Dharma nest six icosahedral solids exactly, from an icosahedron out to the icosahedron φ³ larger; between the truncated icosahedron and the triacontahedron, the layer closes part-way into the DICTO Hecatodihedron. DICTO asked what the cycle looks like turned inside...

The DICTO Hexa: a cube of 8 DICTO Hexa Jewels, the Hexa-Key, and their 5 : 3 checkerboard filling of space

The DICTO Jewel is the regular dodecahedron of the Euclid–Kepler–Pacioli cell carved by the stella octangulas of its six neighbouring cells (volume 12 at cube edge 2); with stella octangulas it fills space. Eight DICTO Hexa Jewels (exact DICTO Jewels on a 2 × 2 × 2 block of cells...

Kagome Hulls: six corner-sharing networks of solid clusters in two space-filling lattices, one rule, and its icosahedral exception

Two space-filling lattices share the same rock-salt arrangement of cubic cells: the Stella–Jewel Lattice (DICTO Jewels on the even cells, stella octangulas on the odd) and the Sunstar Lattice (regular dodecahedra on the even cells, Dogstars on the odd). Grouping even cells into s...

The DICTO Dodeca-13: the 13-dodecahedron cluster made solid, its gaps as two exact pieces, and separable finned units

A regular dodecahedron with a regular dodecahedron on each of its 12 faces is unique (each outer one is the centre's mirror image across the shared face) and is the first shell of the 120-cell laid flat. The 12 outer dodecahedra never overlap; they touch only along the centre's 3...

A Fibonacci Spine in a Five-fold Chorus of Ogdoecontahedron Packings

The DICTO Ogdoecontahedron is a convex, centrally symmetric icosahedral solid with 20 regular hexagons and 60 kites. Its hexagon directions contain five inscribed cubes, and each cube gives a periodic (body-centred cubic) packing in which copies meet face to face on hexagons. We...

The Euclid–Kepler–Pacioli Cell: all five Platonic solids on 13 points in one periodic cell, and a corner-sharing network of Kepler stars and icosahedra

Place Euclid's six roofs on a cube of edge 2 to obtain the regular dodecahedron, then reflect the twelve roof vertices back through the cube faces: they land exactly on a regular icosahedron inside the cube. Repeating the cube by translations of 2 gives a periodic structure, whic...

The DICTO-Star: an icosidodecahedron, 12 regular dodecahedra and 20 tridiminished icosahedra closing exactly in ordinary space

On each of the 12 pentagons of an icosidodecahedron place a regular dodecahedron, and on each of its 20 triangles a tridiminished icosahedron (Johnson solid J63), attached by its one triangle that borders only pentagons. The 33 regular-faced pieces (edge 1) meet only in whole fac...

Theaetetus: an icosahedron held exactly by its Cleo-Icos, grown to the DICTO Hecatodihedron in two exact stories

We hollow the DICTO-ICOSA star (the DICTO-Star with a pyramid on each of its 152 free faces) and find that a core icosahedron resting on twelve inner tips is loose: it turns about 14° before it touches anything. DICTO's remedy is to hold it with Cleo-Icos (an icosahedron's face p...

DICTO ATOM-13×13: thirteen clusters of thirteen, twelve around one, six golden and seven periodic, with the pieces that fill between them

Twelve equal spheres can touch a thirteenth and no more (Schütte and van der Waerden 1952), and twelve atoms round one make the icosahedral cluster of liquids and metals (Frank 1952). We set thirteen polyhedral thirteens side by side, each a centre and twelve around it. Six are g...

DICTO: Collected Geometric Works

DICTO is a geometric sculptor in Osaka whose work grows out of building with recycled PET and from three open apps on one shared geometry: Kaleidohedra, Rhombiverse and Polyhedraverse. This record gathers DICTO's published geometry under one DOI. Each work has its own DOI, always...

Capturing the In-betweens: a Study of Transitional Polyhedral Shells

DICTO's Chain of Karma, Spheres of Dharma and Bodhi Seed are exact icosahedral dissections: each grows a large polyhedron from a small core by adding rings of congruent pieces. Between the familiar solids at the start and end of each step lie the in-between stages, the transition...

The Chain of Karma: the icosahedron and the dodecahedron as endless shells of each other

We show that the regular icosahedron and the regular dodecahedron can each be filled exactly as a shell around the other, and that the two shells alternate without end. A dodecahedron of edge 1 is an icosahedron of edge 1/φ inside a shell of 12 pentagonal pyramids, 30 AXE and 20...

Spheres of Dharma: a second endless golden cycle of the icosahedral solids, through the icosidodecahedron, the fullerene cage and the rhombic triacontahedron

We describe an exact, endlessly repeating nesting of six icosahedral solids (the Ritsudō cycle), each layer filled by a few repeated pieces. With the dodecahedron at edge 1: an icosahedron of edge 1/φ, an icosidodecahedron of edge φ/2 (around it 12 pentagonal pyramids and 20 twis...

Quantitative Bridge or Black Box? A Practice-Based Framework for Statistical Programming Contributions to Population Pharmacokinetic/ Pharmacodynamic Modeling in Regulatory Drug Development

Population pharmacokinetic/pharmacodynamic (PK/PD) modeling now sits at the center of model-informed drug development (MIDD), shaping dose selection, labeling language, and regulatory submissions across therapeutic areas. Less attention goes to the statistical and clinical progra...

A Rule on the Sheet: Price and Date as Sanctioned Properties of the Artwork

Recent philosophy of contemporary art holds that artists fix the properties of their works through public acts of sanction, and that rules are a medium in which such sanction is exercised — rules for how a work is displayed, conserved, and participated in. Price has not been trea...