Star prisms, star antiprisms, crossed antiprisms and their reciprocal duals Six adjustable connected families, in addition to the existing four convex families. Display n = 5–200. Choose integer n,d with gcd(n,d)=1 and 2 <= d < n/2. The base is the regular star polygon {n/d}. Unreduced fractions would give compound polygons and are deliberately excluded here. The formulas continue for arbitrarily large admissible integers beyond the display limit. Families and duals: star prism <-> star dipyramid (congruent isosceles triangles) star antiprism <-> star trapezohedron (congruent kites) crossed antiprism <-> concave trapezohedron (congruent darts) Crossed antiprisms additionally require n/3 < d < n/2, equivalently 2 < n/d < 3. Both the crossed form and its dual retain the base label {n/d}; its triangle connections use retrograde step n-d. Literature also describes this as the {n/(n-d)} antiprism. For example a pentagram {5/2} is the base of both ordinary and crossed antiprisms; the latter uses retrograde 5/3. Zero-height and imaginary-height configurations are not displayed. Primary mathematical references: George W. Hart, Virtual Polyhedra, Uniform Polyhedra, section on prisms, antiprisms and crossed antiprisms: https://georgehart.com/virtual-polyhedra/uniform-info.html Branko Grünbaum, Uniform Polyhedrals, section 2, pp. 3–4: https://faculty.washington.edu/moishe/branko/Uniform%20polyhedrals/Uniform%20polyhedrals%20source/Uniform%20polyhedrals.pdf Antiprism, polygon generator, including the distinct 5/2 and 5/3 examples: https://antiprism.com/programs/polygon.html Original runtime construction (unit uniform edge before fitting the viewer): r = 1/(2 sin(pi*d/n)). Prisms: aligned copies at z=+/-1/2, with corresponding edges joined by squares. Ordinary antiprisms: twist pi*d/n and height H = sqrt(1 - 1/[4 cos²(pi*d/(2n))]). Crossed antiprisms: step q=n-d, twist pi*q/n and height H = sqrt(1 - 1/[4 cos²(pi*q/(2n))]). The latter is real and positive exactly when n/3 < d < n/2. All base vertices and complete faces are retained, including star winding. Uniform counts: prism V=2n,E=3n,F=n+2; antiprisms V=2n,E=4n,F=2n+2. Duals are true geometric reciprocals: the plane a.x=b maps to a/b about the origin. Their complete face loops follow the original edge/face incidence cycles. Angular sorting is not used: it would erase the concavity of darts. All reciprocal vertices are finite in the admitted parameter range. Dual counts: dipyramids V=n+2,E=3n,F=2n; trapezohedra V=2n+2,E=4n,F=2n. Models are fitted independently to the viewer. Edge crossings are not vertices; no triangulation faces or edges are added to the mathematical counts. David McCooey's published pentagrammic metrics independently check dimensions, face types and duals: https://dmccooey.com/polyhedra/PentagrammicPrism.html https://dmccooey.com/polyhedra/PentagrammicAntiprism.html https://dmccooey.com/polyhedra/PentagrammicCrossedAntiprism.html https://dmccooey.com/polyhedra/PentagrammicDipyramid.html https://dmccooey.com/polyhedra/PentagrammicTrapezohedron.html https://dmccooey.com/polyhedra/PentagrammicConcaveTrapezohedron.html Independent offline validation uses 58 source meshes exported by the pinned Antiprism Polygon and polar-dual implementations, revision: 7805b34d3f8860857fb9ac10a1917d1bf4c7888d The original MIT Polygon source is in data/antiprism/polygon.cc and COPYING; QA reference meshes are in data/star-families/antiprism-reference.json. Re-export with scripts/export-star-family-reference.cc, or: python scripts/build-antiprism-export.py ANTIPRISM_CHECKOUT OUTPUT.json --star-families No source mesh or network request is needed at runtime. Tests cover every parameter-pair validity condition through n=200, representative complete geometry and reciprocal incidence, congruent face shapes, independent native vertex/edge distance spectra, true dart concavity, independent 3-D ray picking/depth, filled multiply wound star centres, mobile face inspection, saved choices, presets and actual page handlers.