Triangle Sky

Stars, strands and sheets over a freely movable triangle. Select an apex to inspect its tetrahedron and perpendicular foot.

Index ≤ Marker size 3 px

Drag to orbit · scroll to zoom · click a star to pin

Lower → higher effective index● Constructive◇ Optimization— Power-k– – Eisenstein

Preparing families…

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Base triangle

Strands & sheets

Power-k strand Eisenstein strand Prekite arcs Symmetrical apex

Power k 2

Eisenstein k 1

Power-k sheet · −3 ≤ k ≤ 5 Eisenstein sheet

Sheets use k and h coordinates. Labels k=… [1] and [2] distinguish height graphs meeting at folds. Dashed strands are Eisenstein; thin cross-lines have constant h.

Visibility & sampling

Sheet opacity 42% Window / edge 2.5

Positive power weights only Square grid in ABC plane k, h & graph labels Constant-height contours Project strands onto ABC Dim catalogue stars Orthostar, parabolic & regular stars

Sampling

Triangle centers

Show selected centers Labels

Ctrl/Cmd-click selects multiple centers. Centers at infinity are omitted.

Inspect a strand

Choose a strand component, then move along it.

Catalogue

Constructive Optimization Calculate numerical stars

Select a star (including overlapping markers)

Select a star

Hover over a marker or select a catalogue entry to inspect it. Clicking pins its perpendicular and information.

16 triangle presets · 167 constructive stars · 6 optimization stars · Stellar strands & sheets. For a fixed base, a strand is one-dimensional and a sheet two-dimensional; allowing the three base side lengths to vary gives the four- and five-dimensional families. Catalogue expressions and ETC identifications supplied by the author. Numerical stars are multistart local minima, not certified global optima.

Mathematics & running locally · Original Mathematica catalogue · Julia source · MIT license for the software port