Four numbers. The straight figure is Chiodo's construction: nine triangles built from four heights a, b, c, d on the vertical diameter. Everything else follows by ruler and compass (a circle through three points, the meeting of two lines), so the sliders change the construction itself.
What is exact. The seven places where an apex lies on another triangle's base, and the twelve triples of lines that meet in one point, hold to machine precision at every setting of the four sliders, because each of them is a step of the construction.
Held conditions. Each switch above the sliders adds one condition (the innermost triangle centred on the bindu, the innermost triangle equilateral, or D3's corners on the rim) and satisfies it with the remaining freedom. Four heights can carry at most four conditions, so not every combination of switches can hold at once.
Twelve points, eighteen numbers. The arc figures are built from twelve seed points. Every side is a single circular arc, so its curvature is constant, and a straight side is the case of zero curvature. The eighty-eight incidences hold because each one is a step of the construction.
The grown rings. Outside the nine triangles the figure grows two rings of cells. Here ring 1 is constructed on its circle: the circle is drawn first, the ring's fourteen vertices are taken on it, and the arcs of the yantra are drawn through them. Twelve numbers are free (eleven seed numbers and the radius of the circle), and one more is solved for so that a single coincidence holds.Here ring 2 is constructed on its circle in the same way, with eleven free numbers instead of twelve.Here neither ring lies on a circle, so all eighteen seed numbers are free and nothing is solved for. The figure opens in the shape of the ring 2 figure.
Thirteen corners, twenty-twotwenty numbers. These two figures have eleven triangles instead of nine. The outermost side on each hand is carried on until it closes into a triangle of its own, so that six triangles point down and five up, and the fourteen corners that the other arc figures hold on the circle are released. Thirteen corners of the figure are placed (nine free in the plane, three on the axis and one on the rim)(seven free in the plane, four on the axis and two on the rim), and the other fifty-twofifty-four objects follow from them.
The ring. Between the yantra and the rim the eleven triangles cut a ring of tenfourteen cells, and its outer nodes lie exactly on the circle. This ring is part of the figure, not a grown ring, so these figures have no ring switches.
No arranged coincidence. Each step of these constructions draws a node or a curve from those before it, as a circle through points or the meeting of two curves, and nothing is solved while the sliders move. At every setting every incidence holds, including the outer nodes of the ring on the rim. Because each corner is determined by the ones before it, a single corner moved on its own soon reaches the end of its range; a combination of corners can move much further, and the principal directions provide such combinations.
Principal directions. The parameters form a basis of the family but make poor sliders: equal moves of two parameters change the figure by very different amounts. On the constructed figures the differences are large, because sixty steps of "circle through three points" amplify a change in a seed, and the radius of the ring moves the innermost triangles four hundred times as far as it moves itself. The principal directions span the same space, ordered by how much of the figure each one moves and scaled so that a full slider moves the most affected node by the same distance for every direction.
The bindu. The dot at the centre is the reference point for the innermost triangle; on the straight figure one of the switches places the centre of that triangle exactly on it.
Note. Every figure here is a construction; nothing is fitted. On the two ring figures one number is solved for, because one coincidence has to be arranged rather than constructed, as Chiodo does on the straight figure. Everything else, including the whole ring, follows from the order of the steps.
The paper. The straight construction is Chiodo’s: A. Chiodo, On the construction of the Śrī Yantra, C. R. Math. Acad. Sci. Paris 359 (2021), no. 4, 377–397 (doi:10.5802/crmath.163). He proves that the conditions can be met exactly and shows the essential step to be a circle–line–point problem of Apollonius. This viewer and the arc figures are at github.com/michaelkopp/sriyantra.
The paper. The construction is written up in The circular-arc Śrī Yantra, which proves that the classical concurrency conditions hold exactly, in addition to eight further concurrency conditions that hold all the corners of the Śrī Yantra on the circular rim; it counts the dimension of the family in two independent ways and comes with GeoGebra files for the three yantras above. The paper, the GeoGebra files and this viewer are at github.com/michaelkopp/sriyantra.
The paper. These two are treated in Extended circular-arc Śrī Yantras, which is about the eleven-triangle families. This viewer and the GeoGebra files for both figures are at github.com/michaelkopp/sriyantra.