The DICTO Yantra: Exact Śrī Yantras, a Rational One, and a Perfect Yantra on the Cleo-Antonius

The Śrī Yantra asks its nine triangles to meet three at a time at 18 points (the marmas), to make 43 small triangles, and, in the strict reading, to touch the circle with its smallest top and bottom triangles and to centre the innermost triangle (the bindu). We prove that every rational choice of four natural parameters gives a yantra with all 18 meetings exact in one quadratic field (Theorem 1), and give a yantra with every coordinate a fraction (Theorem 2). The 43 are the pieces lying inside an odd number of the nine triangles (Proposition 3). Spun about its axis the yantra is the axial section of nine nested cones; still tetrahedra show it from three positions narrowed to cos 30° (Proposition 4). The perspectives that keep the circle round keep every meeting and the product of the two great slopes (Theorem 5); in natural coordinates the yantra is one quadratic and one relation of degree 10 (Theorem 6). With one slope for all nine triangles the heights are forced to 2 : 4 : 4 : 4 : 6 : 6 : 8 : 8 : 9 and the great triangles cannot both be inscribed (Theorem 7). Computed to 60 digits, with exact forms open: the DICTO Yantra, on DICTO's Cleo-Antonius pair (slope product 2/φ³), meets every condition, with 43 triangles, 18 marmas and 24 sandhis; it is the Cleo-Antonius seen from one viewpoint. The most even exact yantra keeps the 43 but leaves the circle. Every exact yantra found chooses between its small triangles touching the circle and its pieces being even: the DICTO Yantra is perfect by the rules, the most even yantra perfect to the eye.