# Jean Constant

Mathematics represents universal abstract intelligence at its best and reaches for the most abstract expression of our collective consciousness. It is in itself a fertile ground to celebrate both our collective intellectual achievements and the unsurpassed qualities of its association with our more tangible environment. This selection of 3 digital Sangaku is submitted to the 2012 Bridges conference in conjunction with a short paper and an 8 minutes QuickTime animation on the making of this digital Sangaku series.

From an original Sangaku, Iwate prefecture, 1842. If CB’A If ABC is a right triangle with right angle at A, then the circle touching the circumcircle of ABC internally and also touching AB and AC is twice the size of the incircle of ABC. (Okumura). Texture, color and depth were added to highlight the esthetic quality of the original composition - The contours of a Klein bottle wireframe is reflecting in the shield.

(Tools: 3D-XplorMath, Pixologic-ZBrush, Adobe CS 5)

From a Sangaku by Hirayama and Matsuoka, 1966. In a square PQRS, there are two circles touching SP and the incircle of the square, where one of which touches PQ and the other touches RS. Let A be the point of tangency of QR and the incircle and let the tangents of the two small circles through A intersect the segment SP at B and C. Given the inradius of the square, find the inradius of the circle in the triangle ABC

The answer is that the medium circle is also half the size of the largest circle. The Klein bottle visualizations defining each circle enhances the r dynamic of components interaction in the composition.

From an original Sangaku, Miyagi Prefecture, 1877. Two equilateral triangles are inscribed into a square. Their side lines cut the square into a quadrilateral and a few triangles. Find a relationship between the radii of the two incircles. Elements of a Klein bottle wireframe have been added on the circles and ellipses to add optical depth to the flat surfaces.