2021-09-17 15:47:28
UMBILIC TORUS
A single-ended 3-dimensional shape
Face and edge property
The long edge goes three times around the ring The shape has only one external face.
Cross section property
A cross section of the surface forms a deltoid.
Deltoid
The roulette created by a point on the circumference of a circle as it rolls without slipping along the inside of a circle with three or one-and-a-half times its radius.
Parametric form
x = sin u(7 + cos(⅓u - 2v) + 2cos(⅓u + v))
y = cos u(7 + cos(⅓u - 2v) + 2cos(⅓u + v))
z = sin(⅓u - 2v) + 2sin(⅓u + v)
-π ≤ u ≤ π, -π ≤ v ≤ π
Singularity theory
The umbilic torus occurs in the classification of umbilical points (points on a surface that are locally spherical.) which are determined by real cubic forms
ax³ + 3bx²y + 3cxy² + dy³
The equivalence classes of such cubics form a three-dimensional real projective space and the subset of parabolic forms define a surface — the umbilic torus.
Sculpture
In 2012, celebrated sculptor and mathematician Dr. Helaman Ferguson joined President Samuel L. Stanley Jr., MD, and Jim and Marilyn Simons for the dedication of his piece, Umbilic Torus, at Stony Brook University, New York.
866 viewsMuhannad Qasim, 12:47