Question

what is the center of a 3-sphere passing through the points (0,0,0),(0,1,0),(1,0,0), and (0,1,1)?

Answer 1

well a 2-sphere embedded in 3 dimensional Euclidean space

Answer 2

yes, a 3-sphere needs 4 coords.

You could put the center at (x,y,z)
and define the vectors
center - point
i.e.
(x,y,z) - (0,0,0) = (x,y,z)
(x,y,z) - (1,0,0) = (x-1, y ,z)
(x,y,z) - (0,1,0)= (x, y-1, z)
(x,y,z) - (0,1,1) = (x, y-1, z)

Answer 3

the length squared of each vector is the radius squared:
A x^2 + y^2 + z^2 = r^2
B (x-1)^2 + y^2 + z^2 = r^2
C x^2 + (y-1)^2 + z^2 = r^2
D x^2 + (y-1)^2 + (z-1)^2 = r^2

equation A - eq B gives
x^2 - (x-1)^2 =0
x^2 - (x^2 -2x +1) = 0
2x-1=0
x= ½

Answer 4

you can find y and z using A-C and C-D
or you can argue from symmetry. either way, we get
y=½ , z= ½

radius will be
\[ \frac{\sqrt{3}}{2} \]
see attached figure.