Hypersphere
How to see a sphere in four dimensions.
01 Shadow
A shadow is one dimension less.
3D sphere→2D shadow
θ
02 Shadow, one step up
A 4D sphere casts a 3D shadow.
4D sphere→3D shadow
xw
03 Slice
Cut a sphere, get a circle.
2D
3D sphere∩plane→2D circle
z
04 Slice, one step up
Cut a 4D sphere, get a sphere.
+w−w
4D sphere∩3D space→3D sphere
w
05 Build it
Now build one yourself.
2Ddiameter → circle
3Dcircle → sphere
4Dsphere → hypersphere