Is there a way to describe, mathematically, a hypersphere in $R^4$ such that we can take 3 dimensional "cuts" such that the sphere, which is the "cuts", has constant radius?
If you take an object in $R^4$ and move it around, your perspective in $R^3$ will reveal an object that looks to be a different shape, or suddenly a different object, but as you are aware the remainder of the object is hidden in the extra dimension.
I want to be able to move an object in $R^4$ where it will always be the same size sphere in $R^3$.
I do not know where to start to prove or disprove such an object.
It seems that it could actually require a 5 dimensional sphere:
$x^2+y^2+z^2=R-w^2-u^2$ such that the other variable would have to change such that they cancel out to keep the radius of the sphere constant.
Is there another method without requiring the extra variable?