# Computer Algebra System for TNB Frames (How to write in 4 Dimensions)?

So I am working through a problem and we are allowed to use a computer algebra system to check our answers. I am trying to input a 4 dimensional curve and get mathematica to display the relevant answer. However, I am having trouble trying to figure out how to write this. Any help would be appreciated.

So, for a 3 dimensional curve that was

$$X(t) = (e^t\cos t, e^t\sin t, e^t)$$

I was able to write the following line and have the TNB produced.

Assuming[t>0, Simplify[FrenetSerretSystem[{e^[t]cos[t], e^[t]sin[t], e^[t]}, t]]]

Now in 4 dimensions I am working with the following curve

$$x(t) = 1/\sqrt 2 ( \sin(\omega_0 t) / \omega_0, -\cos(\omega_0t) / \omega_0, \sin(\omega_1 t) / \omega_1, -\cos(\omega_1 t)/\omega_1 )$$

Here is the code I am trying to run

Assuming[t>0, Simplify[FrenetSerretSystem[1/sqrt[2]{(sin[mt])/m, (-cos[mt])/m, (sin[nt])/n, (-cos[nt])/n}, t]]]

I know this is wrong, but I am not sure how to fix it. I am not sure of where to put the $$1/\sqrt 2$$. I also changed the $$\omega_0$$ to a $$m$$ and the $$\omega_1$$ to a $$t$$ because I do not know how to do subscripts.

Any advice?

As a check if anyone runs the code it should give something that looks like this

{{curvature, torsion, fourth dimesnional analog}, {{T}, {N}, {B}, {D}} and T will be

1/sqrt(2) (cos(mt), sin(mt), cos(nt), sin(nt))

## 1 Answer

I didn't realize that Mathematica now had FrenetSerret working in any number of dimensions. Very cool.

Your problem is that you don't know the basics of Mathematica syntax. It thinks that $$mt$$ and $$nt$$ are single variables. You need a space between the $$m$$ and the $$t$$ (which makes it multiply them as independent variables). But Mathematica is not good about performing trigonometric simplifications; you might try putting a //FullSimplify after your syntax.