Can you somehow make left handed helix into a right handed?

[sol47739]

TL;DR

Can you by any type of rotations or transformations turn a left handed helix into a right handed or vice versa?

Can you by any type of rotations or transformations turn a left handed helix into a right handed or vice versa? If yes why? And if no, why not?
For example if you have a 2 d triangle like the one in the picture:

You can turn it in 3 dimensions 180 degrees in a sense mirroring it and you will get back to the original, but then you have to turn it in 3 dimension in a sense lift it up from the plane and turn it.

Can you do the same for a helix or not? If you were able to turn it any way you wanted in 3 dimensions as many times as you wanted could you get a left handed screw to become a right handed?

 

[DrClaude]

Can you do the same for a helix or not? If you were able to turn it any way you wanted in 3 dimensions as many times as you wanted could you get a left handed screw to become a right handed?

If you could, what would be the point of calling it left-handed?

https://en.wikipedia.org/wiki/Chirality

 

[FactChecker]

It's my understanding (admittedly casual amateur) that here are some chemical reactions that do just that. Mad cow disease (Creutzfeldt-Jakob disease) converts the fold direction of a usable prion into one that is unusable and clogs up the brain. I believe that changes it from a left-hand fold to a right-hand fold (or the converse?).
Mathematically, in Geometric Algebra, a reflection can be done this way: Suppose $\hat {v}$ is a unit vector and $\vec {a}$ is a vector. Then $\hat {v} \vec {a} \hat {v}$ is a reflection of $\vec {a}$ across $\hat {v}$. See An Overview of the Operations in Geometric Algebra at 10:26

 

[jedishrfu]

The thalidomide disaster of the 1950's was due to the wrong chirality of molecules:

https://en.wikipedia.org/wiki/Thalidomide

Originally it was thought the one of of the molecules (R version) was safe and that the (S version) was not. However, they later discovered that the body can convert (R to S or S to R) which made the drug dangerous for certain kinds of conditions including pregnancies used by either men or women.

Thalidomide destroyed many medical assumptions of the time ,including the notion that drugs during pregnancy would not harm the fetus as it could cross the placental barrier, and led to safer more thorough drug testing.

 

[Hornbein]

TL;DR Summary: Can you by any type of rotations or transformations turn a left handed helix into a right handed or vice versa?

Can you by any type of rotations or transformations turn a left handed helix into a right handed or vice versa? If yes why? And if no, why not?
For example if you have a 2 d triangle like the one in the picture:View attachment 325219
You can turn it in 3 dimensions 180 degrees in a sense mirroring it and you will get back to the original, but then you have to turn it in 3 dimension in a sense lift it up from the plane and turn it.

Can you do the same for a helix or not? If you were able to turn it any way you wanted in 3 dimensions as many times as you wanted could you get a left handed screw to become a right handed?

View attachment 325220

I believe it can't be done in 3D. I don't know how to prove that though. In 4D it's easy.

 

[TeethWhitener]

I believe that changes it from a left-hand fold to a right-hand fold (or the converse?).

Not quite. The conformational change is believed to involve an alpha helix to beta sheet transition, rather than a change in the orientation of a helix.

There are chemical reactions that invert chirality (SN2 reactions are a prominent example), but a chiral molecule (or any chiral shape in Euclidean space) cannot be superimposed onto its mirror image using only rotations and translations. Mathematically, chirality requires that the shape's point group not contain an improper axis of symmetry.

 

[lavinia]

Left handed and right handed helices have the same curvature but have torsions of opposite sign.

Rigid motions of three space preserve curvature and torsion so opposite handed helices cannot be transformed into each other by a rigid motion (a rotation plus a translation).

At every point of a smooth curve that is parameterized by arc length and whose curvature is not zero, there is a plane spanned by its unit velocity vector and its unit normal called the osculating plane.

As a point moves along the curve, this plane rotates infinitesimally toward or away from the normal along an axis that passes through the direction of motion. The torsion of the curve measures the speed and direction of this rotation. If the torsion is positive, as in a right handed helix, the osculating plane turns away from the normal. If it is negative it turns towards it.

This link has an animation of the moving Frenet-Serret frames of reference (the moving positively oriented frame whose axes are lines through the tangent, normal and binormal) on the two helices.

https://mymathapps.com/mymacalc-sample/MYMACalc3/Part I - Geometry & Vectors/CurveProps/Torsion.html

 

[lavinia]

I was wondering what a natural generalization of a helix to four dimensions might be. A space curve in 4D has two curvatures plus a torsion. One generalization might be a 4D space curve that has constant curvatures and torsion. I wonder what such a curve looks like.

Another might be the 4D lift of an infinitely turning curve on a sphere e.g. a loxodrome in analogy to the helix which is a 3d lift of a curve turning on a circle.

 

[Hornbein]

Another might be the 4D lift of an infinitely turning curve on a sphere e.g. a loxodrome in analogy to the helix which is a 3d lift of a curve turning on a circle.

This clever idea gets my vote. I never would have imagined such a thing.