B Imaginary Pythagorus

[DaveC426913]

TL;DR Summary

Is this geometry valid?

I posted this in the Lame Math thread, but it's got me thinking.

Is there any validity to this? Or is it really just a mathematical trick?

Naively, I see that i² + plus 1² does equal zero².

But does this have a meaning?

I know one can treat the imaginary number line as just another axis like the reals, but does that mean this does represent a triangle in the complex plane with a hypotenuse of length zero?

Ibix offered a rendering of the diagram using what I assume is matrix* notation:

which I assume makes the apparent paradox go away, but does that mean the first diagram is not valid?

* never learned matrices

I suppose it is theoretically possible to have a triangle with zero length hypotenuse if you look at it edge-on in an abstract 3D space - i.e. the two axes are superimposed.

 

[Mark44]

I know one can treat the imaginary number line as just another axis like the reals, but does that mean this does represent a triangle in the complex plane with a hypotenuse of length zero?

A triangle in the complex plane would have the lengths of all of its sides as real numbers; i.e., as the magnitudes of the various quantities. $|i| = 1$.

Ibix offered a rendering of the diagram using what I assume is matrix* notation:

I'm not sure what the equation scrawled in the diagram is saying. Is the left side of the equation $2^{ab}$? If so, I don't understand how that can be equal to a matrix.

I suppose it is theoretically possible to have a triangle with zero length hypotenuse

If both legs are equal in size and one lies on top of the other, the hypotenuse of such a triangle would be zero, but that's not a very interesting triangle.

 

[fresh_42]

The problem here is, that $i$ is not a length. If you write it as real vectors like in the Gaußian plane of complex numbers, then it becomes wrong. If you consider it as a complex equation, then it is correct, but lacks the interpretation in the real world.

 

[kuruman]

You have two vectors in the complex plane:
$Z_1=(0,i)$ and $Z_2=(1,0)$.
Noting that $Z_1-Z_2 = (-1,i)$, the magnitude of the hypotenuse squared is
$|Z_1-Z_2|^2=(-1,i)^*\cdot(-1,i)=(-1,-i)\cdot(-1,i)=1+1=2.$
This is the sum of the magnitudes-squared of the sides, $|Z_1|^2+|Z_2|^2.$

 

[PeroK]

TL;DR Summary: Is this geometry valid?

I posted this in the Lame Math thread, but it's got me thinking.

View attachment 365364

Is there any validity to this? Or is it really just a mathematical trick?

Naively, I see that i² + plus 1² does equal zero².

But does this have a meaning?

That's one representation of Minkowski geometry, where the vertical axis is time and the horizontal axis is spatial. The spacetime distance in $c =1$ units is given by:
$$(\Delta s)^2 = -(\Delta t)^2 +(\Delta x)^2$$This is sometimes represented using "imaginary" time.

PS the paths of zero distance are those followed by light.

 

[DaveC426913]

That's one representation of Minkowski geometry, where the vertical axis is time and the horizontal axis is spatial. The spacetime distance in $c =1$ units is given by:
$$(\Delta s)^2 = -(\Delta t)^2 +(\Delta x)^2$$This is sometimes represented using "imaginary" time.

PS the paths of zero distance are those followed by light.

Fascinating crossover. Can you suggest where I can read up on this more? (I'm not post-secondary edumacated, so the hard maths is over my head, but the physics may not be*)

*yes, the irony of the notion that these can be separated does not escape me.

 

[Ibix]

Can you suggest where I can read up on this more? (

@bcrowell's Relativity for Poets is a decent maths-free resource, and free to download. https://lightandmatter.com/poets/

 

[fresh_42]

This is essentially the difference between $\mathbb{R}^2$ and $\mathbb{C},$ and what distinguishes real from complex analysis: squares are no longer automatically non-negative. This has a significant impact on functions.

If you are interested in Minkowski spaces and a notation with imaginary time, then you should read about the Wick rotation. It translates between the two.