Visualizing the four-dimensional symmetries of the gcd and lcm.

This is a variation of my previous post. The following combination of $4$-tuples provides a better insight of the symmetries regarding the gcd:

$$(a,b,gcd(a,b),sgn(gcd(a,b)) \cdot \lfloor \sqrt{\frac{a \cdot b}{|gcd(a,b)|} \rfloor})$$

where $gcd(a,b) \in \Bbb Z$ (instead of using the definition of the gcd as the absolute value, we are using the alternative definition that lets the gcd to be negative when the sign of the elements $(a,b)$ is not positive) and

$$sgn(gcd(a,b)) \cdot \lfloor \sqrt{\frac{a \cdot b}{|gcd(a,b)|} \rfloor}$$

is the floor of the square root of the least common multiple, letting it be negative (sgn is the sign function) if gcd is negative and positive if the sign of the gcd is positive. I am making the square root of the LCM because it is an approximation to the expected values of its biggest possible pair of multiples $x$ and $y$:

$$\{x,y\ /\ x \cdot y = LCM \} \land \{\not\exists\ x',y',\ x' \cdot y' = LCM , x'+y' \gt x+y \}$$

$x$ and $y$ will be located around the square root of the LCM, so is is a good approximation, and they are located inside the range of the tesseract, so they are very useful in terms of representing the relationship of $(a,b)$ and the gcd using $4$-tuples.

It leads to the following results, first an overview of the projection:

And zoomed:

So it seems better to use directly the gcd and a manipulation of the LCM to observe the symmetries of the tuples.