0
$\begingroup$

What is this three dimensional curve that looks like an infinity sign called? (Is there a known parametric equation for it?)

Rydberg_dihedral_kernel_KPCA_visualization_dim_3

It was generated with this Sagemath - script, where you can zoom in 3d in your browser. The background lies in the following formula of a positive semidefinite matrix, which then is processed with Kernel-PCA to be visualized in 3dim and is related to this question and this question.

In the book "Matrices and Graphs in Geometry" by Miroslav Fiedler, we have the following generalisation of the sum of angles equals $\pi$ in triangles:

theorem_dihedral_kernel

In this case, which we are looking at, we have:

$$G_n^{-1} = (d_1^T,d_2^T,\dots,d_n^T)$$

where

$$G_n = (1/\max(i,j)^2)_{1 \le i,j \le n}$$

is a Gram matrix and $d_{n+1} := \sum_{i=1}^n d_i$. Aftwerwards we treat the entries in the positive semidefinite matrix above as a p.s.d. kernel and thus can visualize its entries $1,\cdots,n+1$ with KernelPCA to get the picture above.

Visualized (with KernelPCA) in 2dim, it looks like this:

Rydberg_dihedral_kernel_KPCA_visualization_dim_2

Edit: Changing the kernel to $K(a,b) = \frac{k(a,b)}{\sqrt{k(a,a)k(b,b}} = \frac{\min(a,b)}{\max(a,b)}$ will give us the following surface in 3d:

visualization_rydberg_angle_kernel_kpca_3dim_pic_1

visualization_rydberg_angle_kernel_kpca_3dim_pic_2

visualization_rydberg_angle_kernel_kpca_3dim_pic_3

I would also be intersted to know what this surface is called, if it has a name.

Improve this question
$\endgroup$
3
  • $\begingroup$ number-theory? physics? applied-mathematics? but not "terminology"? $\endgroup$
    – Gerry Myerson
    Commented Oct 4, 2023 at 0:53
  • 3
    $\begingroup$ @GerryMyerson: Ha, ha, very funny. "number-theory" because of the p.d. kernel $1/\max(a,b)^2$ on natural numbers. "applied-mathematics" because of the applied KernelPCA method and "physics" because of the question linked to the Rydberg formula. I know it is an unusual combination, but I did not know how to better tag it. $\endgroup$
    – mathoverflowUser
    Commented Oct 4, 2023 at 2:16
  • 5
    $\begingroup$ Fair enough. But since your question asks how something is called, I'd encourage you to add the "terminology" tag. mathoverflow.net/questions/tagged/terminology $\endgroup$
    – Gerry Myerson
    Commented Oct 4, 2023 at 5:24

1 Answer 1

Reset to default
3
+50
$\begingroup$

As far as I know, every curve that has the shape of the usual infinity sign is called "lemniscate". Now, we also have a curve belonging to the same family, so we could call it a 3D lemniscate and then we are free to specify something else in order to distinguish it from the rest (e.g., "helical lemniscate", "double 3D lemniscate", and so forth).

Furthermore, the lemniscate has the property to be the inverse curve of the hyperbola with respect to its center (see Lemniscate (MathWorld)) and this is consistent with the 2D projection shown in Figure 2.

Improve this answer
$\endgroup$
3
  • $\begingroup$ thanks , i knew the word lemniscate but i was searching for a 3d version so to say. $\endgroup$
    – mathoverflowUser
    Commented Oct 11, 2023 at 11:03
  • 1
    $\begingroup$ Yes, I know... my point is that as "the lemniscate resembles some toric sections when the cutting plane is tangent to the torus along the circumference of its central hole", you can simply extend the original definition by generalizing the environmental space from $\mathbb{R}^3$ to $\mathbb{R}^4$, getting a 3D lemniscate... then you simply have to coin a proper name into the definition, with the equation, you provide. $\endgroup$
    – Marco Ripà
    Commented Oct 11, 2023 at 11:19
  • $\begingroup$ thanks again for your insight $\endgroup$
    – mathoverflowUser
    Commented Oct 11, 2023 at 12:54

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .