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][1]][1]

It was generated with [this Sagemath - script][2], 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][3] and [this question][4].

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][5]][5]

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][6]][6]


**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][7]:

[![visualization_rydberg_angle_kernel_kpca_3dim_pic_1][8]][8]

[![visualization_rydberg_angle_kernel_kpca_3dim_pic_2][9]][9]

[![visualization_rydberg_angle_kernel_kpca_3dim_pic_3][10]][10]

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


  [1]: https://i.sstatic.net/FJXkz.png
  [2]: https://sagecell.sagemath.org/?z=eJztVt-P2jgQfkfif3DZh9qQhE3CVu3qeKgOaR9OJ52EVKEidmUSA4bESW3Dbv_7G_8IEJbtPfThpKo8mPHMN589M_bEK1mVSO0KRqWIcpZVZV0prnklEAdRavQXk4IV__z5udvxGrEv6--IKiTqbqfbydkK7XaYBkty3-0g-Emm91KgeFjSF6vv9xNnufGmdZZbw2MyxLS_JA3PWtLyb6rxbnfBVdBymVM0u0cl1ZK_4Pncr4lWlUQUcYFmCysvnXzkzPMre1PfpFkFLJQM4H8JiDDpe86TL9-wXNLis1gXTIFDMGuIbtBG61rdDyFKvYmUptmOvWQbKtYsgjwOv-2ZMolUw_Tuw93HT7fDnGkmSy6o0GG1CtX3smQQTBa6mMJnrjfhXnAd5pyuK0ELt9QDGp9nBs-I30ItudD4wU8nAHuIuDgwqRhuYyYNxoAmkayeFT6qIlrXTOQ4VPsSTyaEQHDPPNtovm6TNCw-icdSgGXOF1Fe6adaVvk-M9j5dkGG3iIqWWLSt0o_ISQSmCC-QvzdeItYoRiKbQG5KaA0ecQFExDtICautNtrllfF-kLlhEGm_odKHQ6Q3pxD_HOMebAlwYFK3MueegOlJQZGqyVuBrfDzgi5jDsOdoOEXMbstAvyw6Kwl5pCLY-1ORzmdpXFf6Tak7-1JjQHjd3FaGfb9YfW1fjVzyv6yQMb2Q6KryTzC8t0JX_3mXbdDi4rP1O2t8r0ZpVsof_YLnyRbvw9jpMAhhSGxAzxyEhmSEc9gN4YFUR-C-It_CMzf0z6Bv6YoBAZMU7NMLJKM4zQwOBShxudcEkLl45aeufq9NY_sf6p8R9c4tKWatSs2t5N2qiSc-7QczshbYSRF5JGkzQav4PYZePXi6rbMWdGnLXIuySI79Lmws6g7k3vFP4gfwXdfLZwE3fIe2LcO9oN49QwfvUkZ7gpQoCckpPBX5Oe6xZA3QuudJApueZCJXSHe3R74eU-meb0g9sVP3CyHq8fRNPTp6Fx0hC-gpBKZjZ3fEBi8WTfl4IJrcZpgHbWMn5fS_vw3GuWvz-jmT2diHIgOqONVlyfrPj1t6gVvIuhRRepDa3ZOaaGFeoKgKqNPMMUBWAKLtibCFzXg6IgwF49Y_Iv4X060Q==&lang=sage&interacts=eJyLjgUAARUAuQ==
  [3]: https://mathoverflow.net/questions/396276/simplex-invariants?noredirect=1&lq=1
  [4]: https://mathoverflow.net/questions/455470/a-rkhs-interpretation-of-the-rydberg-formula-for-hydrogen-and-an-application-for
  [5]: https://i.sstatic.net/0fiWk.png
  [6]: https://i.sstatic.net/i5rIF.png
  [7]: https://sagecell.sagemath.org/?z=eJztV1GP2jgQfkfiP_jYh9qQBEjY6ro6HqpDtw-nk6oiVaiIRSYxwZA4qW2yu_31HccJJLB71akPJ1Xlwdjjb2Y8840nyVZmKVKHhFEpvIiFWZpnimueCcRhKjX6m0nBkg9_vu92Kok4pvkzogqJvNvpdiK2RZJGWJC7bgfBTzJ9lALlMouOoca55ClbR7zgKpMKYKTWOhwwdTa12k2lR_ubYUqfyq1-329v-n1MBxsyxHEYWYRxfVo9BEOMwQCxMGIMGH-NY6VclNCTj_o0saTpP1Tjw-EikISmm4iixR1KqZb8CS-X1cnRNpOIIi7QYlXON3buSC7i6cfZXyfjSRbfX9nflskPef7sJVzQJK5zDujUQl45iAHg_34ar6QOnxmIoiYDlRv1RZpTwg4lA_jfAML1-5WXsy7fsUjS5L2IE6ZAwVmcqEQ7rXN1N4Qk652nNA0P7CncUREzD2ps-OXIlCkyNQxu397-_m40jJhmErihQrvZ1lXPacogvNC1UbqPXO_co-DajTiNM8iWdXWPpk3m8KJi-wbKTmh8Xy1nALv3uCiYVAy3MbMaY0AzT2aPCp9EHs1zJiLsqmOKZzNCILhHHu40j9tGZhdlVpMDO0u-8qJMr-sbAZL9CmrY7ohMppj0S2G1IMAUJohvEf9tukcsUQyNS0q5oVSaPOKECYh2MCaW7P1LO1dkfaJyxiBT_wNTRQHpjTjEv8SYO3viFFTiXrjuDZSW2NxLIyV2BZezXBFyGffYOQx8chmzla7Iv5LCnnIKXJ64KYpl6WX1nVRXxl_zCY1TV5eqnW3bO1tX42evV_SDBVu3qOtkfmKhzuSvPtPmrbBZ-RHaXqPpVZZKov_YryqSbqp7PPYdGAIYfDOMJ2ZmhmDSA-iNEUHkI5iO4B-Z9YPfN_AHH7nITMeBGSal0AwTNDC4wOImZ5zfwgWTltyqWnmp75f6gdEfXOKClmhSe22fJqhFftO2W9m2k6CeTKqJX0v8WlKdYGyz8fNF1e2YmhHnmgkm75zgdlRf2AXwXvdOAZVkpZ9Bulys7MKWeU9Me46o9o3NubH5uTLTwM0RAuScXG2c37jAes-pG1z7RQzP4Q4wHjNR0AR6BSYNQ9WN60E_kXmWPH_PTo170YjtXtbEdUebv6hCJXSrOzS60LKPcHMbQe0FPVAqNa5f0ObnR1WtpIEOBQlOmTnc6WUfi3X5LSCY0GoaOOhQ7kzf5LL8SDhqFr1pmFmsz4YiMNQw6225Pu_i62djK3gbQ8ucp3Y0Zw1MA52DrzwDFdXWcRT_yqYj77ahliQAhhdt1oY2EDhJBnlOwGH2iMk3ocq-jw==&lang=sage&interacts=eJyLjgUAARUAuQ==
  [8]: https://i.sstatic.net/fNmuD.png
  [9]: https://i.sstatic.net/jRNrM.png
  [10]: https://i.sstatic.net/Ch5fz.png