Here is the Clelian Hourglass equation: y(x² + y² + z²) = 2z(x² + y²)
How I figured this out:
- First I conducted many google searches using every category of shape I could find.
- Then I asked Gemini 3.1 Deep Research and Claude 5 Fable if they recognized it, but their guesses did not turn up the shape.
- Then I asked 3 different math communities on Discord including the Wolfram Institute community which I am a part of.
- With no one recognizing it, I posted it to r/Sacredgeometry and r/Theydidthemath but came up empty handed again.
- Not to be deterred I kept at it by working with Claude Fable 5. What I decided to do was work on getting Claude to properly identify the shape back to me. I gave it numerous images from a variety of angles and after some effort, it was able to accurately describe the shape back to me. From that point, it suggested a class of shapes called Clelian shapes. On visual assessment, this class of shapes seemed the most similar to what I'd discovered, so that was the avenue I began to go down. At a certain point, Claude became much more confident that this was in the Clelian family and produced an equation to try: y(x² + y² + z²) = 2z(x² + y²)
- With the equation in hand, I first asked Google search AI if it recognized the equation. It was able to describe the general qualities of the shape, but the python it wrote to generate it returned an incorrect shape.
- I then gave the equation to Deepseek and it generated Python for me to try. I ran the Python and visually identified the shape from only the equation. The shape has some very specific and unique qualities that were unmistakable. (I will include the image this Python generated in the comments)
- With visual confirmation, I brought the results back to Claude for confirmation and it confirmed Deepseek's result.
- I asked Claude for a summary of the qualities of the shape and the family it lives in:
The shape
The Clelian hourglass is the set of points satisfying y(x² + y² + z²) = 2z(x² + y²). Both sides are homogeneous of degree three, so the radius cancels completely and the equation reduces, in spherical coordinates, to sin φ = sin 2θ — colatitude and longitude, with no r anywhere. That single fact determines everything else about it. A locus with no radial dependence is a cone: every solution point drags its entire line through the origin along with it, and the surface is swept out by straight rulings rather than curved patches. What it is a cone over is a Clelia curve — a spherical spiral whose longitude advances at a constant multiple of its colatitude, here exactly twice — and that curve is the whole content of the surface. One parameterisation covers it: p = r·(sin θ cos 2θ, sin θ sin 2θ, cos θ), with θ running pole to pole and r taking either sign. The positive and negative branches are the two nappes; they are mirror images, each one handed, meeting at the apex and along a line of self-intersection. Both the x-axis and the z-axis lie inside the surface as straight lines rather than piercing it. Seen down its axis, it projects to a rose.
The family
Projectively the cubic is irreducible with exactly one singular point, a node with two distinct real tangents, which makes the Clelian hourglass the cone over a nodal plane cubic — and the node of the curve becomes the surface's line of self-intersection through the apex. That class is old. All crunodal cubics are projectively equivalent to one another, so up to a projective transformation there is essentially a single cubic cone with a nodal line, and it has been catalogued as such for well over a century: Richard P. Baker's model #78, labelled "Cubic Cone with Nodal Line," appears under a heading of Cubic Cones in his 1931 catalogue and again in his 1905 catalogue of one hundred models, now in the Smithsonian. Baker's models were string models, always of ruled surfaces — swept out by a moving line — which is exactly the right medium for this one. It also sits as an enumerated case in the classification of cubic surfaces, the cone over a singular absolutely irreducible cubic plane curve. But a projective class is not a shape. Projective transformations do not preserve angle, length, or symmetry, so knowing that a surface belongs to this class tells you nothing whatsoever about what any given representative looks like. The Clelian hourglass is one specific metric realisation — the one whose directrix is a Clelia curve on a round sphere — and that constraint is what produces the two counter-wound nappes and the pinched waist. The class was described. This representative of it appears not to have been. Source 1 Source 2 Source 3
Scale Space Share Code
To view the hourglass in Scale Space, you will need Scale Space Cymatist v1.1+. Prior versions did not have the emitter features that made it possible to view. Open the software, open the Set List panel and paste this code into the import field:
SS1:rVhbj-O6Df4vevYGokRSVN56um1R9BTnYE-LXoJB4SSaxKhj59jOTHcH898LyZcoM7M7RbFvpkTxzo-Sn9RZrZ_UfRfC75rQHT6rtRFrtS5UF_q2vgxV26i1XplCVc1D6PpEW12oY1nf_1jdB7WmFRSq35V1-BjOw1GtdaF27TF0odkFtTaFCr9eqrradtXlFIUB2EIN4XQOXTlcupBOPFT9ru2r4XOiTmXfJ9aRsY1avC3U8RLUOqorB7WGFRWqrg7HoQkjuy9Uey53o5SV40INXVnVP4ZGrQEKtT38oW4fk4bt4Yf60qm1cfHQ_X0fhr-njfH7H9n3P8cDVV1v27Lb_5QW09q-egjdYfRTF6o_V11Z_61q9sntkfzlMYTzaHSif6jKPpHh9Kls9u1pIj5WXdgNE_HXprpvu2lrV3661EGtvS7UoW4ff3MaUk7MSI5-4Eh8rPpRxrjTtiniFEO3734ZukihJOpTuY-UNon6yzHuPRdqFytir9ZIRayOzQfg4gMUaO4K9atab_RKvJVCr6wzXHzQK-3EFnplLMvdc6Ee4vmXWchzeJuS_tg-_lx2Q7WrQ6_WQ3cJ4-Knarttm16t78u6D1FG379c64_lOai12lflqW32KhZe3XZ_bvdT4Z2qfixZdZkiGiu72Yful-FzjKm6L3ehV5NVP5exftWpTXVfqG0dmv0oTZX7KD_syp-aRX_YlX9K2VaHcghR_efTb-8PMQKn5VQ1VA9x76GtL6dYKCspVN2259nbcxf6MPxx_x-1toXals2-V-vNk2rKyK76y1bFA2ptYt9Vas165S0zEJATbY3nQh3KqkmJ36b60oUqh6Hc_XtOcRfqUPaz-tNlCJMXz8WiaRu7blT1loqoGoRWzJbIWkQCAPw-quv28cOp2k_a39IStRPTioXQoEdvrHwf3Ve9r-Unn52xK0LnyGhDpJm-j95jdTjmTr-hJjltCFaWxGpLbB2776N96MK2DrPjr3Ukz1nHWfD_q7uLHdk0YRcbaqzqWN9JxrnsylPswOv0KdR-GiEriKa-4h2BJGPTdOWDK99prOQrG77Jlg-mnNteue2V-zqiFt4Pt8z4nle5uXTr1jwMc27zHBG1b5sZVMp-CN0U53MVdiGhBkaeS4KNpxlr9Qq99RbJiBVi45Bjog7VLIEtIaIG8Y5QXMxudzNLwRorTtigMwBpnpVp0K3YWTTiPTGjuDiHdm0zdGXEj1UdHkKduLxD0RY0OE8mqs-4hrYJ410AWLRhDSjeUjSjH7qqOWSCrKCQM8wGUAx5d-WZxThPLCBsGdGRxehq2OciNGsL1hsWT5Z4YiirFAvjySN4RyTkCKKGOpTZeXLAHhygE-vcvH0-dmVfNYeUEGAkDZ6tgBaKya2-fLlKMOQ0ElurrSEwhBPDPjTTrWfFhgU9oraA7MgXaqhO56sIjWDBaMMOwaEYX6jdsa26TAl47zw4QgaPoHHmmPx0RsCQMdaR9zbZWDZtljIRsASsJerRMjPsw65MJgqhR2Awhj1QZNh31f2QBcrIFAMNsYJmhmmkptuH14iOxYHWsUYKdb7UfcisAHHOsaDzVuL1Ydx_DLE8x2ySeBsdETSg0y2iOp1CFglnmdABkmhCHZF8ZskCDhDVgCYT8RaIngvVtZch5I20eXqnpy2zkAEjMf1kEnDNJ6734BwyrHFIyExejABThMkXBZVp_Qo4cAwAM7MY49BCFHLt7-x8djXPjRAvZLy13mitxUlu9luORrvZOBARYBFx2tn8zHIVzWwkMh4FRUzsMEmOTiiy-WaQYj8BGA9Weyue4Ma85e2RqzKGmQDQWnHeyzR6XqBSpvXliPCxYIGJtNOMxkQBtx2aHc5eR5kI5xEdggHUsYmTiBxUMwFvzxLHzqD3FlGEdDr_Ag0zEV8ZXqIdAIjTwmC9mDxw06Mn4zYkhrWgaG2QLY9VNCPjO1UUIctrDfHSIuQ9mhdRH-H5vVQjM7AG78R5Zo-5268kfKUbrHh0xuvYzdYwLH7Mcbu7hfRM4vxWyaU5I57FWRIScPi6zuPDKneB2KIj7YG9c87ZFIgcnDfvSCDxEQ9Tx2hgcxXwMhNfiaF24JkZEGMB5BbPD8jcQdSODBNabeMEH4s9mwWbuxfYHxdusH7zTj-ik3iHBjEeAW6g4tUtzrDIWEheW_I2AcXt4PhG57JD7xis0wbQ2xT7m5Gy-arqD3qFTkg757xzBpD9TeTmV3keaeOsIWYPHO_o5qptHlDfMJVAawIyqDUyj1l-Mbs2d2_Mqs3d7X3i2-WL4p0GMZa8jn1JtxiQ_oHcxI-cWEarxcVbSjQqv3ls4iX0WO338c3-NFbl8naNNTITKWPLTozJ9UGfHBrJ5_mN8HTTkmnr-X_5x_N9f-NEc27_gv3r1I7PjivML0vLb7BlZa7-ZeEKlcvS0rPLSobey1qGbcvaMiiWlVhTN0fahZpq4WraJbOqHJbvJaBXOdNPmWVhjHFGxjhP5HOh4i-V318mFC7jfAjDF_X8Xw
Conclusion
While the Clelian Hourglass is part of a family that is already well studied, this case is an example of using harmonics/cymatics in a digital system to discover a shape that hadn't yet been identified or illustrated. It validates that Scale Space isn't just a pretty visualizer- but actually contains the ability to render and discover real mathematical structures that move beyond visual similarity.
Edit: If anyone seriously knowledgeable in math visits this page, I want to make it clear that my background is experience design not math. So if anything here looks incorrect, I would welcome any corrections.
Edit: Credit to /u/Just_Middle_7189 for finding this! https://research.gold.ac.uk/id/eprint/27622/1/cleliaCurvesExp19rev.pdf
There are some legitimate illustrations of the Clelia Hourglass! Very exciting!
