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I connected five and six square paper sheets (which are all initially flat and have the same dimensions) using tapes to create two smooth saddle surfaces (see below), but I couldn't figure out the analytical/numerical geometric shapes of these two surfaces?!

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I apologize if 'saddle surfaces' have more rigorous definitions in mathematics. In my opinion, these two 3D shapes are ruled saddle surfaces (because the line segments connecting the center and each point on the edges are all straight), and I can obtain the entire surface once I have all the vertcices and curved edges determined. Furthermore, all the curved edges can be determined by calculating the geodesics connecting those vertices. So the question boils down to the coordinates of all vertices in 3D space. Please correct me if I am wrong. I do not have a strong background in differential geometry.

If you don't have an answer for analytical shapes but have some thoughts on solving it numerically, that would also be great!!

Any help will be appreciated!! Thank you so much for your time!!

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    $\begingroup$ I would instead recommend that you conceptualize and construct these surfaces by gluing together some number of quarter circle sectors, rather than squares. This would ensure that the boundary is a constant distance from the center, and the idealized surface would not depend on where the glued lines occur. $\endgroup$
    – heropup
    Apr 13 at 0:11
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    $\begingroup$ @heropup Thank you for providing this idea!! I agree. That would be a good way to approach this question, but I don't think that will essentially change this question as the squares are just an extension of quater circle sectors? $\endgroup$
    – Wayne
    Apr 13 at 0:23

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If you conceptualize the surface as being composed of some number of quarter unit circle sectors, it's clear that the total arclength is $n\pi/2$ for some integer $n \ge 4$, and that the boundary of this surface is embedded in the unit sphere. Moreover, the boundary uniquely determines the surface, because as you noted, it is ruled: every point on the surface lies on a straight line segment from the origin to a point on the boundary. So we have some (presumably) symmetric closed curve on the sphere, but now the problem is that such a curve is not uniquely determined. We could choose any smooth closed curve with no self-intersections, so long as the total arclength equals $n\pi/2$, and this generates a surface satisfying the criteria you set.

Think of the case $n = 5$. You could have a relatively gently undulating boundary curve with four extrema, but you could also have a surface that has many "ridges," in which the boundary curve fluctuates up and down many times as it circles the sphere. Being a ruled surface, the paper can do both. In general, the center will not be a point of differentiability. Is there a boundary for which the surface is differentiable at the center? I leave this question open.

If we want such a surface to obey some additional constraint, such as having minimal total curvature, it still might not be unique. One may need to consider some kind of "energy-minimizing" condition that reflects the behavior of a real-world object.

Additionally, we can see that for a sufficiently large $n$, the boundary curve would need to have many turning points in order to avoid self-intersection.

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  • $\begingroup$ Nice analysis!! A few comments on your thoughts: 1) The boundary is indeed NOT unique. This type of saddle surface origami is actually multi-stable which means it has more than one stable configurations (I am a mechanics researcher so this is something we are interested in). That being said, I don't think the 'shape' ever changes, what changes is just the relative positions of the vertices? 2) You are right about the 'energy-minimizing' strategy. We already computed this shape based on this 'mechanics' principle, and now we are looking for a pure geometric description as a reference. $\endgroup$
    – Wayne
    Apr 13 at 3:01
  • $\begingroup$ The surface in question is a cone whose generators connect the boundary curve to the center. Being built from paper, the surface is a developable ruled surface. The curvature and torsion of the boundary curve in space is explictly related to the rate at which these generating lines, drawn on the flat square, change direction as you slide along each square edge of the paper. $\endgroup$
    – MathFont
    Apr 28 at 17:14
  • $\begingroup$ On the Geodesics and Geodesic Circles on a Developable Surface Author(s): William Caspar Graustein Source: Annals of Mathematics, Second Series, Vol. 18, No. 3 (Mar., 1917), pp. 132-138 Published by: Annals of Mathematics Stable URL: jstor.org/stable/2007118 $\endgroup$
    – MathFont
    Apr 28 at 17:15
  • $\begingroup$ As you observed, the straight edges of the square are indeed geodesics on the curved surface. That is one good reason to work with squares. $\endgroup$
    – MathFont
    Apr 28 at 17:19
  • $\begingroup$ @MathFont Thank you for adding your thoughts!! Though the edges are geodesics, I don't think the surfaces are developable? If you cut them, you cannot flatten them on a plane? $\endgroup$
    – Wayne
    May 1 at 2:28

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