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The Nash-Kuiper $C^1$ isometric embedding of flat torus into $\mathbb{R}^3$ has recently been spectacularly visualized by the Hevea Project. This suggests two questions.

Q1. What is the area of the smallest square that realizes the corrugated Nash-Kuiper embedding?

The Hevea project shows this image of how the square maps to the torus after the 4th corrugation "wave":


Hevea
And here is the 3D embedded torus, overview (left) and closeup (right), which appears to be approximately $R$-major / $r$-minor axes, $R=2$ / $r=1$: Torus
Independent of the above, one can "gift-wrap" non-flat surfaces with paper or foil, as was studied for wrapping a ball by Demaine et al.1

Q2. What is the area of the smallest square that can gift-wrap a torus, in the sense of completely covering its surface? For specificity, let the major axis (center of tube to center of torus) be $R=2$ and the minor axis (radius of the tube) be $r=1$. Thus the aspect ratio $R/r=2$. (When $R>r$, it is a ring torus. As j.c. says, $R=2$ / $r=1$ is known as the "unit ring torus.")

If this torus were a present, one might wrap its convex hull, but here I mean truly covering all of the surface with paper, so that the wrapped result looks like a torus (with many crinkling creases).

From the Demaine et al. paper, here is a definition of what constitutes a wrapping:

Definition of wrapping. A wrapping is a noncrossing contractive mapping of a piece of paper into Euclidean 3-space. The contractive constraint means that every distance either decreases or stays the same, as measured by shortest paths on the piece of paper before and after mapping via the folding. Contractive mappings are called short or contracting by Burago and Zalgaller [BZ96] and submetry maps by Pak [Pak06].


1 Demaine, Erik D., Martin L. Demaine, John Iacono, and Stefan Langerman. "Wrapping spheres with flat paper." Computational Geometry 42, No. 8 (2009): 748-757. (PDF download.)

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    $\begingroup$ For Q1. Maybe I'm missing something but since the Nash Kuiper embedding is $C^1$ and isometric, its area is the area of the flat torus you start with, and by Nash Kuiper, every flat torus is isometrically embeddable, so I don't really see what Q1 means. $\endgroup$ – Thomas Richard Sep 23 '15 at 15:19
  • $\begingroup$ @ThomasRichard: That's remarkable, with all those corrugations. $\endgroup$ – Joseph O'Rourke Sep 23 '15 at 15:27
  • $\begingroup$ Just to be clear, is a "gift-wrapping of a torus" a wrapping whose image is some particular embedding of a torus given in advance? $\endgroup$ – j.c. Sep 23 '15 at 15:35
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    $\begingroup$ The paper by Borrelli et al explains this fairly clearly. They begin with a wrapping $f_{init}$ from the flat torus to the embedded torus, and (via their corrugation procedure) generate a sequence $f_{k,3}$ of wrappings from the flat torus to an embedding of the $k$-corrugated torus in Euclidean space. This sequence is constructed so that the "isometric default" (the pointwise difference between the inner product on the flat torus and the pullback of the inner product by $f_{k,3}$) converges to zero as $k$ goes to $\infty$. (cont'd) $\endgroup$ – j.c. Sep 24 '15 at 0:11
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    $\begingroup$ To attempt to address your question: the image of $f_{init}$ is just the area of the initial embedded torus they chose; as you say in your edit, it may well be the "unit ring torus". The area of the $k$th corrugated torus (the image of the flat torus under $f_{k,3}$) converges to that of the flat square torus. $\endgroup$ – j.c. Sep 24 '15 at 0:15
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This answer addresses Q2, as Q1 was more or less answered in the comments. Edit of Oct 21, 2017, an upper bound and a big correction.

You can get a lower bound on the area by the fact that the diameter (in the sense of metric spaces) of the starting square must be greater than or equal to the diameter of the final embedded torus with its intrinsic metric.

I think the diameter of the standard embedded torus is realized by the distance between two points on the outer equator that are opposite each other.

I was able to extract the following from the note Geodesics on the Torus and other Surfaces of Revolution Clarified Using Undergraduate Physics Tricks with Bonus: Nonrelativistic and Relativistic Kepler Problems by Robert Jantzen.

For a standard torus with major radius 2 and minor radius 1 (what Jantzen calls the "unit ring torus" and what I'll denote $T_{emb}$), Jantzen computes that the shortest path connecting two antipodal points on the outer equator is of length approximately 15.26/2 = 7.63. Below is his picture (Fig. 11) of the "[1,1;0]" geodesic that connects them:

Fig. 11 in Jantzen

The path is depicted in blue. Somewhat confusingly, a meridian is also colored in blue in this image; that should be disregarded.

The diameter of an open square of area $a^2$ is $\sqrt{2}a$, realized by a diagonal joining two corners.

The square with diameter 7.63 has side length $a=5.39$ and area 29.1, which gives a lower bound on the area of a wrapping square.

This is about 1.47 times the area of the unit ring torus $T_{emb}$ $Rr\pi^2=2\pi^2\approx19.739$ (formula from MathWorld).

Now, this is a pretty awful bound since it's intuitively obvious that if we try to wrap $T_{emb}$ with a square so that the diagonal of the square lies along the above-depicted diameter, there's a lot of the torus that won't be covered (more than half, it seems). This does beat the trivial "area" bound though.


I now consider a different problem: that of finding a map from the flat square torus with "side length" $a$, $T_{flat}(a)$, to $T_{emb}$ which is a wrapping in the sense of the question.

Note that every wrapping from $T_{flat}(a)$ to $T_{emb}$ induces wrapping maps from a square with side length $a$ to $T_{emb}$, so it is possible to get upper bounds for the area of the smallest wrapping square by constructing a wrapping from a flat torus.

For instance, we can get an upper bound by the following obvious map from $T_{flat}(a)$ to $T_{emb}$: let the square fundamental domain of $T_{flat}(a)$ be parametrized by $(x,y)\in[0,a]^2$. Then we map this to the embedded torus so that constant $x$ circles go to longitudinal circles and constant $y$ circles go to meridians.

For what values of $a$ is this map a wrapping? All meridians have the same length $2\pi$ which means that we must take $a\geq 2\pi$ for a wrapping. The longest longitudinal circle is the outer equator of the torus, which has length $6\pi$, which means that in fact we must take $a\geq 6\pi$. One can check that this is also sufficient for the map to be a wrapping (by computing the Jacobian of the above map and seeing that its operator norm $\leq1$ precisely when $a\geq 6\pi$). Therefore:

The above map is a wrapping from $T_{flat}(6\pi)$ to $T_{emb}$ and so an upper bound for the area you're looking for is $36\pi^2\approx 355.3$, which is 18 times the area of $T_{emb}$.

Unfortunately, results relating to this problem don't obviously give lower bounds to Q2. This is because wrappings from a square to $T_{emb}$ don't have to "glue up" to maps from $T_{flat}(a)$ to $T_{emb}$. [In earlier versions of this answer, I did not realize this!]

Nonetheless, I think this is an independently interesting problem (maybe a worthy "Q3"?) and perhaps the results below inspire some other ideas. In particular, I find it unlikely that the best wrapping surjection from a square has a smaller area than the lower bound given for $T_{flat}(a)$ below, though I don't know how to prove it. Indeed, I would be surprised to see any wrapping that does better than the one given as an upper bound above!

In what follows I'll discuss a lower bound for wrappings from $T_{flat}(a)$ to $T_{emb}$ using the idea described in the first part of this answer. The diameter $T_{flat}(a)$ is $a/\sqrt{2}$, realized by a point in the "center" of the square fundamental domain and a vertex.

The flat square torus with the same diameter as $T_{emb}$ has $a=10.79$ and area 116.4, which gives a lower bound on the area of a wrapping flat torus.

This is about 5.90 times the area of $T_{emb}$, $Rr\pi^2=2\pi^2\approx19.739$.

This bound is very unlikely to be sharp for this problem. Note that the diameter of $T_{flat}(a)$ is realized by 4 paths (the segments of the "crossed diagonals"). See the green paths in this figure, which is an edit of this file from wikipedia.

flat torus cartoon

The diameter of $T_{emb}$ is also realized by 4 paths (the "top" and "bottom" of the [1,1;0] geodesic depicted above, as well as the reflections of these through the $xy$ plane). A crude cartoon is depicted below, where the green paths represent the geodesics (but were not actually computed as such).

embdded torus cartoon

However, the graphs formed by the union of these 4 paths in the two spaces have topologically inequivalent embeddings in their respective tori! One way to see this is to consider the (topological) circles formed by the edges in the graphs. For the graph in $T_{flat}(a)$, such a circle never bounds a disk, but it can (in multiple ways) for the graph in $T_{emb}$.

In the following figure, the left panel shows the previous sketch of the diameters on $T_{emb}$ with a red longitudinal circle and blue meridional circle highlighted. If we imagine cutting $T_{emb}$ along those circles and unwrapping it, the graph of diameters will be isotopic to the one in the figure in the right panel.

embedded torus picture cut open

I don't know how to show that this precludes the existence of a wrapping of this size but it is certainly boggling my mind.

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  • $\begingroup$ Nice observation re diameter to achieve a lowerbound. $\endgroup$ – Joseph O'Rourke Sep 23 '15 at 17:29
  • $\begingroup$ "boggling my mind"---Indeed! :-) $\endgroup$ – Joseph O'Rourke Sep 25 '15 at 21:00

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