The riveting tale of my doubly twisted adventure

Subhranil De

It all started when sometime back in March, during my late-night grocery shopping at the local grocery store, I saw a certain table lamp. It had a stand that looked like what you obtain by twisting a rod around itself. There was something simple yet quirky about that stand that surely intrigued me. The next day, I went back and purchased it. In figure 1, the left panel shows a horizontal view of it, while the right one is closer view and captured from an elevated angle. 

Figure 1. The twisted lamp stand (from two different angles)—the thing that started it all!

Let us get to the essence of its geometry. A more regular, pristine version of that twisted stand with a uniform width would be a ‘helical prism’ with square cross section, as depicted in the left panel of figure 2 [photo courtesy: freepik.com]. It is obtained by displacing a horizontal square at a uniform rate along a vertical axis (call it the z-axis) passing through its center, while also rotating the square at a uniform rate as well. The tantalizing surfaces thus produced on the sides are call ‘open normal ruled helicoids’ [1].

For such a helical prism, it is time for some definitions that govern its specifics. Let us define the twist rate  as the angle of rotation of the square cross section per vertical drift z, i.e.

                 .                                                                                               (1)

Now, if the side length of the square cross section is a, a dimensionless twist rate  can be defined as

    .                                                                                               (2)

The convenience of the pure number  is that its value does not change under uniform expansion or compression of the helical prism in all directions. For example, as perceived in a standard enlarged or reduced photograph of a given helical prism, the perceived  value will remain the same!

At this point, I simply felt a helpless, inexplicable whim of trying to make my own helical prism. Somehow, playing cards came to mind, and I purchased several decks of them. They are rectangular instead of square, but it would be something close in shape, I thought. I erected a straight tower first, and then with some clumsy maneuver with my hands I managed to twist it, somewhat. What I achieved is the structure in the right panel of figure 2. A rather crude attempt, indeed, but perhaps with a certain rustic allure. Some people said it looks as if carved from wood.

Figure 2. Left: a computer-generated 3D image of a helical prism of square cross section [photo courtesy: freepik.com]; Right: My first attempt at building my own helical prism using playing cards.

By now, I was obsessed. I wanted to do better. At one point, it dawned on me: sticky notes! How did I not think of it before? Indeed an obvious choice, being a common square-shaped object in our day-to-day lives. The regular 3in3in size, of course, is what I was thinking.  

Thus started my grand affair with sticky notes. I purchased many, many of them, and started building. First, I tried to use the whole standard pad, consisting of some 80 sheets, as a building block, so that the thickness  of each block is the thickness of a whole pad. I tried to maintain an angle of  between every two consecutive building blocks—a value I somewhat randomly chose. Of course, I also tried to ensure that their centers lie on the same vertical axis, i.e., the z-axis. What I obtained was the structure at the left of the panels in figure 3.   

Figure 3. An approach to smoothness, from left to right. The top panel in original color is an almost horizontal view, while the bottom panel in black-and-white is more of a top-view.

Needless to say, this hardly looks like a smooth helical prism! However, now I tried a building block that is half the thickness, consisting of some 40 sheets each, while at the same time trying to maintain an angle of  between consecutive blocks, consistent with an unchanged value for . This is the structure in the middle of the panels—it is somewhat smoother, exuding an alluring serrated grace, with the helical nature more prominent as well. Well, next, I made the building block much thinner, all the way to 10 sheets per block, making the attempted  value accordingly smaller, in an attempt to still retain the same value for . This gave me the structure at the right of the panels—much closer to a smooth helical prism, albeit still retaining a fine, serrated disposition.

At this point, I ask you, the reader, to continue with this process in the realm of the imagination. As the thickness   of each square-shaped building block is made indefinitely smaller, i.e. , (accordingly with , with  fixed!), the structure we obtain in that limit is precisely a smooth helical prism with square cross section and the chosen value of ! Indeed, this thought experiment, along with the depiction in figure 3, is reminiscent of figures accompanying introductory calculus explanations found in textbooks, in the context of, for example, volume calculation of solid objects.

Only naturally, I became curious about properties like the volume and the surface area of our helical prism! I performed some calculations, and I will share what I came up with, but I will not share the full-fledged calculations. Rather, I will share insights that relate to the said properties and will leave any complete calculation to you the reader as a little challenge involving geometry and calculus.   

First, consider the surface area of the helical prism. Ignore any truncations at the top and the bottom, and just imagine the structure of infinite extent. In that light, it makes more sense to talk about the surface area per vertical height. This means that, if the surface area is A, we are interested in calculating dA/dz. And already, at this juncture, I ran into a subtle point that happened to engage my attention to a good extent.  Let us look at the three structures in figure 3 again, made with progressively thinner building blocks from left to right. Consider the total exposed surface area for each structure, ignoring the very top and the very bottom, of course. The actual, exposed surface area, mind it! Pay heed to the fact that the said exposed surface area is always a combination of vertical surfaces and horizontal surfaces. Also, notice that the four vertical sides of every building block are always exposed, no matter what the twist rate is. However, for the horizontal surface area, a greater contribution is contingent on a greater value of twist rate; conversely, a zero twist rate exposes no horizontal surface between successive building blocks, and consequently the helical prism is rendered a straight cylinder of square cross section with its four purely vertical faces. It turns out that for a fixed, given value of , if we are approaching the smooth helical prism by indeed using progressively thinner choices of building blocks (i.e. ) like in figure 3, then the actual exposed surface area, vertical and horizontal combined, approaches a finite limit. Let us call it the proper surface area, . It is given by:

                               .                                                                       (3)

To obtain the above expression is already a sweet little exercise in calculus. To shed more light to it, let me point out that the contribution to the expression above from all the vertical surface area and all the horizontal surface area are 4a  and , respectively. If , we are left with the vertical contribution only, pertaining to the four faces of the square cylinder.

Now, let us get to the curved surface area of the smooth helical prism. Let us call it the smooth surface area,. Interestingly, this surface area will be different from the proper surface area given by (3), and it will actually be smaller in value [Think Pythagoras!]. Setting up the calculation for this one is slightly more involved, and so is its mathematical expression, as given below:

                     .                                   (4)

Well, now to the volume. As we continue the process depicted in figure 3 indefinitely as mentioned, the total volume of the structure made of the progressively thinner building blocks simply approaches the volume of the corresponding smooth, elusive helical tower. Hence, it is not hard to conceive that the proper volume and the smooth volume are the same here, both given by:

                                    .                                                           (5)

The above expression is the same as that for a straight square cylinder, implying that the twisting does not affect the volume—a fact that is charming but not surprising. However, let me tell you here: in case you think the proper volume and the smooth volume having the same value is obvious and trivial, just wait till we get to the next phase of our adventure!

Next, I pursued a playful idea with the structure farthest to the right in figure 3, which is awfully close to the elusive, smooth helical tower. Because of its (almost) helical geometry, a pure rotation about its axis is expected to evoke a sensation of motion of the whole structure along the axis. For example, if it is rotated clockwise (as viewed from above), it should give off the pleasantly deceptive feel of vertically upward drift. For this purpose, I purchased a ‘lazy Susan’ turntable. I placed the tower on it, took a video of the slow rotation, and then sped up the video using a software. ‘Ahh…,’ I exclaimed, when I saw the result. Below is the YouTube link for that video:

https://www.youtube.com/watch?v=HVc0fGkfKcY

For a brief few days, I thought that my playful adventure on the theme of the helical prism had reached a conclusion. Then one day in April, while spending an afternoon hour in a forest glade close to home, I saw something on the forest floor. Among the newly sprung radiant blue wildflowers of spring, there were still countless fallen leaves from last autumn, and one particular leaf caught my attention. It was a simple, shriveling leaf. However, the curvaceous contours of its veins and the dips between them were reminiscent of the surface of a helical prism containing the pointed helical contours and the helicoidal surfaces between them. I did not forget to take a photograph. 

For the leaf though, the curving on the two sides of the mid-rib are naturally in different directions, as you can see in figure 4. A quirky idea came to my mind. How about incorporating the two opposite directions of curving in a single helical structure, perhaps using two different colors of sticky notes, so that the two directions almost seem superposed together?  

Figure 4. Left: the fallen leaf that sparked the doubly twisted episode; Right: once again, my own helical structure with the thinnest building block used.

At first, I was having a hard time conceiving what such a doubly helical structure would even look like. Then, I simply started building. I used the thinnest building blocks again, consisting of 10 sheets each. However, I used two different colors this time for two opposite directions of rotation. Pink, and blue. I started with a pink block, then placed a blue block above it and aligned with it. Then, a second pink block, rotated counterclockwise by some small angle  with respect to the previous pink block! Then the second blue block, rotated clockwise by the same  about the previous blue block! As if two helical prisms, pink and blue, are rising independently through each other, twisting in opposite directions!

Finally, when I was done, I could not help but marvel at the devilishly tantalizing structure I had concocted. Because of all the void spaces between the successive pink building blocks (or separately, the successive blue building blocks), the structure has a certain comb-like (ctenoid) disposition. From certain angles, the feel of pink helicoidal surfaces passing through blue helicoidal surfaces is more apparent. From other angles, there seem to be leaf-like protrusions, the two veinous sides of each leaf being of two different colors. I felt a hint of ludic gratitude toward that shriveled leaf on the forest floor that had given me the first idea of the double twist.

Figure 5. Left: me reveling in my doubly tortuous concoction; Right: An almost top view of the doubly helical structure.

At this juncture, once again, let us perform a limiting process consisting of progressively thinner building blocks in the realm of the imagination. This time, as the thickness  of each square-shaped building block (both pink and blue) is made indefinitely smaller, i.e. , (and accordingly with , with  fixed for both the pink part and the blue part of the structure, despite the opposite twisting directions!), the structure we obtain in that limit is precisely a combination of two smooth helical prisms with square cross section, twisting in opposite directions, each with the same chosen value of . But what do I mean by the combination of the two helical prisms in this context? Let me try to explain as simply but precisely as possible. Imagine the two helical prisms, twisting in opposite directions, being present independently of each other in their respective spaces, with a due amount of successive overlap. By the combination I mean the set of all points in space that belong to the volume of either of the helical prisms or both!

Now, let us focus our attention to the surface area and the volume (per vertical height!). Unlike the case of an individual prism, for the doubly helical structure, the cross section keeps periodically changing depending on the vertical location, since the angle between the two square cross sections of the two different colors keeps changing along the vertical (z) axis . Hence, it makes sense to talk about the average value of any of these quantities, where the average is taken over the whole length along the z-axis before the structure repeats itself!

Look at figure 5 one more time. Pay heed to all the void spaces between successive building blocks of the same color as I already mentioned. Each of these void spaces are bound by exposed surfaces above and below, much of it may be hidden from sight depending on where you look from. Each of these surface areas is a finite fraction of the surface area of each square. However, all of these contribute to the actual exposed surface area. When , the number of building blocks within a given vertical height increases indefinitely, and the number of finite contributions of exposed surface area coming from the intermediate void spaces increases indefinitely as well, thereby leading the total exposed surface area to diverge. Hence, the proper surface area diverges, i.e.,

            ,                                                                                               (6)

where the notation < > denotes the average.

Now to the curved surface area of the limiting shape that is the smooth doubly helical structure. The calculation of this part was the most involved in the course of this whole adventure, and it did not yield a closed-form expression. Below is what I got.

                                                                                                            .                        (7)

Oh, well…

And now to the volume. Consider, yet one more time, the void spaces between the successive building blocks of the same color. The volume of each void space, irrespective of how small  is, always remains a finite fraction of the volume of an individual building block. However, in the course of the limiting process with  , the void volumes do contribute to the volume of the limiting structure that is the smooth doubly helical body, while the calculation of the proper volume, involving only the actual volume occupied by any part of any building block, excludes the void volumes. Hence the smooth volume is greater than the proper volume in this case, unlike what we had for the singly helical case! The expressions are below:

                                                                        ,                                   (8)

while

                                  .                                   (9)

Notice that the above is independent of the twist rate! Why is that? Think about it.

At this point, let us look one more time at (7), the formidable expression for the smooth surface area. It turns out that for the smooth doubly helical structure, if we consider progressively slower twist rates (i.e., , and equivalently ), the limiting value of the smooth surface area has a neat, closed-form expression after all:

                            .                                   (10)

Let me point out to you that the value above is also identical to the average value of the perimeter of the cross section of the smooth doubly helical structure! But why is that? Once again, give it a thought!

At this point, I went ahead with the idea of rotating my own doubly twisted structure with the lazy Susan turntable, just the way I had done with the singly helical structure. Since the twists are in opposite directions for the pink and blue components, the expectation was the illusion of the two colors drifting through each other in opposite directions as well. With utter anticipation, I completed the needful. The YouTube link is below. As you can see, the pink appears to drift upward, and the blue downward. A mesmeric deception indeed, with an almost capricious air.

https://www.youtube.com/watch?v=DGF_m6SIm34

At the end, a couple of friends asked me to try to 3D-print a doubly helical structure. That could indeed produce a next to flawless version. I was tempted, but then decided against it. For now, let the pristine essence and any elusive perfection remain sheltered in the realm of the imagination. Meanwhile, let that hint of crudeness in my hand-made structure scintillate as a mark of precious human touch. My human touch.

References

[1] https://mathcurve.com/surfaces.gb/helicoidregle/helicoidregle.shtml