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Hexagon Weave

On weaving only an array of hexagons
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Quite a few years ago there was some furor in the tennis world regarding tennis racquets. New ones had entered the market, including a larger-than-normal size, and various folks thought the sport was being fouled.

However, they apparently were legal according to the rules of the game. While I haven't paid much attention since, to see if they have locked down some racquet-variety rules, to prevent similar problems in the future, I did happen to notice that the main reason the new racquets had been introduced was because of something known as "the sweet spot", the best part of the racquet for controlling a tennis ball.

A larger sweet spot means a better chance to control the ball, provided you actually make the racquet meet the ball while playing the game.

Well, an Idea came to me: What If The Strings Of The Racquet Formed An Array Of Hexagons Instead Of Squares? See, hexagons are more naturally circular than squares, so it might logically follow that such an array would fit/grip the curved ball-surface better, and therefore equate with a larger sweet spot.

The only problem is, every intersection of a grid of hexagons has exactly 3 lines connecting to it. A grid of squares has 4 lines connecting at each intersection, and a grid of triangles (I THINK I saw a racquet once that was strung with a grid of triangles) has 6 lines connecting at each intersection.

It seems physically impossible, at first glance, to weave a grid of hexagons. After all if every string that enters an intersection must exit again, then you have to have an even total number of lines at each intersection, and a hexagon grid still has only 3.

Note that there are ways to cheat. See the first link, which actually yields a grid of triangles and hexagons.

Nevertheless, I thought of a way, and no, it does not involve big fat knots at every intersection. What you do is visualize the grid of hexagons as having two parallel strings at every line where you would normally see just one string. This means that each intersection can have 3 strings going into it (half of each parallel pair), plus those same 3 strings going out (the other half of each parallel pair).

All that remains to be done is to figure out how to lace them AT the intersection. And, no, it STILL is not a knot! See the second link, showing 3 connected rings. If you imagined that the outside of each ring was cut, and they were made of string, you could pull their connection point tightly together, and they would stay well-connected. You could also pull on the strings such that they would be approaching that intersection at 120- degree angles, exactly like the intersection of 3 hexagons.

So all you have to do, to make a Hexagon Weave, is to ensure that EVERY intersection connects like that one. Which turns out to be slightly more complicated. The parallel strings have to cross each other allong every one of the straight-side lengths of the hexagon array.

Nevertheless, It Appears To Be Workable In Theory. When the strings are all pulled taut, a true grid of hexagons should be produced. Perhaps a tennis racquet featuring a Hexagon Weave can be made yet! Note that ordinary weavings don't pull all the strings taut, and so something like a "doily" would look more like a mass of tangled strings than a nice hexagon pattern. Still, there are a few possibilities, after the example of "fish- net stockings". Since said netting is under tension when worn, its threads are taut. Which means "chicken-wire stockings" should be quite possible....

I've created an image (3rd link). You may have to load it into an image viewer and enlarge it to see the intersection details. But you should be able to see that the green strings zigzag mostly horizontally, always crossing on top of the blue strings; the red strings zigzag from upper-right to lower-left, always crossing on top of the green strings, and the blue strings zigzag from upper-left to lower-right, always crossing on top of the red strings.

Vernon, Oct 31 2011

"Hexagon weave" https://geometricol.../08/tesselation.jpg
As mentioned in the main text, this is a "cheat", since actually you get a mixture of triangles and hexagons. [Vernon, Oct 31 2011, last modified Jul 16 2015]

Borromean Rings http://en.wikipedia...iki/Borromean_rings
As mentioned in the main text, this (first image on the right) is how the middles of three strings can lock together at an intersection. Note that the image in the "Hexagon weave" link just below uses the same three colors, which overlap each other the same way as here (described in last paragraph of main text). [Vernon, Oct 31 2011, last modified Nov 01 2011]

Hexagon Weave http://www.nemitz.n...on/HexagonWeave.png
You should be able to see some of the details, as described in the main text. [Vernon, Oct 31 2011, last modified Nov 01 2011]

Chicken wire http://en.wikipedia.org/wiki/Chicken_wire
Braak! [pocmloc, Oct 31 2011]

Hexagonal weave http://www.absolute...cs/Hexagonal_tiling
[xaviergisz, Nov 01 2011]

Chair Cane Weave http://www.chairsca...ep_strand_cane.html
Octagonal holes from hillbillies. [baconbrain, Nov 01 2011]

3D hexagonal Kagome weave structures http://www.ifam-dd....img/Drahtgitter.jsp
[sqeaketh the wheel, Nov 02 2011]

HexWeave http://www.nemitz.net/vernon/HexWeave.PNG
A "baked" image, as mentioned in an annotation. It occurs to me that there appears to be a visual pun here, involving the idea of an infinite mesh.... [Vernon, Nov 14 2011, last modified Nov 15 2011]

Racquet Rules http://www.livestro...nnis-racquet-rules/
To the extent that that page is accurate, a hexagon weave should be legal, if surrounded by an acceptable frame (the frame of the test-model I made is too physically weak, I think, for actual use in tennis). [Vernon, Nov 21 2011]

Tennis Ball impacting Racquet http://cellar.org/pictures/tennis.jpg
I've been looking for a picture that better shows the strings during this event. But even in this image it should be clear enough that the strings are NOT currently going exactly straight across the racquet body. [Vernon, Nov 21 2011]

Distorted racquet strings http://www.tennisin...ng_tension_giv.html
This article has an image that nicely shows some degree of string-distortion when a ball impacts an ordinary racquet [Vernon, Jul 16 2015]

Simple hexagon grid https://encrypted-t...iHBbsL3jIN73iaIww5_
Mentioned in the 5th paragraph of the main text. [Vernon, Jul 16 2015]

Triangular grid, with hexagons data:image/png;base...asAAAAASUVORK5CYII=
Like the simple hexagon grid, there are still 3 strings at each intersection point. [Vernon, Jul 16 2015]

A string-tension study http://www.ncbi.nlm...gov/pubmed/16195027
Lesser string tension seems associated with higher ball speeds and better control. That could be a "plus" for this Idea, since it may not be easy to get high string tension. [Vernon, Jul 16 2015]

Twist analysis http://s1199.photob...Weave2.png.html?o=0
Balanced left- and right twists [neutrinos_shadow, Jul 16 2015]


       [+] for raising the question of weaving hexagons, and all the topological wonderment thereof.   

       Does the tennis rule-book say that there have to be strings? Why not a single-piece mesh, stamped from a sheet of some guttish material?   

       Alternatively, start with a tall rectangle of material and make rows of vertical slits, suitably offset in adjacent rows. Then, when the material is stretched sideways, it will form a hexagonal grid.
MaxwellBuchanan, Oct 31 2011

       Or! Make the tennis balls superconducting, and place powerful magnets around the rim of the otherwise empty tennis bat. The ball will try to exclude the magnetic flux-lines, leading to a stringless-string effect. The use of electromagnets would allow the bat to be tuned.
MaxwellBuchanan, Oct 31 2011

       To a perfect pitch I suppose?
pocmloc, Oct 31 2011

       I think in tennis it's called "serving"; you're probably thinking of golf.
MaxwellBuchanan, Oct 31 2011

       I never think of golf if I can help it.
pocmloc, Oct 31 2011

       //See, [hexagons] are more naturally circular than squares, so it might logically follow that such an array would fit/grip the curved ball-surface better, and therefore equate with a larger sweet spot.//   

       Sweet spot is due to the node of vibration of the strings on a raquet (accoring to a quick internet search). You could argue that strings with 3-way symmetry (hexagons) gives a bigger/better node of vibration than 2-way symmetry, but I'll leave that speculation to someone else.   

       If you've invented a new type of weave, great, you don't need to find a purpose for the new type of weave.   

       I still can't see your illustration (and I can't figure it out from the text alone), but how is this different from regular hexagonal weave? (see link)
xaviergisz, Nov 01 2011

       My canoe has what is called woven-cane seats (also known as cane-bottom). The weave starts at right angles to make squares, then goes across diagonally to make octagons.   

       There are lots of octagonal openings, which look circular. And which provide fair grip for wet nylon shorts. That should serve for a racquet. (Link.) (It works for snowshoes, too.)   

       The hexagon weave and the chicken wire are not stable or solid. There are no direct lines running out to the frame, and there are no internal triangles. As a racquet weave, it's going to be mushy as heck.   

       Back when I was building chicken pens, we could stretch and flex the chicken wire like crazy, and it would always end up sagging. I assume that tennis racqueteers want the opposite of that.
baconbrain, Nov 01 2011

       Chicken wire can be held taut and flat inside a complete frame; it only tends to sag and stretch if it is attached on 2 or 3 sides only. Similarly, a hexagonal net would be fine in a tennis racquet.
spidermother, Nov 01 2011

       Folks, as I write this, the 3rd-link image described in the main text is available for viewing. It has a background-grid of hexagons, with the weaving pattern covering part of it. The hexagons aren't so obvious amidst the weave. If the strings were thinner or the hexagons larger, the pattern would be more obvious.   

       [xaviergisz], the first weave that I saw at your link uses really thick "strings", and the size of the hexagons is strictly related to that. The weave I'm proposing can allow arbitrarily large hexagons, regardless of string-size.   

       And the second weave I saw there creates a mixture of hexagons and triangles. This weave yields hexagons only.
Vernon, Nov 01 2011

       Are you talking about the Hexagon Weave at nemitz.net? The light purple background? http://www.nemitz.net/verno n/HexagonWeave.png?   

       Because that is just like chicken wire, as far as I can tell.   

       I'm still saying it isn't going to be rigid. I'll go look for something to test or illustrate.   

       Meanwhile, I'd take your hexagon weaving loom in the lapweaving link, and add another set of pegs to make another set of triangles, weave that all together to make hexagons, and forget binding the layers together.   

       Standard tennis rackets don't interlock the strings in any way, they are just woven.   

       If you really want to bind it all together, do what the snowshoers do, and varnish the strings after assembly.
baconbrain, Nov 01 2011

       [baconbrain], yes, that's the image. But if you studied how chicken wire is made, you would see that this must be different. Chicken wire makes use of the fact that steel tends to maintain a bent shape. So, certain lengths of the middle parts of two parallel strands are bent around each other multiple times, forming one side of that hexagon. Two sides of each hexagon are made that way, but the other four sides of each hexagon are single strands of steel.   

       Here, EVERY side of every hexagon involves pairs of strings, and they are not wrapped round-and-round each other. There is a single half-twist that allows them to enter (in the drawing) each intersection neatly. The interlacing at the intersections is what holds this weave together.   

       And, yes, I know that tennis racquets involve straight strings only, and that this pattern could be problematic for that use, unless the strings were VERY strong and VERY taut. I don't know that it is impossible to achieve.
Vernon, Nov 01 2011

       Well, after close scrutiny, I'm fairly sure that Vernon's weave can produce a flat, rigid net, as long as the terminus of every strand is rigidly held in the frame. It has no degrees of freedom; if the strings don't change in length, a node can only move if an adjacent node moves; by an informal use of mathematical induction, it appears to me that no node can move unless a string changes length or one of the attachment points slips.
spidermother, Nov 01 2011

       Thanks, [spidermother]. As I wrote in the main text: "When the strings are all pulled taut, a true grid of hexagons should be produced." Its rigidity depends on the strings being taut.   

       It should be possible to construct this weave in a frame using just one long long string (which is actually normally done in tennis-racquet frames). After the string crosses the framed gap, it doubles back to cross again in a different place, over and over again.   

       Also, the first two sets of crossings (say, the horizontal crossing and the upper-right-to-lower-left crossing) are easy, since all the strings of the second set are always "above" the strings of the first set (they don't interweave like in a normal square-pattern tennis racquet).   

       But that third set of crossing strings (upper-left-to-lower-right) is a nightmare, unless you employ a template holding the first two crossings in zigzags. And even then it will be tedious; all the intersection-lacings must be done exactly right.   

       If you actually do this with just one long long string, the final nightmare will be getting all the slack out EVENLY, after all the weaving is done. Good Luck! --which probably explains why this Idea is Half-Baked! Although if the Hexagon Weave is used to make something like "chickenwire stockings", then the strings become taut when a leg is inserted, and they use stretchy string to make sure it becomes taut. Possibly much more Bake-able than a tennis racquet....
Vernon, Nov 01 2011

       Hexagonal Japanese Kagome basket weave is long known The link shows pictures of 3D hexagonal Kagome weave structures for other purposes. [+]
sqeaketh the wheel, Nov 02 2011

       //Thanks, [spidermother].// No worries. I just tells it like I sees it.   

       If you want to try this, I suggest making a hexagonal grid of nails driven part way into a board. Tie the strings to the outside nails. Weave one colour at a time, pulling the strings uniformly tight if possible, and tie off the loose end. Once all the strings are in place, you should be able to pull out the nails (except those around the edge).   

       You probably want string with a little stretch, to take up the small amount of slack generated as each nail is removed. Ordinary knitting wool might be good. I'd be interested to know how it works.
spidermother, Nov 02 2011

       If a hexagon is good, an octagon should be better as it even more closely approaches a circle. This opens the potential for an arms race among tennis players as each new fabulously strung racquet offers polygons with more and more sides and consequent tennis superiority.
bungston, Nov 02 2011

       [squeaketh], like chicken wire, those Kagome structures appear to depend on the fact that metal can usually hold its shape after being bent. And the basket weave yields hexagons that are very restricted in size, related to the diameter of the strands of material being woven. The strands themselves occupy enough space as to distort the overall grid (it's not pure hexagons; it includes triangles of overlapping strands, too). This Idea features a different weave that yields hexagons that are not size-restricted (so the strands don't have to occupy so much of the overall space of the grid).   

       [bungston], sorry, the main text explains what I thought was true about tennis-racquet sweet spots at the time of the furor, when I originally thought of this Idea. It appears that the sweet spot depends on a different factor. So, it remains to be determined whether or not a racquet strung with this Hexagon Weave will have a larger sweet spot.
Vernon, Nov 02 2011

       I'm guessing that the "sweet spot" is just the addition of the tensions of the strings... so it shouldn't be that difficult to figure out if it gets bigger or smaller (or remains the same).
FlyingToaster, Nov 02 2011

       I'm guessing that the "sweet spot" is where the mesh isn't distorted by edge effects. If that is so, the interlocking strands of this hex weave would be a good thing.
baconbrain, Nov 02 2011

       how about non-regular patterns, like the quasicrystals the recent nobel prize winner discovered   

       I know it isn't feasible, just a funny idea/discussion to me.
EdwinBakery, Nov 02 2011

       Folks, I've spent some time constructing an actual hexagon weave. Since I was thinking in terms of tennis racquets, I got some 50-pound-test monofilament nylon fishing line and planned for an average width of the zigzagging lines to be 1/2 inch (a little more than a centimeter).   

       The lid of a 5-gallon plastic bucket got a big circle cut out of it, and a lot of small holes were drilled around the edge of the big hole.   

       Since the circular cut-out was the right size, and made of easily scratch-able plastic, I marked it with lots of lines in three directions, and nailed about 400 "wire brads" into appropriate intersections, creating a basic hexagon grid. The plastic alone wasn't strong enough to hold the nails solidly, but a piece of plywood behind it worked fine.   

       As previously indicated, the first two "layers" of fishing line were easily zigzagged among the nails, since the second layer was placed everywhere on top of the first. The third layer was not quite as nightmarish as I originally thought it would be, because the actual intersection is a bit simpler than portrayed in the "Borromean Rings" link.   

       It still took a couple of weeks of spare time to get it done, using one long long continuous strand for all three layers of zigzags and intersections.   

       I did what I could to make sure all the strands were pulled taut, but when the nails were removed, some slack was introduced. Nevertheless, the weave is holding its shape quite well, despite a bit of flexibility. (While the overall construct is about the size of a tennis racquet "head" --there is no handle-- the strands currently are nowhere near taut enough for the thing to be used for that game.)   

       Anyway, the finished thing was placed on a flatbed scanner. I had to manually press the mesh onto the glass to get an image that was in focus (would not have been able to do that if the slack hadn't been there!). The scanned image is over 4 megabytes in size, so I clipped out a portion of it for posting on the Web ("hexweave" link).   

       The scanned image is about 3 times larger than life-size. This means the thickness of the strands is also about 3 times life-size. The hexagon pattern is NOT QUITE as obvious as it would be if the mesh had been made wider than 1/2-inch, or if I had used thinner fishing line. But it is obvious enough to prove that it can work.
Vernon, Nov 14 2011

       That's beautiful, [Vernon]. How well does it serve?
swimswim, Nov 14 2011

       (Square) tennis racket weave has no angles to speak of, at least none that rub. You've a plethora of 60deg angles just waiting to wear and break... maybe a badminton racket.   

       Considered lacrosse ?   

       Good job, by the way; I know who I'll be going to for chainmail.
FlyingToaster, Nov 15 2011

       Bun just for the conviction and dedication to see this through.
AusCan531, Nov 15 2011

       [Flying Toaster], I was wondering when someone would mention the issue of strands rubbing against each other. The angles are more like 120 degrees, though. At least when the mesh-hole-size is larger (maybe 90 degrees in the one I constructed), or the strands thinner. On the other hand, either of those things would make it less suitable for use in tennis.   

       The best answer to the rubbing problem may be the first anno on Nov 1 by [spidermother]. That is, the more taut the strands, the less "degrees of freedom" they will have in which to move. And obviously, the less they can move the less they can rub.   

       Now, I've seen the high-speed photos of the impact of a tennis ball with a racquet, and know full well that even very taut straight strands will either stretch or otherwise exhibit some freedom to move under that circumstance. Logically, however, it means that even a standard square mesh will have strands rubbing each other. Which means the wear-and-tear problem isn't actually quite as great a problem as you might think it is.
Vernon, Nov 15 2011

       I agree, [Vernon]. Rubbing won't be a big problem with this weave.   

       The strings will be weakened where they cross, more than in the case of a square weave due to the sharper angle, but again, not enough to matter.   

       A tennis racquet strung like this would, I predict, work perfectly well. It would be a little softer for a given string tension, and the forces would be closer to radially symmetrical, which would change the response somewhat - quite likely for the better!
spidermother, Nov 15 2011

       [Vernon], that's beautiful. [+]
baconbrain, Nov 15 2011

       Nicely done, [Vernon].
If I may make a suggestion; alternate the twist at each successive junction, so you get a "left over" then a "right over" around each hex. If they are all the same, it may impart a spin on the ball, and possible a twist to the frame too. By alternating, I think it will "sit" more neatly.

<Later> I don't think it is actually possible to do what I've suggested... Damn 3-way symmetry!
<Even later> Haha, figured it out. See linky. Equal numbers of left twists and right twists (even if, at first glance, it doesn't seem so).
neutrinos_shadow, Jul 16 2015

       [neutrinos_shadow], please look at the 3rd link of the group, to clearly see which strings cross which other strings. I suspect the crossings you suggest will unravel the weave. One thing I found out, when first contemplating the weave, is that there are really only two ways for the strings to cross at each hexagon- intersection.   

       That is, consider the red and blue stings in the linked image mentioned above. At the intersection, the red can cross the blue, or the blue can cross the red --and the other two crossings, of green-and-red or green-and- blue, are forced. So, only two ways to do the intersection.   

       The entire weave uses only one of the two different intersection-crossings; trying to mix them doesn't work because it leads to the equivalent of "ring mail", a whole lot of closed loops, instead of cross-fabric strings.   

       I invite you to draw some red, green, and blue lines of your own, to attempt to devise the alternate crossings you would like to see.
Vernon, Jul 16 2015

       One thing I didn't really think about when I constructed that hexagon weave using 50-pound nylon line, was that if every side of every hexagon basically consists of 2 stings, then a lesser-strength line would probably be fine, for a tennis racquet. The intersections would be a bit tighter and the strings would be bent slightly less, passing through each intersection.
Vernon, Jul 17 2015


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