By Glen Whitney for the Museum of Mathematics 
We’re in the midst of putting four-bar linkages through their paces. See the introductory post for this series for the MoMath Linkage Kit, an introduction, and general instructions. Last time we saw the complexity of what seems like a pretty simple mechanism: four bars, connected at their ends in a loop. What can one do with that complexity? Well, if you are designing a mechanism, you might want one piece of that mechanism to visit various locations in turn, as the mechanism executes its function. The mathematical question becomes how you determine the linkage that visits the desired locations. So let’s take a look: Two locations
It turns out that if we just specify two locations for a bar, there is always a simple rotation of the plane which maps the one location to the other, as this diagram (and Geogebra worksheet) show. (Technical note: it might happen that it’s actually a pure translation that moves the bar from one position to the other. But that can be thought of as a rotation about a point “infinitely far away”, so it’s not really that much of an exception.) 
Translating this to our four-bar linkage, we can connect that point of rotation to the two ends of the moving bar, and the fourth, fixed bar of the linkage shrinks down just to a fixed point. We apply this result to the upright and the crossbar of a large letter “T” in the next recipe: “4-bar” T linkage
Ingredients: A 27-bar (A), 34-bar (B), and 12-bar (C); two two-ply linking sticks and one three-ply linking sticks; and one spacer. Directions: Link A to B to C and back to A. Note you will need a spacer at the A-C joint in order for both bars to be planar. (The spacer is at the same height as bar B, with one of A or C above and one below.)
Your linkage should look like this: 
How to use: fix the A-C junction at a point so it is free to rotate. Position the linkage so that the 34-bar is perfectly horizontal. Now rotate it 90 degrees to the right. The top end of the 34 bar is now exactly at the midpoint of its former location, i.e. the two positions of the 34 make a letter “T”. The bar lengths were computed, using the construction above, to arrange that letter T. Here are the two positions of the linkage, superimposed with the GIMP: 
OK, if we only used three of the four bars to achieve two positions, we clearly have capability to spare. Can we achieve any three arbitrary positions for one of the bars, which we will call the “floating bar”? For example, could we make the floating bar successively take the positions of the three strokes of a letter A? Indeed, we can. The key in this case is to recognize that with one fixed bar, the free endpoints of each of the other two bars which are linked to the fixed bar must travel in a circle (that’s the “siamese compass” mentioned last time). Therefore, we need to find one circle that goes through the three corresponding positions of one of the endpoints of the floating bar, and another circle that goes through the three positions of the other endpoint of the floating bar. Since there is a circle (or technically, sometimes a straight line) through any three points, we can always do this, as the following diagram and Geogebra worksheet show: 
And here is the diagram with the segments moved to mimic the the three strokes of an “A” (note that points B and E are now superimposed): 
This leads to our next recipe: 4-bar A linkage Ingredients: A 16-bar (let’s call it Z this time, to avoid confusion with all of the other letters floating around), two 40-bars (Y and W), a 60-bar with a 30-hole (X), four linkers, and a pen. Directions: Link Z to Y to X0; link X60 to W; and link W to the free end of Z, crossing over Y. Optionally put a pen through X30. Your linkage should look like this: 
To use: Fix Z horizontally, and rotate Y to move X successively into the three positions shown above. Here they are superimposed: 
Incidentally, if you put a pen in the center hole of the floating bar and trace it all the way around, you produce this lovely cardioid: 
Next time we’ll try to push the limits of the four-bar linkage. More: Filed under: Design, Science         
 
And we’re back with our thirty-ninth installment of Your Comments. Here are our favorites from the past week, from Makezine, our Facebook page, and Twitter. 
In response to Snake Cake! Run for your Lives!, P. Burmese says:
As striking as it appears I'd recoil from a second helping.
In the piece Beauty's Bionic Bald Eagle Beak, James Patrick remarks: It a fantastic accomplishment. I wonder if one could sedate the animal and scan its beak in 3D to create a more accurate, custom prosthesis.
In response to Haunting Drone Instrument, user Eric Freeman says: i'm telling you, send this link to Bear McCreary (composer, Walking Dead, Battlestar Galactica) … maybe he can mix this instrument into one of his scores…
In the piece Back Yard Roller Coaster with 12′ Drop, Clifford Paul says: This coaster looks great and is nothing in terms of pure mortal danger; in Chelyabinsk, Russia our family beheld a dollar (please read 28 rubles) carnival of DIY created "Big Wheels" for the children to participate in a what might be envisioned as a Roman arena chariot race. No horses for these chariots, but these tricycles were complete with unpadded seats repurposed from old 1940′s tractors, adorned with jagged metal shards and welded joints, guaranteed to fail rusted through frames, and bike wheels from old Soviet-era bicycles. Parental terror could not be derived from these vehicles, nor seeming disinterest in the eat-or-be-eaten manner by which most of the unsupervised tots were colliding, nor the seeming lack of triage for the hapless participants. No, the terror was not with the rides, riders or audience – it was the man arc welding 20-foot I-beams with the electrodes and power source just inches from the Russian Circus Maximus. Given a choice between a cleanly designed PVC track or dismemberment/ electrocution in a Russian playground I know what every parent would choose.
In the article Lampshades from Coffee Grounds, user trkemp writes: One of the questions I have when I see a "new" material, like wood putty made from coffee grounds, is: what is the long term stability and safety of it? Will it retain it's color? Will it disintegrate? I know that as a child I often built things out of unconventional materials. When I went through boxes of my old stuff from my parent's I discovered that many of my early creations looked awful or were falling apart. Just because this is "patented" doesn't mean that any long term testing has been done on it. Will it out-gas something when the light bulb warms it up. Will sitting in the sun make bits flake off and fall into your Cheerios. Using traditional materials and finishes may seem less wonderful in some way, but you know what the end results will be.
Like these comments? Be sure to sound off in the comments! You could be in next week’s column. Filed under: Your Comments         
  Continuous casting copper disc (99.95% pure), macro etched, ∅ ≈83 mm, by Heinrich Pniok.  Well, what’s left of August is Metals Month, I should say. A broad subject, to be sure, and with only a couple of weeks to explore it, I want to be fairly strict about focusing on interesting and unusual metals themselves (and processes for working with them) rather than more general “cool stuff made from metal.” Even limiting the scope this way, our archives are rich with relevant content, and my first priority is to round up the best of it. Look for that tomorrow, and in the meantime, as always, if you have suggestions for subjects you’d like us to cover, under this banner, please let us know below.
Filed under: Announcements, Chemistry, Earth Science         
 It won’t be long before our college-aged readers head back to school for yet another year of studying and general devotion to all things academic. To make the next couple semesters a little more geeky, we’re rounding up our favorite hacks that would be perfect for the residence halls. #10  A pair of MIT students created the “iDoor.” Not only can you activate this hydraulic door mod with an iPhone, but it also has a secret knock detector for admittance.
#9  Another pair of MIT students tricked out their dorm room with MIDAS, a centralized dorm room automation system. It’s complete with an over-the-top “party mode,” demoed in the video above.
#8  Our friend Becky Stern shows you how to trick out your boring couch with some EL wire.
#7 
Space is scarce in many dorm rooms, so why not make the best of it? This bunk couch, which was recently spotted on i3 Detroit’s Facebook page, would be perfect for video gaming or movie watching. Just watch your head.
#6 
What better way to show off your impeccable taste in music than using this speaker to blast out spherical sound waves? The project is from our own Sean Michael Ragan and it also serves as a nice piece of decor.
#5  Of course, since there’s always some academic interest in the consumption of alcoholic beverages, here’s a robot that pours the perfect glass of beer. Just don’t let your RA find out about it.
#4  When you’re cramming for an exam or trying to get some sleep and the party is happening right outside your window, this automatic window closer can take care of the situation without any intervention from you.
#3 
If you’re looking build some of your own inexpensive furniture, the Rok-Bak chair is the perfect choice for studying in comfort.
#2 
Here’s another stylish chair designed from plywood.
#1 
Living with others is a fact of life for many college students. Make sure that everyone in your suite does their fair share of the dishes with this BeagleBone-based dirty dish detector. Filed under: Education, Home and Garden         
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