A Counting Balance
Counters (weights) and Lever Arms (moments)
Add, Subtract, Multiply Small Integers
A balance designed and made for my granddaughter Anna Plumlee for her class at the Montessori school in Brooklyn Heights, New York (1998).
A Counting Balance
Counters (weights) and Lever Arms (moments)
Add, Subtract, Multiply Small Integers
A balance designed and made for my granddaughter Anna Plumlee for her class at the Montessori school in Brooklyn Heights, New York (1998).
In a darkened room suspend a point of light on a thread from the ceiling; put a camera on the floor; give the source a swing; and open the shutter for a while. The result is a recording of the path of the luminous point.
A simple pendulum, as in a clock, would produce only a straight line, but if the bob can swing in all directions straight lines, circles, and ellipses can be the result:
A Lissajous suspension produces more interesting results. The pendulum has one length in one plane and a shorter one in a swing plane at right angles to it. The result is the family of Lissajous figures:
And, just for laughs:
After having worked out a design I set up a rudimentary production line and made about sixteen of these rubber band guns, mainly for my grandchildren—then under ten. Every kid has to have two because it’s no fun when you have a friend over if you’re the only one armed. Two of them became a wedding anniversary gift set packaged as a dueling pair. One I kept for myself after having discovered its deadly utility as a fly swatter. Imagine a swatter that doesn’t pull down the curtains or knock over the crockery!
I stacked the thin wooden sheets and cut them all at once on a bandsaw. Needing a spring to return the sear after each release led to the making of a simple spring winding mechanism for steel piano wire. The “barrel” needs to be long enough to store a useful amount of energy. U. S. Post office munitions (#64 calibre x 63mm) pack a punch and are free: .
When spring finally comes to Ithaca after a cold and dismal winter the students at Cornell shed their winter gear, bid farewell to the endless grey above and turn their faces at last to the sun. It is said that Ithaca sees more cloudy days per year than the Olympic Peninsula, the record holder.
April’s social centerpiece then was spring house party weekend the Saturday of which was Spring Day, a more or less pagan celebration featuring weekend long “dates” (“blind” or otherwise); women allowed—under some sort of chaperonage—upstairs in the fraternity houses; more alcohol than might be prudent; and widespread organized inanity. Among these: pie throwing contests, the hotel school’s Waiter’s Derby, the architect’s Dragon Day parade entry, fraternity and sorority floats and parties, and in 1948, the recently established Inter-fraternity Crew Race.
A fifty yard course on Beebe Lake crossed above Triphammer Falls. The rules were simple: human effort only (no motors, no wind), and five crew members. This last to eliminate sophisticated racing shells. Entries comprised anything that would float from rafts of beer kegs to brass bedsteads made seaworthy.
We at Sigma Chi decided to build a paddlewheel boat. The naval architecture and ship building expertise fell to me and to Al “Oop” Thomas.
We first conceived a Mississippi riverboat arrangement with a paddle wheel on each side but soon abandoned that idea as impossibly unstable; even at small departures from the vertical we couldn’t get the center of gravity below the center of buoyancy. We had thought at first that one boat and two wheels would be easier to make. And so it became a center-wheeler with a pontoon boat on either side.
Al proved expert at building these using pine freeboards and a galvanized sheet iron bottom with carefully fashioned lock-seam joints. The wheel, eight feet in diameter, had eight paddles driven by a crank handle on either end to be turned by four of the strongest among us.
In deep secrecy we took the parts down to Cayuga Lake Inlet and reassembled them there for trials. It was windy and cold and the water choppy so we spent less time evaluating results than we might have. It seemed fast enough but the enthusiastic man-power overcame some of the structural elements which had to be re-detailed for strength. One crucial aspect of the design’s shortcoming went unnoticed.
We christened it the “Big Red Wheel” in homage to the world of Cornell “Big Red” sports and—incidentally of course—to the bawdy barroom song. The engine room comprised Bill Konold, Bob Rath, Ed Rorke, and Al Thomas.
Reassembled on Beebe Lake before dawn on the day of the race it passed a strength test; we were ready.
The race was almost an anti-climax. We churned forward way ahead of the competition. But owing to unexpectedly large counter-torque the stern was sufficiently depressed that we took on water at a rate large enough to cross the finish line essentially submerged. The rules had not addressed this submarine possibility and we were adjudged the winners—both for speed and originality of design.
That was the year we stayed for summer session and Bob Rath and I painted the Lansing high school building. The Big Red Wheel spent all summer on the lake gradually falling into disrepair at the hands of whoever could manage to keep it afloat.
Wm. C. Atkinson, 2016
NOTE: If you wish you may go directly (back) to my original ROWING site.
Rowing was big at Cornell when I was there as a student in the forties, as it had been previously at the turn of the twentieth century. My parents, both Cornellians of that era (Classes of 1912 and 1918), spoke fondly of the winning crews of yore and of their legendary coach Charles E. “Pop” Courtney. And so we modern students paid enough attention to Cornell rowing to have often watched the races on Cayuga Lake and to have had time to observe the rhythm of the stroke.
As a student in Mechanical Engineering and, more particularly in Fluid Mechanics, I often puzzled over the vagaries of the motion of the shell. It is noticeably uneven. In the drive, as the crew applies effort to the oars, the boat seems counter-intuitively to slow only to leap ahead again as the oars and rowers return for the next stroke. This is not an illusion. The shell is feather-light compared to the weight of the crew so that as they propel their mass stern-ward in the reach for the next catch the boat is sent sharply in the opposite (bow-ward) direction.
It is well known in fluid dynamics that the power required to maintain a boat’s motion against fluid friction increases as the cube of boat’s speed through the fluid. Thus, the energy expended to maintain a high velocity for a time can never recovered in the same time spent maintaining a lower velocity.
So, for the same amount of effort, how much faster might a racing shell go if its velocity were somehow kept constant?
As I discovered in later years finding the answer is a daunting task if all you have at your disposal is a pencil, a slide rule, and a piece of paper. Every year or so I would cast my mind back over this problem, sit down in a comfortable chair with my rudimentary tools and have a go at it. My calculus being by then sufficiently rusty meant that even writing down the basic equation of motion for the shell/rower system was an uncertain task. And so I would abandon the effort for better luck another time.
As time went on I had various occasions to dip into the calculus for my career engineering work and in 1964 I discovered the computer—a behemoth known as the CDC 6600  occupying an entire floor of a large building on Boston’s miracle circumferential highway: Route 128. I bought a copy of Daniel D. McCracken’s a guide to FORTRAN IV programming and taught myself to write code. It helped that I was, by then, in the computer division of my employer’s consulting engineering firm. Eventually I became a half decent programmer.
After retirement in 1994 I ran across an article  about a new competitive oar blade design that seemed to be winning races: “Just two years ago top college crews were winning races in … five minutes and 45 seconds. [Today] that time has dropped [by] 15 seconds.”
And at about this time (1995) I had a single shell in temporary “storage” in my yard—a birthday gift from a friend to his wife. I was inspired to measure it up for a potential computer description. I realized that now I could assign pencil, slide rule, and paper operations to a machine and so I returned at last to the rowing problem.
First I made a schematic diagram and then worked out a force balance and the differential equation of motion involving the excursions and velocities of the separate masses of crew and boat. From my professional days I had inherited an extensive library of indispensable FORTRAN routines greatly simplifying the writing of code for a great variety of useful functions related to data entry, reading, and checking; sorting; interpolation in data sets; and the control of organization and pagination in final output printing—so-called “housekeeping” routines.
Next, for rowing eights, I designed a seating arrangement that would permit a crew to make eight individual strokes in sequence without physical interference in the boat. Sequential strokes were essential to the aim of reducing the large velocity excursions to eight smaller “ripples”. This new geometry required a slight lengthening of the shell from its traditional sixty-two foot base. In practice it would also require the coxswain to broadcast a “beat” to a headset worn by each rower so that each had his own private stroke timing.
After several months enough code had been written to permit the production of rudimentary graphs of speed, forces, and motion depending on a very large number of input variables. Ultimately the model had more than eighty input variables having to do with boat, oar, and rower characteristics, environmental data (wind speed, water temperature, etc.), initial conditions, and the lift and drag characteristics of oar blades. The model currently has many thousands of lines of code and is still a work in progress.
The rower has two points of force contact in the dynamic system—at the oar handle and at the footstretcher. With one exception  no rowing modeler had previously considered forces at the rower’s feet in relation to their possible effect on the net advancement of the shell. The forces there are as large as those at the oars. They produce the large shell excursions seen. But, because nothing other than fluid friction connects the boat to the water (no “blades” involved), the net effect on speed was, I think, assumed to be nonexistent or too small to be of importance.
The program divides the stroke into one-thousand increments each of which is calculated in sequence and then repeated (iterated) with re-adjusted starting conditions until the calculated shell advance per sweep is equal to the theoretical (geometric) advance—often a process of more than a hundred iterations. In addition the shell speed at stroke’s end had to be continuously adjusted to equal that at the beginning. In 1996 when I started coding on my PC this process took agonizing and frustrating minutes per iteration. Today one thousand “external” iterations (plus many hundreds of thousands of internal ones) are completed virtually instantaneously.
It turned out to be too daunting to write code for eight separate rowers whose strokes were offset with respect to one another by one eighth of the cycle. But I finally hit upon the way to model a shell speed with constant velocity—my initial aim. It was as simple as assigning a near infinite inertial mass to the shell (its speed can’t be changed no matter what the size of the forces on it) but without changing its displacement or the hydrodynamic resistance forces—something a program is eminently capable of doing.
The result was surprising. The average shell speed for the same rower power expenditure was less than that with the original shell speed excursions. The net work done on the boat at the foot board at the varying speed turned out to be positive and not to be done without. A related observation: I think if one sits in a shell (no oars) and shifts his weight, quickly one way and slowly the other the shell will (very inefficiently) advance in response to the quick effort. Children commonly can do this in go-carts, which have greater friction effects than does a boat on water.
So, I had answered my question. In rowing there is virtually nothing to be gained by eliminating the velocity excursions of the shell.
But there were other things to ponder. Now that I had a working model I decided to improve it to a point where it might actually be of use to rowers . Too, there was much rowing lore and conventional wisdom to look into, some of which confused and baffled me for long periods of time.
Rowers spoke of the various effects of this or that on the “force at the blade”; things like the blade’s size or its efficiency, or the degree to which it was submerged, or the lever ratio (“gearing”). In fact, I finally realized that the force at the blade is nothing more than the result of its balancing the rower’s effort at the handle, having nothing whatever to do with the blade’s shape or size.
It took ages to resolve the mystery of oar blade slip in the water. Observed from above, as from a bridge, the blade seems to leave the water at almost exactly its entrance point—it doesn’t seem to slip at all. What was happening? I fixed a rectangular board to the end of a broomstick “foot-wise”, noted its area, weighted it a known amount, and timed it as it sank a measured vertical distance in a deep pail of water. Crude to be sure, but the result of calculation from the known resistance to flow of flat plates told me that oar blade slip must be hugely greater than anyone seemed to think. But, how was I to determine it? There’s no fixed catch reference point left behind in the water.
Then I saw that a blade with no force on it was the only condition under which its slip could be zero. What is the path of an unloaded oar blade? All one has to do, from a coasting shell, is to drop the blade into the water and let it drift freely stern-ward. The drifting blade center follows a path well known in analytical geometry—the tractrix. And thus the mathematical solution of the path of the tractrix enables one to find the true blade slip which, in practice, is huge—as much as a meter in many cases. The slip is the longitudinal separation between the actual blade path and the zero-slip path.
The simple tractrix is the path of a straight, flat blade—one without a cant angle or “spoon”. The conventional six-degree canted blade follows a modified path that I have chosen to call an “oblique” tractrix. Unfortunately the solution to the differential equation for this path was not direct (it involved arc functions) and had to be iterated by trial and error using Euler’s modified method.
Working out the complex vector diagrams of the blade forces and velocities was difficult owing to the dynamic nature of the lift and drag forces. The slip of the blade depended upon the lift and drag forces which, in turn, depended upon the slip and so yet another iterative process was introduced to solve every instant of the stroke. My general “take away” from all of this is that “Nature is inherently iterative”.
I chose, too, to iterate (by Euler’s method) the main solution to the equation of motion, although this is not required; a direct solution involving the velocity squared is easily made. My reasoning was that I was not entirely sure that the valid velocity exponent is an integral 2. Many hydrodynamic relationships use an exponent near two (1.85 for instance) and in trying to see whether ROWING would produce the same or similar performance as some real world data sets that I had available (from Kleshnev ) I thought to see whether exponent changes might account for interesting differences. The velocity exponent is part of my input data set. There is no way of dealing with non-integer equation of motion exponents without an iterative process.
Eventually I discovered others who had written a comprehensive rowing model. One, in MatLab, is Marinus van Holst of the Netherlands . Among other things he and I once agreed on a data set with the same content for each model and showed that each produced almost the same result in speed and rower effort; a strong indicator of the validity of each model. Others are Sander Roosendaal  and Leo Lazauskas .
The drive has three phases: force buildup, force variable, and force declining which can be defined in terms of straight lines defining absolute force and force duration.
I divided the stroke into eight “regimes”, four in the drive phase (including the arms bend) and four in the free return permitting study of various patterns of slide and torso motion: pause, onset, steady, and close. Thus, especially, the free return can seem to be “coached” in its four phases. However, contrary to the conventional wisdom, the model shows that nothing one can do in the free return can directly alter the (average) speed of the center-of-mass of the boat-rower system. Nevertheless, it is possible to save some small amount of energy in the free return—by encouraging “float”— which could be used elsewhere as slightly increased stroke rate or greater effort at the oars, but without increase in total effort.
Since, on average in the steady state, the shell is not accelerating it must be that the instantaneous forces on it sum to zero over the stroke. This requirement is a powerful check on the accuracy of the computations. The same is true of the reaction forces on the body of the rower.
Much was made in the rowing literature of the supposed beneficial effect of hydrodynamic lift, but I could not envision clearly how this might operate on a surface with such wild variations in angle of attack as compared with the relatively stable airplane wing.
The lift and drag characteristics of an oar blade have no more effect on the rower’s propulsive force than the lift characteristics of a wing have on the weight of an airplane.
Lift and drag are important only insofar as they affect inefficient blade slip. And, in this regard, drag rules.
One aspect of the work done by a rower (or any athlete) is the work done on his own body—what I call the internal energy. It is the work done (above resting metabolism) to warm and tire the muscles, and to increase the breathing and heart rates. There seem to be many, especially in the world of rowing ergometers, who do not take this effort into account—it doesn’t (can’t) show up on the erg display or in its calculator. It seems to me that if a rower on an erg were to urge himself forcefully back and forth with his leg and torso muscles (feet in stretchers and without the fan wheel handle) he would soon tire. Its calculated value in ROWING is not inconsequential, maybe as much as twenty percent of the rower’s total output. The model calculates it and adds it to the rower’s total effort where it shows up in total rower power and efficiency.
In general the model seems to show that a trade-off between stroke rate and rower work, in which increasing effort at the expense of stroke rate (at equal total rower power), is advantageous. This shows up in several ways. In its simplest form—simply pull harder (but less often) at a reduced rating—it is analogous to shifting a car into higher gear. Also it can be achieved by small increases in oar lever ratio in order to decrease rating without pulling any harder. Furthermore, my pages at oarlngth.htm and at forcratg.htm show that high (“stiff”) lever ratios can be advantageous. Much points toward pulling a bit harder and easing up on the stroke rate . It might be a good thing if, during a race, a rower could easily vary his lever ratio; high for a tail wind, low for a head wind; low for the starting strokes and high for winning in the long haul.
The model indicates, too, that narrowing the angle between oar and shell at the catch can help; and puts to rest the myth of the dangers of the “pinch point”. Catch (bow) angle can be reduced by rigging “through the work” (sculls) or by changing the cant angle of the blade (sweeps). See bladangl.htm. Dreher at Durham Boat  did some experimenting with negative sweeps cant angle but was unable, I think, to put into it the immense and chancy effort required to prove the idea.
The advantage of large blade surface area [bladarea.htm] seems so clear that I am baffled as to why area has not increased since the introduction in 1994 of the “Big Blade”. Blade slip is the single most important variable in the definition of blade efficiency—slip decreases markedly with area increase. To be sure, at the same lever ratio, a big blade feels “heavier” than a smaller one, but it is all a part of the same idea of pulling with the same force at a reduced rating. I think blade makers err in increasing area and then, at the same time, turning around to reduce the lever ratio so that the oar again “feels” more like what rowers are used to. The reduced ratio simply reintroduces the unwanted slip that the area increase was to have reduced. Obviously, of course, there is an area too large for any rower to deal with, but I don’t think that point has yet been reached.
If a rower wants seriously to go faster he will have to train himself to accept and to perfect performance with a new and unfamiliar “feel”, whether it is the result of a better blade or of a new, more deliberate style.
On a personal note:
My ROWING website at http://www.atkinsopht.com will not be “up” forever. It has now been on the Web for almost fifteen years. I am ninety-three years old and the time will soon arrive whereon I will no longer be able to maintain the site, nor to answer the many interesting queries I receive in my e-mail at Atkinsopht@Gmail.com. My best wishes to all of the many who have shown an interest in this site.
It can always be found on the WayBack Machine.
 CDC 6600: https://en.wikipedia.org/wiki/CDC_6600#Operating_system_and_programming
 Langston Gantry Jr., “The Technology Crew”, Cornell Magazine, June, 1994.
 Valery Kleshnev: http://www.biorow.com His work is valuable because of its empirical nature; much harder to make lab and field measurements than to make computer computations.
 The ROWING abstract: http://www.atkinsopht.com/row/rowabstr.htm
 Marinus van Holst: http://home.hccnet.nl/m.holst/RoeiWeb.html
 Sander Roosendaal: SanderRoosendaal.wordpress.com
 Leo Lazauskas: https://www.boatdesign.net/threads/free-internet-rowing-model-2-32-released.52866/
 Brown, Daniel J., “The Boys in the Boat”, Penguin, 2014 (Paperback) pp. 99, 105, 116, 168, 247, 254. The author’s race histories (if accurate) seem to support this idea.
 Dreher Oars: http://www.durhamboat.com/oars
Working for Henry Dreyfuss  at first seemed glamorous. World famous boss; office on West 58th Street in the Paris Theatre building, near the Plaza Hotel, around the corner from Bergdorf’s; surrounded by classy shops, galleries, and restaurants; Central Park’s horse drawn carriages parked outside; business lunches in the Oak Room; and the new MoMA only blocks away.
I began working there in 1952. But, after seven years (itch?), I began to have misgivings. I had advanced to the project management level and there was probably no reason that I could not have stayed on for a long career in design. In fact, not long after my announced departure, I was invited into a partnership with one of the principals who himself was leaving to set up his own shop. However, what had seemed alluring at first about industrial design—the freedom to create “exciting” new concepts—had become increasingly fraught with disillusion. “Form follows function” it was said, but often the form seemed lacking in support of the function.
I found myself in client meetings with engineers at Minneapolis Honeywell, Crane Co, Mosler Safe, Mergenthaler Linotype, etc. and gradually began to realize that I sat on the wrong side of the table. Too often it seemed to me that the engineers had a better hold on how best an item should work or be received or manufactured than we did, and our arguments as to how it should look seemed increasingly contrived. I remember an internal argument over whether a screw-head should be concealed beneath a more expensive, brushed metal cap and heard myself saying “What’s so bad about the fact of an honest screw?” And so, in 1959, I opted out. Henry gave me a nice watch.
Fortunately I had a line on a new job through a man I had met socially. Mihai Alimanestianu , who with his brother had escaped Ceausescu’s Romania by leaping into the night from a moving train, had invented an automated system for parking cars. He had financial backing and had an agreement with the Otis Elevator Co. to manufacture the lateral transfer machinery. He had ideas for a new version of his current device and took me on as machine designer and draftsman.
He called his operation “Speed-Park.” One of my first assignments was to design his logo and letterhead as well as the advertising brochures. In addition I made several perspective cut-away renderings of the garage to make clear the mechanical principles involved.
The building of the first garage was underway on West 42nd Street. On either side of a two-lane driveway elevators moved up and down in towers which, at the same time, moved longitudinally on rails. On the elevators a comb-like forklift transfer device could pick up cars and deliver them automatically to parking niches, one-deep, on either side of the tower runway. After the motorist had left the car parked in place safety barriers rose and the car was lifted a few inches onto a grid shaped to interweave with the fingers of the fork-lift. The garage operation required only one attendant who oversaw a computer console that printed the receipts, directed the elevator to the closest available stall, and—at the end—calculated the parking fee. Owners could lock their cars.
Mihai’s new device parked cars not one—but two deep on either side of the tower runway. This was more difficult because the forklift arrangement could no longer be simply cantilevered from the elevator platform; it had to roll back and forth sixteen feet on wheels. He had worked out the general mechanical arrangement and so I set to work with manufacturer’s catalogs and my college texts to make the calculations and drawings necessary for the building of a prototype—at Dreyfuss I had become a competent draftsman.
We occupied a small office on the East Side near 42nd Street. The job was interesting. It comprised structural, mechanical and hydraulic components operating in conditions of severely restricted space. Strength and deflection under load, cable and sheave arrangements, “bureau drawer” slides and squaring shaft , hydraulic extension and jacking—all had to be addressed. Office conditions, however, were stressful. Mihai’s secretary yammered incessantly on the phone and played the radio, snapping her chewing gum the while.
Eventually, to my relief, we moved to more elegant digs on East 57th Street in a building owned by Huntington Hartford, the A&P heir who was the venture’s principal backer. I had my own office and got a raise even though I had to pay for a new drafting table myself; Mihai explaining that he would compensate me later when money became less tight.
The construction of the machine prototype was now in progress at the Link-Belt plant in Lansdale, PA. I made many visits there to nurse it along. We had trouble with cycling speed because as the hydraulic oil changed temperature and density. I hadn’t the background in control theory to fix it, but Otis stepped in and added a modification of its elevator floor leveling controls.
The 42nd Street Garage opened late in 1962. It seemed to me that it would be good to have a means of estimating the rate at which cars could be parked and un-parked using the known speeds of the various mechanical components and depending on the degree to which users might cause delays. I found a way of finding average parking rates using the volumes of overlapping prisms and pyramids whose heights were times in seconds and whose bases were dimensionless numbers of levels and bays. It was a way automatically to handle cases where elevator and tower moved simultaneously, where one had to wait for the other, and where the efficiency of the motorist became a factor.
Mr. Pinto at Otis Engineering was very interested in this work. At 42nd St. I had made stopwatch observations of my own closely confirming its results. In retrospect it would have been an ideal application for a computer program; the calculations were tedious and there were unique cases to consider. In this respect we were only a few years too early.
Mihai wanted a promotional film and so I enlisted the services of a photographer friend Peter Pruyn and an actress friend Naomi Thornton who played an attractive motorist. We made a five-minute film. Mihai was instantly dissatisfied with the result and demanded endless and frivolous changes—finally he rejected it and refused to pay the principals, and so we agreed that the footage would be returned unused to Peter.
By early 1963 some long periods of not-much-to-do ensued. I used the time to make the design and construction drawings for my eclipse spectrograph .
It gradually became evident that this enterprise was not going to take off. Also evident—not only to me but to others: Mihai was by nature a micro-manager and somewhat of a charlatan. For example, I packaged up the film for return to Peter and discovered too late that Mihai had secretly stolen it back from the outgoing mail. He never paid Peter or Naomi. Later he did grudgingly pay me for the drafting table, but only after having discovered that my new employer in Boston was to be a consultant on one of his future projects.
And so—time to move on. The garage operated for a year or so more but following a major tower failure—forcing drivers to wait weeks for their cars—it closed and was demolished.
Wm. C. Atkinson, 2013
“Dad, would you make me one of these?”
So queried my accomplished daughter, Meg (artist, painter, and educator), on a visit several years ago to her home and studio in Brooklyn’s Boerum Hill neighborhood.
What she had in mind was Marcel Duchamp’s famous Bicycle Wheel. The original—lost when he moved to New York in 1915—was subsequently recreated and is on display at New York’s Museum of Modern Art  [1a].
“Sure,” I said. And so, in the ensuing months, I started thinking it over. I pictured Duchamp in his Paris studio rising from his chair or stepping over to his wheel from time to time to give it a spin, only to suffer its gradual coast to a stop after a few moments. What fun was that? It seemed that what really should happen is that it rotate continuously on its own.
But wait… if the wheel is to rotate continuously on its horizontal axis why not on its steering axis as well? Beaucoup plus intéressant, non? Thus, the die was cast and my path became clear.
While mulling over the difficulties of building such a confection I made for Meg a different bicycle wheel-related gadget. Duchamp played with plumbing via his controversial Fountain  of 1917. He called these sculptural creations his Readymades.
At my local landfill I found an abandoned bicycle and a friend produced a kitchen stool from hers. Now I had two of the crucial elements. The third requirement was a suitable source of motion. As luck would have it my workshop crate of old electric motors yielded up a small one of 60rpm output.
I can no longer reconstruct the sequence of steps that led to the final design. It was a mix of trial and error, false starts, scrapping and redoing, changes of mind, and above all—serendipity. Without my son’s drill press it would have been impossible and throughout a blistering August it confined me to the blessed coolth of my basement workshop.
A continuous cord around the bicycle wheel leads off to a smaller wheel—tangent at ninety degrees—with a six-tooth, one-inch pitch pinion at its lower extremity. The pinion engages the inside of a one-inch pitch crown gear composed of finishing nails set upright in the top of the stool and is the driving force of the rotation of the vertical axis. The cord then rounds the small driving sheave on the motor shaft and thence, passing a tension idler, returns to the bicycle wheel. I solved the problem of the continuous twisting of the electric power cord by constructing a rather clunky, but happily invisible, commutator arrangement under the stool top.
The pinion is the inspiration of Arthur Ganson whose fanciful machines depend on gears made of bent wire. There is a comprehensive exhibit of his work at the MIT Museum in Cambridge, MA. He sells a DVD of his creations and, at the end, shows how to make a wire-bending jig to produce working gears out of clothes hanger wire .
Notwithstanding his fame for the Nude Descending a Staircase there are other fanciful elements in the sum of Duchamp’s creativity. He is known for a painting in yellow of a Chocolate Grinder  in which the active elements are rendered in white thread applied with a needle to the original canvas.
Between 1915 and 1923 Duchamp created his Large Glass . Among its elements are a re-rendering of the chocolate grinder and the appearance of his “Scissors” above on the grinder’s vertical axis. Thus I had the idea of incorporating these two additional elements in my kinetic design. The grinding cones are of stiff paper around which is wound white sewing thread as in Duchamp’s original painting. The scissors are supported on chopsticks.
Cut into pieces, a print of the Nude Descending is affixed in spiral array to the spokes of the wheel. Thus, metaphorically at least, she descends as the wheel turns.
The Machine resides at my daughter’s in Brooklyn where she awakens it occasionally to keep it limber and to bask in its soothing ambiance.
Duchamp Bicycle Wheel: Kinetically realized
Hommage á Duchamp, Tinguely, et Ganson:
The wheel rotates continuously about its hub while the fork rotates about the bicycle steering axis, driven by the six-toothed pinion engaging the internal gear (finishing nails) on the stool top. Two commutator (slip) rings under the stool top provide the required 110volt power to a 60rpm synchronous motor- thus the wheel can be started in either direction.
The Chocolate Grinder harks from Duchamp’s painting of the same name and both it and the Scissors are elements in The Large Glass.
The Chocolate Grinder drums are “driven” by the rotation of the pinion sheave- its axial “cap” in counter rotation- and the Scissors are activated by a small crank on the motor shaft.
As the wheel turns the fractured Nude endlessly “Descends” her Staircase.
 What my daughter asked for:
[1a] The original work had a straight fork (impossible for riding) like those used in repair shops for re-spoking wheels.
 Duchamp and plumbing:
 Chocolate grinder:
 The Large Glass:
 Making wire gears:
Machine designed and built by W.C. Atkinson (2005)
Work owned by Margaret K. Atkinson, painter, Brooklyn, NY
Stool and base courtesy Cohasset, MA landfill
Fork and wheel courtesy Weston, MA landfill
Pinion design concept courtesy A. Ganson
Steps of Poured Concrete with Balanced Reinforcing
At the east end of our old house on South Avenue was a set of rotting wooden steps giving access to the outside door of what we called the Downstairs Cold Room—from the fact that, in our first year or so, that wing of the house had no heat and, in winter, the broken windows upstairs allowed snow to pile up on the floor.
Under the old steps was Meg’s “house” and she wasn’t happy to hear that progress was about to change it.
The existing steps at the front and kitchen doors (and the two terraces) were monolithic blocks simply poured against the foundation and, unfortunately, directly against the wooden siding of the house itself. Already, in a previous year, I had had to rebuild the front entrance owing to moisture induced rot in the sill and the consequent depredations of carpenter ants. I vowed that new steps would not so compromise the structure of the house.
In contrast to the solid mass of the existing steps I thought to make something “lighter” and more graceful, and so I set to work at the drawing board to see what might be forthcoming.
Beams of concrete alone, if relatively thin, cannot be employed in long spans. This is because concrete, while strong in compression, has only a small ability to resist corresponding tension. Gravity bends a simple beam under load downward; the beam resists the load by developing compression in its upper surface (chord) and tension in its lower chord. In the extreme the concrete cracks at the bottom. The solution to the weakness is to embed steel in those parts of concrete structures under tension. And this leads to a structural concept called balanced reinforcing. Concrete is used in compression up to its design load limit
on top, and embedded re-bars in tension to their limit at the bottom. That is, under extreme load, one could not predict for sure whether the concrete would crumble on top or the steel bars below pull apart. Achieving such a balance yields the most economical use of material in the structure as a whole.
Fortunately my engineering handbooks provided sufficient technical detail to permit creating a balanced design using 3/8” rods (re-bars). The straight rods had to be bent and for this I made a jig embedded in a sturdy block of concrete buried in the ground with two bending posts protruding at the surface.
At the top a 4-way* slab for the landing, so called because of its proposed support at the four corners producing bending in two directions. I assumed as design load a 250 pound mover in the center carrying half a piano. The steps going down are the same as the slab only narrow. In the center the re-bars are in the lower chord and, at the overhangs, in the upper chord because here the bending tension has moved to the top.
Slab and step spans are three inches thick at the center, tapering to two and a half at the ends to give a light effect.
Slab and steps are supported on two arches leaning against the house foundation.
I assembled the wooden concrete pouring form first for the left arch. After that pour having set we stripped it and reassembled it for the right arch. Matthew helped me mix concrete and aggregate in the wheelbarrow. The form for the landing slab included pockets for the end posts. Re-bars wired in place and supported on “chairs” of small stones become incorporated into the final mass. The form for the narrow steps was a strip cut out of the slab form. By the time we poured the bottom step that form had been stripped and reassembled twice and was almost a wreck.
We cast the date (1971) into the upper slab and all three children added their hand prints. When one jumps on the slab and steps they have a nice stiff “ring.” And Meg got her “house” back. Today the steps are forty-five years old; a bit weathered, but as stiff as ever.
*Technically not quite true.
Never an accomplished player myself I had long admired the game of chess. Occasionally I might win a game—I was once trounced by an opponent who sat with his back to the board.
Everyone is familiar with the traditional set of chessmen—the crowned king and queen, mitred bishop, archly maned knight, castellated rook, and the lowly and knobby pawn: the Staunton Pattern.
Perhaps my inspiration for a new design originated in Brancusi’s “Bird in Space” which I had seen recently at MOMA, or maybe in roaming the aisles of Bonniers on Madison Avenue at lunch hour; around the corner from our office in the Paris Theatre building on the Plaza. Bonniers then led the New York retail world in offering a new, clean Scandinavian vision. I was then newly a part of that world having, in 1952, joined the office of famed industrial designer Henry Dreyfus.
I set out to create a design for a set of elegant chessmen. Ultimately, their form came to dominate the necessary distinctions made to show their game function and so, with some irony, my result seemed at odds with the current design mantra: “Form follows function” .
After having defined my idea on the drawing board I moved on to three-dimensional rendering of the pieces . At Dreyfus I worked occasionally with John Amore , our product design model maker, a master in the rendering of objects (radios, telephones, control knobs) in plasticine and plaster of Paris. From John I learned how to “turn” volumes of revolution in plaster on a small hand-cranked “lathe” made of coat hanger wire and sheet zinc [pic]. I produced a set in plaster; and a few pieces in lead using plaster pieces in making the casting molds. This work I managed to do in my apartment at 21 Jones street.
I was sufficiently taken by the result gradually to entertain the idea of a commercial retail venture. With a list of upscale New York City retailers I went around looking at non-traditional chess designs; there were many, and I didn’t think any of them were as nice as mine . I found a shop in Germany that would make a sample set of pear and alligator wood and one of Dreyfus’ local makers for the metal set; to be gold and silver plated. A local shop made me a wooden box and another an elegant velvet lined box for the metal set.
A Cornell classmate in WDC arranged to get me a design patent .
Thus armed I made appointments with retail buyers around town. What I discovered was that, while impressed with the design, retailers were not positioned to underwrite inventory. And so I would have to undertake that myself with no guarantee that the product would sell. My sample makers all gave me production estimates but, in the end having no real resources myself, I decided that I wasn’t rich enough or sufficiently brave to take the plunge.
So now I have two lovely sets. Chess anyone?
 “Form [ever] follows function”.
 How nice it would have been to have had a three-dimensional printer!
 John Amore, American Acadamy, Prix de Rome (sculptor) 1940. Well known medalist.
 No Google then, but just Google images for “chessmen” now!
 US Patent Office, Des. 178,946, Patented Oct. 16, 1956 “SET OF CHESSMEN”