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Special Content: Repraps for Education

This is part of a series of posts about starting and facilitating a project-based 3D printer club at a local elementary school, with the ultimate goal of replicating the program at schools everywhere. We'll be posting as many details as possible, including lesson plans and supporting materials. For more information about the entire project, including a listing of posts related to it, please visit the 3D Printer Club for Schools project page. 

3D printers are an exciting new technology just starting to gain a foothold in schools. Once built, a printer facilitates many creative endeavors and links to math, science, art, and more. Building a 3D printer is a great project for a school club or a project-based classroom because students have to integrate STEM skills and work as a team. Students who build a printer from scratch (as opposed to just buying a kit or a completed machine) will have a detailed understand how it works and will be able to properly maintain and operate it.

Here we are developing and making available a 3D printer club curriculum that any school can use. We’re developing these lesson plans as they are tested in an elementary school club, with members in grades 2-5. This page will be updates with the latest lesson plans and resources as they are developed.

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I built a framework for robot wheels a while back using OpenSCAD and used it to create a few different wheel styles. I recently decided to combine them all into one massively configurable wheel model, and add a number of new features as well. The result is an OpenSCAD file with 46 parameters that provides a limitless set of combinations and wheel designs. I call it One Wheel To Rule Them All.

It includes twelve tread patterns (all configurable in often surprising ways), eight core spoke patterns (also highly customizable), configurable support for o-rings, bands, and even optical encoder timing slots (directional and non-directional), and a lot more. Plus, I’m still adding features as I think of them.

First, here are a few images of the types of the various basic elements, as well as a few variations that show the flexibility of the designs. For example, as shown in these images, the Spiral style can be used to create a variety of interesting designs that you might not think of when you think “spiral.” After the images you’ll find full details on the parameters.

It’s important to note that you can configure this wheel to such an extend that it may not be printable at home with extruded plastic printers. In these cases, services such as Shapeways could come in handy.

The source file is available at, or on github at

Tire Parameters

Often wheels are built around the tires. In this section, specify the properties of the tires you’re using, and this will define the diameter of the wheel. If you’re using o-rings, the tireCSDiameter should be the cross-section diameter of the o-ring, or if you’re using some other flat tire material (such as rubber bands), jsut specify the its thickness. If you’re not using any tire at all, set the tireCSDiameter to zero.

  • wheelWidth: The width (or thickness) of the the wheel
  • tireCSDiameter: Cross-sectional diameter (CS) — How thick is the tire rubber?
  • tireID: Internal diameter (ID) — How wide is the inside opening?
  • tireStretch: Circumferential stretch percentage (usually 1 + 0-5%) — How much to you want to stretch it to get it on?

Rim properties

The rim sits at at the outside of the spokes and supports the tires or added treads. Installed tires (such as o-rings, rubber bands, etc) are set into grooves carved out of therim, while trads are added onto it. Keep this in mind when you’re using tires — as an example, the rim height should not be smaller than the radius of o-ring tires.

The rim also supports rotary encoder timing holes for wheel feedback. Use the padding parameters to adjust the location of those holes. See the compiler output for helpful information about the distance indicated by each timing hole. Directional timing holes will produce a second set of holes that are 90 degrees out of phase with the first. This allows you to stack sensors at the same location over the wheel instead of trying to position them along the circumference. Directional timing holes essentially double the resolution. You can also double resolution by looking for both rising and falling edges.

  • rimHeight: The height of the rim portion of the wheel
  • timingHoles: The number of timing holes to carve into the rim
  • timingHoleInPad: The inside padding for the timing holes
  • timingHoleMidPad: The middle padding if direction timing holes is selected
  • timingHoleOutPad: The outside padding for the timing holes
  • directional: A directional encoder renders two sets of slots, 90 deg out of phase

Tread Parameters

In this section, specify the properties of the tire tread you want to render. If you’re using a wheel (e.g. o-ring, rubber bands, etc), then use either the “o-rings” or “slots” settings, which will cut a groove (or grooves) in the wheel rim to fit the tires. The othertreat styles will render a tread pattern protruding out from the tire surface by the amount you specify in third part of “knobSize”.

Imagine the tire is mounted on a robot and facing straight at you. The “knobSize” parameter defines the size and shape of knobs in an [x,y,z] format, where x goes across the rim, y goes up and down along the perimeter of the wheel, and z protrudes out from the wheel toward you.

The “staggerOffset” parameter allows you to stagger knobs across the tire by an amount you specify. Set this to zero if you want all the knobs lined up along the perimeter and aligned with the edges of the rim.

“numberOfKnobs” specifies how many knobs there are across the tire, and “lineThickness” specifies how thick the lines are from “drawn” tire styles, such as “x”, “cross”, and “zigX”. You can use these pameters together in creative ways — for example to extend a single tread profile across the width of the tire, or to create a contiguous zig-zag.

Finally, “radialTreadSets” defines how many sets of treads are rendered around the wheel. Each set contains two rows in order to create the staggered effect.

Tread styles are:

    • none: No tread is rendered
    • cross: Each knob is the shape of a plus sign with the specified lineThickness
    • o-rings: Grooves are cut into the rim to accept o-ring tires
    • squares: Each knob is a rectangle, whose size is specified by knobSize
    • spheres: Each knob is a smooth bump, whose size is specified by knobSize
    • cylindersX: Each knob is a cylindrical shape running across the wheel, whose size is specified by knobSize
    • cylindersY: Each knob is a cylindrical shape running along the perimiter of the wheel, whose size is specified by knobSize
    • cylindersZ: Each knob is a cylindrical shape protruding from the surface of the wheel, whose size is specified by knobSize
    • spikes: Each knob is a cone or spike protruding from the surface of the wheel, whose size is specified by knobSize
    • slots: Grooves are cut into the rim to accept flat tires, defined by numberOfKnobs (number of grooves), the first and third numbers in knobSize to define the width of the slots and the depth, and spaceBetweenTires for the distance between the tires and also from the outside edges to the first slots.
    • x: Each knob is in the shape of an “x” protruding from the surface of the wheel, whose size is specified by knobSize
    • zigX: Each knob is in the shape of a zig-zag protruding from the surface of the wheel, whose size is specified by knobSize
    • v: Each knob is in the shape of a “v” protruding from the surface of the wheel, whose size is specified by knobSize
  • treadStyle: none, cross, o-rings, squares, spheres, cylindersX, cylindersY, cylindersZ, spikes, slots, x, zigX, v
  • knobSize: The size of each knob [across wheel, along the perimeter, prodruding]
  • radialTreadSets: How many sets of treads to render around the wheel (2 rows per set).
  • numberOfKnobs: The number of knobs to render per row.
  • staggerOffset: A distance to offset the staggered rows.
  • lineThickness: The line thickness for “drawn” styles, such as “x” and “zigX”
  • maxTires: For o-rings, the maximum number of tires per wheel
  • spaceBetweenTires: For o-rings, the space between each tire, if there are more than one

Spoke-related Parameters

This section is used to define the spoke style of the wheel. Some of the properties are only applicable to certain wheel types, and these properties can be used together in creative ways to create a wide range of tire designs.

The “proportion” property affects how some spokes are rendered. The first number is the proportion of the design from the center of the wheel to the inside of the rim, and the second number is the proportion of the width inside of the wheel. For example, to create spokes that are roughly in the shape of a “U”, you can use a “circle” style, and set the proportion to [1.5, 1.0], for cirle spokes that are 150% as long as the distance from the center to the inside of the rim, 100% as wide.

Use spokeInset to specify the inner and outer inset of the spoke area from the inner and outer faces of the wheel. You can use a negative number to make the spoke area stick out further than than the rim. The hub position will be based on the inner surface resulting from this inset.

The spoke styles are:

    • biohazard: A biohazard logo-inspired design. Set numberOfSpokes to 3 to mimic the logo.
    • circle: Spokes in a circlar or oval form, defined by spokeWidth and proportion.
    • circlefit: The maximum number of circles that will fit between the center and the rim, with a set of smaller outer circles specified by outerHoleDiameter.
    • diamond: Spokes in the shape of a diamond (rhombus), defined by spokeWidth and proportion.
    • fill: Fills in the spoke area with a solid cylinder.
    • line: Straight line spokes, like you would see on a typical wagon wheel.
    • none: Leaves the spoke area empty and does not make for a very useful wheel.
    • rectangle: Spokes in the shape of a rectangle, defined by spokeWidth and proportion.
    • spiral: Spokes in the shape of a semicircle, defined by curvature, reverse, spokeWidth.
  • spokeStyle: none, biohazard, circle, circlefit, diamond, line, rectangle, spiral, fill
  • spokeInset: The [inner,outer] inset of the spoke area from the surface
  • numberOfSpokes: Number of “spokes.” Set this to three if you’re doing the biohazard design
  • spokeWidth: This is how wide each spoke is.
  • proportion: proportion to rim, proportion of width
  • curvature: For “spiral”, this is how curvey the spokes are. >0, but
  • reverse: For “spiral”, setting this to “true” reverses the direction of the spirals
  • outerHoleDiameter: For “circlefit”, the diameter of the outer holes, or zero for non
  • concavity: Concavity distance of spoke area for [inside, outside] of wheel

Hub Parameters

These properties define the hub — or how the wheel connects to the motor. The default values for the captive nut are precise for a M3 nut and will make the nut a very tight (if not impossible) fit. I prefer this because it allows you to “melt” the nut into place with a soldering iron. However, if you don’t have a solder iron or prefer a looser fit, then just adjust the nut diameter and thickness. (M3 hardware is, by default, set to 3mm screw diameter, 5.4mm nut diameter, and 2.3mm nut thickness.) Similarly, the holes for the motor shaft and grub screw are also precise. This allows the holes to be drilled out for a more precise fit. Again, you can adjust these to suit your needs.

The hubZOffset can be used to “sink” the hub into the wheel, and it defaults to half the wheel thickness. For example, when the hubHeight is 10 and the hubZOffset is -2, then the hub will protrude 8mm from the wheel, but the shaft hole will be 10mm deep. The set screw will still be positioned in the middle of the exposed vertical height, and the fillet/chamfer will also be rendered in the correct position. This property is also useful if you want to poke a hole entirely through the wheel. (e.g. If the wheel is 6mm thick, set the hub height to 16 and the hubZOffset to -6, and you’ll get a hub that protrudes 10mm from the wheel surface with a hole that extends all the way through the wheel.)

To mount a servo motor, set includeHub to false, set shaftDiameter so that the hole will accommodate the servo horn screw and any bit that protrudes from the top of the servo horn. Then set the servoHoleDiameter to the size of your mounting hardware, and set servoHoleDistance1 and servoHoleDistance2 to the total distance between mounting holes on your servo (not the distance from the center). These sets of mounting holes will be rendered at 90 degree angles from one another. If you only want one set of holes, set one of the values to zero. Adjust the angle of all the holes to avoid openings in your wheel design if necessary using servoArmRotation.

Use innerCircleDiameter to specify a solid inner circle to use as a base for the hub. This can be useful if you need a a solid surface for servo mounting hardware or for the base hub fillet/chamfer.

Use outerNutTrap to create a nut or bolt head trap on the outside (bottom) of the hub area. Used in conjunction with shaftDiameter and false for includeHub, this will create a wheel that can drive a bolt much like the large gear on Wade’s Extruder. (This feature is inspired by that design.)

Use servoNutTrap to create nut traps for bolts used to mount the wheel onto servo arms. This feature was suggested by AUGuru.

  • includeHub: Set to false to remove the hub and only include the shaft diameter hole.
  • hubDiameter: The diameter of the hub portion of the wheel
  • hubHeight: The total height of the hub
  • hubZOffset: The Z position of the hub, negative numbers from the surface of the wheel
  • shaftDiameter: The diameter of the motor shaft
  • innerCircleDiameter: The diameter of the solid inner circle under the hub, or zero for none.
  • setScrewCount: The number of set screws/nuts to render, spaced evenly around the shaft
  • setScrewDiameter: The diameter of the set screw. 3 is the default for an M3 screw.
  • setScrewNutDiameter: The “diameter” of the captive nut, from flat to flat (the “in-diameter”)
  • setScrewNutThickness: The thickness of the captive nut
  • baseFilletRadius: The radius of the fillet (rounded part) between the hub and wheel.
  • topFilletRadius: The radius of the fillet (rounded part) at the top of the hub.
  • chamferOnly: Set to true to use chamfers (straight 45-degree angles) instead of fillets.
  • servoHoleDiameter: The diameter of servo arm hounting holes, or zero if no holes
  • servoHoleDistance1: Distance across servo horn from hole to hole (0 to ignore)
  • servoHoleDistance2: Distance across servo horn from hole to hole, rotated 90 degrees (0 to ignore)
  • servoArmRotation: The total rotation of all servo holes
  • servoNutTrap: Size [indiameter, depth] of servo arm captive nut, or 0 (any) for none.
  • outerNutTrap: Size [indiameter, depth] of a captive nut, or 0 (any) for none.

Quality Parameters

  • $fn: Default quality for most circle parts.

Math Monday came early this week, and we had some fun exploring Reuleaux triangles. Next to a circle, they’re the simplest curve of constant width you can make — that basically means you can roll a plank on top of them as smoothly as on wheels. The kids didn’t expect that one! (They also would make great manhole covers!)

Although they roll very smoothly, they do tend to wobble around a bit, so I built a little gutter on the plank to make it easier for the kids to experiment with. It was a quick project, and lots of fun for the kids. You don’t have to use a band saw or even the sander; all you really need is a jigsaw or coping saw, some scrap wood, and some nails.

Music credits: “260809 Funky Nurykabe” by spinningmerkaba (CC-BY 3.0)


To mark the 63rd anniversary of the first stored computer program to run (June 21), I rewrote the original code in C# last night, both using the original algorithm (that was never intended to be efficient) and using a more modern structure. The Kilburn Highest Factor Routine finds the highest factor of 2^18, and completed with the correct answer in 52 minutes with 3.5 million operations. It was written by the late computer pioneer Tom Kilburn and run on the Machester SSEM (“Baby”).

The code is written in lines 1-19, lines 20-22 contain some control flow values, lines 23 and 24 contain 2^18 and the current factor to test, and lines 25-27 are working memory.  Click for a larger version of the original image from The National Archive for the History of Computing.

Kilburn Highest Factor Routine

Kilburn Highest Factor Routine

The last three columns (labelled 13, 14, and 15) are the instructions, and there are only a few of them. They are:

  • 100 – Jump forward or backward in code by the amount at the address given.
  • 010 – Load the negative of the value at the given address into the accumulator.
  • 110 – Store the value in the accumulator to the given address.
  • 001 – Subtract the value at the given address from the value in the accumulator, and keep the result in the accumulator.
  • 011 – Skip the next line if the value in the accumulator is greater than zero.
  • 111 – Halt

The first version in the listing below follows the original algorithm (and yes, you can use “goto” in C#). There’s a lot of subtraction involved because if you’re short on space and only want to implement one arithmetic operator, you’re better off implementing subtraction, because you can so addition with it, too. “S” is used here to indicate the Storage tube, which stored both the program and the working memory. The accumulator (represented by the variable “acc”) and the program counter used different tubes. The program counter is what kept track of where the computer was in the program during execution.

The only slightly optimized C# version at the bottom takes less than a millisecond today.

Stopwatch sw = new Stopwatch();
int s20_jump_rel = -3;     // unused in code, provided for completeness
int s21_const_1 = 1;       // the amount to decrement the number being tested
int s22_jump_addy = 4;     // unused in code, provided for completeness
int s23_number_neg = -262144; // the number to solve (find the factor of)
int s24_div_init = 262143; // the initial number to test

int acc = 0;

acc = -s24_div_init;        // S01 - Load and negate the initial number to test
int s26_div_neg = acc;      // S02 - Store that number into S26

acc = -s26_div_neg;         // S03 - Load and negate the negated current number to test

int s27_div_pos = acc;      // S04 - Store that number into S27

acc = -s23_number_neg;      // S05 - Load and negate the number to solve

acc -= s27_div_pos;         // S06 - Subtract the current number to test

if( acc >= 0 )           // S07 - If the accumulator is > 0...
goto S06;                   // S08 - ...go back to S06

acc -= s26_div_neg;         // S09 - Subtract the negated current number to test
int s25 = acc;              // S10 - Store that number into S25
acc = -s25;                 // S11 - Load and negate the number in S25

if( acc >= 0 )           // S12 - If the accumulator is > 0...
goto HALT;                  // S13 - ...End

acc = -s26_div_neg;         // S14 - Load and negate the negated current number to test
acc -= s21_const_1;         // S15 - Subtract 1

s27_div_pos = acc;          // S16 - Store that number into S27
acc = -s27_div_pos;         // S17 - Load and negate the number in S27
s26_div_neg = acc;          // S18 - Store that number into S26

goto S03;                   // S19 - Go back to S03

Console.WriteLine( "Elapsed={0}", sw.Elapsed );  // Write the elapsed time
Console.WriteLine( s27_div_pos );                // Write the result

sw.Restart();                // Restart the stopwatch for a more modern version
int i = 262143;              // Set the initial number to test
while( 262144 % i-- != 0 ) ; // Decrement until the number to solve modulus zero is zero
Console.WriteLine( "Elapsed={0}", sw.Elapsed );  // Write the elapsed time
Console.WriteLine( i + 1 );                      // Write the result


This makes a great gift, t-shirt, mug, etc. for any computer or math geek, friend, child or baby! Write you name or message in up to 100,000 of the infinitely non-repeating digits of pi. PiShop is a free online application that allows you to generate customized image to use as a t-shirt transfer, banner, or whatever else you want to use it for. It’s free for personal use, but if you like it, I encourage you to make a little donation for my kids’ college funds! ;)

PiShop: Your Name with Digits of Pi

PiShop: Your Name with Digits of Pi

The application is available on my site where I keep programming-related content. Here is the direct link: The introduction is on Page 1, the application itself on Page 2, and tips on making the shirt on page 3.