Object of Constant Width

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An object of constant width. This property, usually thought to be unique to spheres, is the same diameter all the way around. This allows an object placed on top, along with two other objects of constant width, to roll smoothly. This provides the appearance of balls beneath the object, though the objects are certainly not. Print and teach students some of exploits of geometry. Print Settings Printer Brand: MakerBot Printer: MakerBot Replicator (5th Generation) Rafts: Yes Supports: Yes Resolution: Unknown Infill: Unknown Notes: While the supports may not be strictly necessary, they or the raft is. The object has no flat surfaces, and so will always be in one point of contact with the raft at one time. Supports are thus recommended. Print at a fairly high resolution, and scale as desired. Post-Printing Remove the raft and/or supports. Try to make the surface as smooth as possible, though do not make it perfect. Slight imperfections allow the object to gain traction on carpeted surfaces. Use the aforementioned to demonstrate the quality of these objects. To ensure that the object is level, use 3 of the same scale of the object. Demonstrate to students, and explain to them the properties which make the object work. How I Designed This This began as a simple 2D sketch in Onshape. First, an equilateral triangle was drawn. (Note: The size of this triangle does not truly matter; it should not be given a dimension.) Then, 2 sets of 3 arcs were created, alternating arcs having equal radii, and all tangent to adjacent arcs. The centerpoints of the opposite arcs (e.g. very top and bottom arcs) should be coincident, and the center point should also be on a vertex of the inner triangle created earlier. For example, in the image below, the centerpoints of the top and bottom arcs are both coincident with the top vertex of the inner triangle. This should be done with each set of opposite arcs. The arcs should also all be tangent, so that it creates a smooth surface. Then, add dimensions to the radii of the arcs or the height of the object. In this case, the smaller arcs are set to a 5 mm radius, and the height to 50 mm. This also means that the larger arcs have a 45 mm radius. Do note that the smaller arcs are not actually necessary, but make the edges less sharp and defined. Once this is done, a Revolve function was used to create the round object. If an error appears, then the sketch likely needs to be cut in half so that the function does not intersect itself. Once this has been done, all that remains is for the model to be exported and printed! The basic sketch of the object of constant width. Why it Works Each arc is 60 degrees, such that each arc is in contact with the ground for the same amount of time. The top arc is in contact with the top object for the exact same amount of time, through 60 degrees of the rotation. Because the large and small arcs add up to the same height, in this case 50 mm, the object on top remains level. The height can remain constant when the arcs are added up because the center points of the arcs are in the same place, meaning that throughout the rotation on the arc, the line will always remain the same distance through the center point of the arc. The tangents simply provide a smooth transition from each arc to the next. Project: Recreating Objects of Constant Width Objectives The objective of this project is to teach people how to use Onshape, and how they can use some basic geometry to create objects considered illogical or impossible. Audiences The audiences of this project may be anywhere from someone in middle school to an adult, wanting to learn sketch-based CAD. Preparation A basic understanding of sketches, constraints (namely coincident, tangent, equal, and dimension) and the Revolve function, not to mention a free Onshape account. (Note: This should work with just about any sketch-based CAD.) Steps Get a concept of the project you are tackling. Think especially about what makes spheres roll. 2 things to be kept in mind: That this project can only be done with a triangle as a base, and that only a third should be done at a time. Create a sketch of the outline of the shape; make sure that the inner triangle frame is equilateral, the proper arcs set to equal, the arcs given the tangent, etc. Check that the width is constant. Create a line between two opposite arcs, terminating on the arcs. Constrain each end to their respective arcs with the Normal constraint, and add a driven dimension. This means the dimension only displays the length of the line, and does not control it. The dimension should be the same as the height that was set, or the radii of the opposite arcs added together. The line should be able to be rotated along the arc, remaining the same length. If it checks out, move to the next step! Revolve the sketch. Specifically, half the sketch. If you get an error stating that the model is intersecting itself, either set the revolve to 180 degrees or cut the sketch in half, and use a full revolve on one half. Export and print! Print three times for the best effect. Place a flat object, such as a binder or folder, and spin it around a few times. There should be almost no perceivable wobbling. Any inconsistencies are the result of the printer, and cannot be wholly avoided. Results The students should have a working object of constant width, and the new knowledge that they can do incredible things with CAD!