Mathematics

Mathematics

Answer the following on a separate sheet.
1. In the reading assignment from Flatland, Square says,
Let me recall the past. Was I not taught below that when I saw a Line and inferred a
Plane, I in reality saw a Third unrecognized Dimension, not the same as brightness,
called “height”? And does it not now follow that, in this region, when I see a Plane and
infer a Solid, I really see a Fourth unrecognized Dimension, not the same as colour, but
existent, though infinitesimal and incapable of measurement?
Discuss in a paragraph or two your interpretations of what he means here, especially as it pertains to how we
might consider additional dimensions in addition to the 3 physical ones we identify around us. For example, we
know that we live in 3 physical dimensions; what are the implications, then, for a possible fourth dimension?
You may want to refer to our lecture, readings, video sources, or all the above in your response.
2. When we measure length, width, and height of an object, we usually use rulers and other similar devices. Let’s
suppose that the measurable extension of an object into the Fourth Dimension is called “longth”. Describe the
process of measuring an object’s longth, and what units might be used to describe this property.
3. Consider an actual device that has exactly three mechanical degrees of freedom. Perhaps you interact with this
device each day, and perhaps not; however, it must be real. Name this object, describe its operation, and
explain clearly why you believe it to have only 3 DOF. You may provide sketches, as well, to support your
description.
4. You have been charged with the task of designing a robot for the purpose of spray-painting the inside of a
room. The room is 10’x10’ square, and has 10’-high ceilings. To perform this task properly, the robot would
presumably (but not necessarily) be sitting in the center of the floor, and reach into all the corners to finish the
job (it doesn’t need to paint the floor). Because of technical constraints, only two types of “joints” are
available: the kind that extends (or translates in straight lines) and the kind that rotates (like an axle in a bearing,
or a hinge). Furthermore, cost constraints restrict the maximum number of joints in the robot to 3; you may use
no more than that. Note that the illustrated robot at right has six. Present a sketch or diagram of your robot,
indicating the range of motion available to it; that is, how does each “joint” allow it to move? In addition,
answer the following questions:
a. How many mechanical DOF does each joint have?
b. What are the difficulties, if any, associated with building a robot for
this task having all one type of joint (that is, all rotational or all
translational)?
c. Is it possible to design a robot to accomplish this task that has
fewer than three joints? Why or why not?