![]() We can even look from 3d to 4d, notice that on the 3-sphere any line in xyz-plane (through the origin) is an axis of rotation for the 3d-ball (this is not true of the 2d-ball, a line parallel to the ball is decidedly not an axis of rotation). But in 2d-ball to 3d-ball we rotated partially parallel, this is basically where the missing π is. Notice that in the case of the 1d-"ball" to the 2d-ball we rotated orthogonally, this added a π to our result. rstuvw-hyperplane is orthogonal to the xy-plane, as such we need a line orthogonal to all of those planes, the only choice is one of the lines within the xy-plane itself. This of course generalizes to absurdity, every line in the. Suppose the disc is in the xy-plane, then z is orthogonal, what is to stop us choosing the w axis for example? Well the answer is actually quite simple, with regard the xy-plane the z-axis is no more special than the w-axis, the w-axis is equally orthogonal to the xy-plane as the z-axis is, in fact any line in the wz-plane is orthogonal to the xy-plane. Of course this is only directly obvious in 3d, where there is only a single plane orthogonal to the rotational axis of the disc. if our rotational axis for the 2d-ball is the z-axis then the rotational axis to produce the sphere needs to lie in the xy-plane. In 3d the choice is forced, we have to pick some line that is actually parallel to the disk (either the x-axis or y-axis), i.e. orthogonal to 1d axis of rotation) (otherwise we will get a cone-like thing). (Supposing we restrict ourselves to axes, then if the line is on the x-axis, then either the y-axis or z-axis are appropriate orthogonal axes of rotation).īut then consider the rotation axis for a 2d ball to a 3d ball, we need to pick some direction that is orthogonal to the orthogonal line with the plane (i.e. You can see this in lower dimensional cases.įor example if you start with a 1d "ball" (line), then to get a 2d ball (disc) you need to rotate it around some axis, that axis needs to be orthogonal to the line (otherwise you get a cone). Re-read your post before hitting submit, does it still make sense.Show your work! Detail what you have tried and what isn't working.Use proper spelling, grammar and punctuation.Give context and details to your question, not just the equation.Help others, help you! How to ask a good question Asking for solutions without any effort on your part, is not okay. Beginner questions and asking for help with homework is okay. Post your question and outline the steps you've taken to solve the problem on your own. Do not use ChatGPT in a question or an answerĭon't just post a question and say "HELP". ![]() Do not solicit or offer payments to complete your assignments or tests.No cheating - do not post questions from exams, tests, midterms, etc.No post flooding - Limit your posts to 2 or 3 questions a day.Don't be a jerk - don't be obnoxious or rude.Homework policy - asking for help is okay, asking to be given the solution is not.Make your question clear and concise - include steps you have tried. ![]()
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