tag:blogger.com,1999:blog-5983978585907824595.post1072287763783598609..comments2015-12-12T17:45:19.163-08:00Comments on Nature of Mathematics: Aristotle's Wheel UnraveledBrandonPiotrzkowskihttp://www.blogger.com/profile/01936520590401419379noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-5983978585907824595.post-23938402867676573692015-12-12T17:31:38.729-08:002015-12-12T17:31:38.729-08:00Excellent post, I especially enjoyed all the diffe...Excellent post, I especially enjoyed all the different approaches. Mathematically what's going on is that the circumference of the small wheel has the same number of points as the the large circumference, and the picture is essentially making a 1-1 and onto map from the small circumference to the large. That's why it's on the day we were doing infinity!<br />5Cs +John Goldenhttps://www.blogger.com/profile/18212162438307044259noreply@blogger.comtag:blogger.com,1999:blog-5983978585907824595.post-63246334675281481062015-12-06T20:57:26.454-08:002015-12-06T20:57:26.454-08:00Thank you! I added another section in the end to t...Thank you! I added another section in the end to talk about this very fact. Something to keep in mind is that the inner tire can trace out its circumference but<br /> then the outer one can't. If two concentric wheels have different radii, then one must roll with some slipping. Again if they both traced out their respective circumferences, they wouldn't go the same horizontal distance but different ones. Again thank you for the feedback.BrandonPiotrzkowskihttps://www.blogger.com/profile/01936520590401419379noreply@blogger.comtag:blogger.com,1999:blog-5983978585907824595.post-127587764504703202015-11-29T16:19:26.055-08:002015-11-29T16:19:26.055-08:00Brandon, I really appreciated your dive into physi...Brandon, I really appreciated your dive into physics to set up this problem! I am by no means a physics expert, and I learned a lot from your explanation. <br />I am not sure how I feel about the hubcap example at the end because as I understand Aristotle's wheel, the inner wheel does not catch a different thing than the wheel. I totally agree with you that some slipping must be occurring by the inner wheel, but am not sure your curb explanation fits that conclusion. The picture and explanation make it seem as though the friction of the hubcap on the curb causes the inner wheel to trace out the same circumference as the outer wheel rather than a slipping that occurs within the tire at the rubbing between the axle and the rim. I may just be confused, but I think you may want to elaborate on the last example more. <br /><br />Otherwise, an outstanding post! You broke down the rotation of a wheel very well and I liked your proof by contradiction!K4https://www.blogger.com/profile/07945522281014826345noreply@blogger.com