On 1/21/2017 2:10 PM, Scott Hemphill wrote:> > Write all the thetas as > > \theta_k = 2\pi k/n + \theta_cI've never seen this notation before. What do the back slashes mean? Is the forward slash still division? -- Rick C
Hah! Why 5-lug wheels balance
Started by ●January 20, 2017
Reply by ●January 22, 20172017-01-22
Reply by ●January 22, 20172017-01-22
On 2017-01-22, rickman <gnuarm@gmail.com> wrote:> On 1/21/2017 2:10 PM, Scott Hemphill wrote: >> >> Write all the thetas as >> >> \theta_k = 2\pi k/n + \theta_c > > I've never seen this notation before. What do the back slashes mean? > Is the forward slash still division? >It's how you write maths meant for input to TeX/LaTeX, Knuth's typesetting package. Meant for producing beautiful mathematics. \theta and \pi produce greek letters, the underscores are for subscripts.
Reply by ●January 22, 20172017-01-22
On 1/22/2017 1:36 AM, Fred Smith wrote:> On 2017-01-22, rickman <gnuarm@gmail.com> wrote: >> On 1/21/2017 2:10 PM, Scott Hemphill wrote: >>> >>> Write all the thetas as >>> >>> \theta_k = 2\pi k/n + \theta_c >> >> I've never seen this notation before. What do the back slashes mean? >> Is the forward slash still division? >> > > It's how you write maths meant for input to TeX/LaTeX, Knuth's > typesetting package. Meant for producing beautiful mathematics. > \theta and \pi produce greek letters, the underscores are for > subscripts.Maybe you can translate this as well... sum_{\theta} cos(\theta) = 0 What do the curly braces do? -- Rick C
Reply by ●January 22, 20172017-01-22
On 2017-01-22, rickman <gnuarm@gmail.com> wrote:> On 1/22/2017 1:36 AM, Fred Smith wrote: >> On 2017-01-22, rickman <gnuarm@gmail.com> wrote: >>> On 1/21/2017 2:10 PM, Scott Hemphill wrote: >>>> >>>> Write all the thetas as >>>> >>>> \theta_k = 2\pi k/n + \theta_c >>> >>> I've never seen this notation before. What do the back slashes mean? >>> Is the forward slash still division? >>> >> >> It's how you write maths meant for input to TeX/LaTeX, Knuth's >> typesetting package. Meant for producing beautiful mathematics. >> \theta and \pi produce greek letters, the underscores are for >> subscripts. > > Maybe you can translate this as well... > > sum_{\theta} cos(\theta) = 0 > > What do the curly braces do? >Group things. If you wanted to typeset \sum_{n=1} you'd need them to get all the n=1 centred under the large sigma (summation) sign. With just sum_n=1 you'd only get the n under the sigma sign, with an ordinary sized =1 alongside it. You need them in the example above to ensure the system doesn't throw an error after running into the backslash of \theta. TeX/LaTeX is a wonderful system for document preparation, the only one I'd use for anything serious, but you have to remember it came about in the era of 24 line, 80 column ascii VDU terminals, way before GUI's.
Reply by ●January 23, 20172017-01-23
rickman <gnuarm@gmail.com> writes:> On 1/21/2017 2:10 PM, Scott Hemphill wrote: >> >> Write all the thetas as >> >> \theta_k = 2\pi k/n + \theta_c > > I've never seen this notation before. What do the back slashes mean? > Is the forward slash still division?What Fred said. And yes, the forward slash is still division. It's an in-line division that typesets as a slash as opposed to: "{2\pi k \over n}", which typesets as a horizontal bar, with "2 \pi k" above the bar, and "n" below the bar. I answered this question in pseudo-TeX notation, because I thought it was asked that way. Scott -- Scott Hemphill hemphill@alumni.caltech.edu "This isn't flying. This is falling, with style." -- Buzz Lightyear
Reply by ●January 23, 20172017-01-23
On Sun, 22 Jan 2017 01:05:37 -0500, rickman wrote:> On 1/21/2017 2:10 PM, Scott Hemphill wrote: >> >> Write all the thetas as >> >> \theta_k = 2\pi k/n + \theta_c > > I've never seen this notation before. What do the back slashes mean? > Is the forward slash still division?Pseudo-LaTeX. \theta renders as the Greek letter, a_b renders as 'a' with a subscript 'b', etc. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com I'm looking for work -- see my website!
Reply by ●July 18, 20172017-07-18
On Saturday, January 21, 2017 at 11:51:40 PM UTC-8, Fred Smith wrote: (snip)> > Maybe you can translate this as well...> > sum_{\theta} cos(\theta) = 0> > What do the curly braces do?> Group things. If you wanted to typeset \sum_{n=1} you'd need them > to get all the n=1 centred under the large sigma (summation) sign. > With just sum_n=1 you'd only get the n under the sigma sign, with > an ordinary sized =1 alongside it. You need them in the example > above to ensure the system doesn't throw an error after running > into the backslash of \theta.I haven't thought about this recently, but I don't believe that they are needed for \theta. It goes in as one token, so it isn't a problem. They are needed for n=1, though. In the \theta case, it might be easier for people to read, but I am pretty sure TeX can read it just fine.> TeX/LaTeX is a wonderful system for document preparation, the only > one I'd use for anything serious, but you have to remember it came > about in the era of 24 line, 80 column ascii VDU terminals, way > before GUI's.It has the big advantage over GUI in that you can make small changes that have large effects. One that I remember from the transition days is WYSIAYG: What You See Is All You've Got. With TeX, in many cases one macro change will reformat the whole document, in ways that would take individual reformatting each paragraph in WYSIWYG systems. It was in the mid 1980's that TeX started to get wide usage, about the time that SunView, X11, and Macintosh started to get used, though it took a little while for the latter to get wide use. I suspect that it is similar to the verilog vs. schematic capture, where you have to be able to think in visual terms, without the visual aid of the display. (and in earlier years, between RPN and TI algebraic calculators.) Usually I can write verilog while looking at a logic diagram, much faster than I could get it into a GUI schematic capture tool. Similarly for TeX vs. GUI math tools.
Reply by ●July 18, 20172017-07-18
On Friday, January 20, 2017 at 3:31:33 PM UTC-8, Tim Wescott wrote:> Quite some time ago I handed y'all a quandary, to wit, proving that > sum_{\theta} cos(\theta) = 0, when \theta is evenly distributed on a > circle and there are an odd number of them.In the case of wheel lug nuts, you want them evenly tightened around the wheel. If you tighten them sequentially to full torque, one side will be much tighter than the other. You still don't want to go full torque at once, but with a star pattern over five bolts, and maybe all to half torque and then all to full, it is pretty even. This works best for an odd number, but if you do six in a 1 4 2 5 3 6 pattern, it probably isn't so bad. The five nut 1 3 5 2 4 pattern is nice, as you just advance two each time. With four nuts, I suspect 1 3 2 4 pattern is fine, but you have to be a little more careful not to overtighten the earlier ones. Maybe tighten to 1/3, then to 2/3 and finally full torque.
Reply by ●July 18, 20172017-07-18
On Friday, January 20, 2017 at 3:31:33 PM UTC-8, Tim Wescott wrote:> Quite some time ago I handed y'all a quandary, to wit, proving that > sum_{\theta} cos(\theta) = 0, when \theta is evenly distributed on a > circle and there are an odd number of them.Oh, the usual laboratory centrifuge has 12 holes that you can insert test tubes into. This allows for between 2 and 10, or 12 to be balanced. Three balance easily (holes 1, 5, 9 for example). Or two in holes 2, 7. The combination with five tubes in holes 1, 2, 5, 7, 9 balances. The only ones that don't are 1 and 11.
Reply by ●July 19, 20172017-07-19
On Tuesday, July 18, 2017 at 7:15:33 PM UTC-4, herrman...@gmail.com wrote:> On Friday, January 20, 2017 at 3:31:33 PM UTC-8, Tim Wescott wrote: > > Quite some time ago I handed y'all a quandary, to wit, proving that > > sum_{\theta} cos(\theta) = 0, when \theta is evenly distributed on a > > circle and there are an odd number of them. > > In the case of wheel lug nuts, you want them evenly tightened > around the wheel. If you tighten them sequentially to full > torque, one side will be much tighter than the other. > > You still don't want to go full torque at once, but with a star > pattern over five bolts, and maybe all to half torque and then > all to full, it is pretty even. This works best for an odd > number, but if you do six in a 1 4 2 5 3 6 pattern, it probably > isn't so bad. The five nut 1 3 5 2 4 pattern is nice, as you > just advance two each time.this is what i had always thought about it. it's how i would tighten lug nuts when changing a tire. i've thought that 3-blade windmill propellers were more stable than 2-blades. as for how an odd-number balances, you can make an argument from symmetry. if the 5 lug nuts are equally spaced around the circle, how is the direction that the balance tips to be predicted? why should it tip in one direction rather than another? r b-j
