tag:blogger.com,1999:blog-46417169123610253642016-08-20T09:32:59.207-07:00zome blogWe will use things like Zome to talk
about fun math factsmskmoorthyhttp://www.blogger.com/profile/04521212767155230436noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-4641716912361025364.post-57056649696108617172008-01-31T07:24:00.000-08:002008-01-31T07:37:08.107-08:00Golden Ratio and Right Angle TriangleI truly enjoy helping high school and middle school students reach their potentials in solving Mathematical Problems. In this process, I also learn and relearn.<br /><br />I was working on a right angle triangle problem with some high school students. I realized that there are infinitely many triplets (a,b,c) of side lengths which are in arithmetic progression - for different increments - such as (3,4,5), or any integer multiple of this. We can also get rational increments - again an infinite number of rationals.<br /><br />On the other hand, for (a,b,c) to be in geometric progression, the ratio can be only one - namely the square-root of (\phi) (or 1.2720196492 - up to 10 decimal places) <span class="nfakPe"></span><br /><span style="color:#888888;"><br /></span>mskmoorthyhttp://www.blogger.com/profile/04521212767155230436noreply@blogger.com2tag:blogger.com,1999:blog-4641716912361025364.post-46704565975187193402008-01-14T06:35:00.000-08:002008-01-14T06:38:08.375-08:00Resistances and Golden Ratio<span style="font-family:Times New Roman;font-size:78%;">We are given infinite amount of unit resistors (each resistor has a<br />resistance of 1 Ohm). We can connect these resistors in series or/and in<br />parallel resulting in a resistor network. For example, you can get a<br />resistance of 2/3 Ohm by connecting two 1/3 Ohm Resistor networks in series.<br />To get a 1/3 Ohm Resistor network, we can connect three unit resistors in parallel.<br />One puzzle is to realize a resistance whose value is the exact golden ratio.<br /><br />The infinite ladder resistor (with end points A and B) network having a unit resistor<br />from point A to say Point Y and connect a unit resistor from Y to B. Now repeat the<br />above construction starting at Point Y. This leads to an infinite ladder network.<br />and the resistance between points A and B will be the golden ratio.<br />(as the resistance between A and B is the same as the resistance between Y and B)<br /><br /><br />To see this, consider the resistor network of unit resistor in serial connection<br />with a parallel resistor network of 1 Ohm and x Ohms. Let us denote the<br />resistance between A and B is<br />x = 1 + x/(1+x)<br /><br />or x-1 = x/(x+1)<br /><br />(The author proposed a variation of this problem to IEEE Potential in 1982.<br />A selected collection of problems including the variation<br />problem 3-17 appears in<br /> "The Unofficial IEEE Brain-buster Gamebook : Mental Workouts for the Technically<br />Inclined" by D. R. Mack, Wiley-IEEE Press, 1992.)<br /><br />With duality (changing resistors in series to parallel and parallel to series) and<br />scaling, we can realize resistances of all integer powers of the golden ratio as<br />well as the integer powers of reciprocal of golden ratios!<br /></span>mskmoorthyhttp://www.blogger.com/profile/04521212767155230436noreply@blogger.com0tag:blogger.com,1999:blog-4641716912361025364.post-38355289822996723242007-05-19T13:59:00.001-07:002007-05-19T14:03:29.108-07:00Redundancy, Robust Design, And Mathematics<br></br> <div xmlns='http://www.w3.org/1999/xhtml'><br></br><p>Zome<br /> Systems builds upon the magic of the golden ratio.<br></br>Its<br /> design reflects redundancy at many levels, a feature that<br /> is<br></br>usually common to well-designed systems, and a feature<br /> inherent<br></br>to anything mathematical.</p><br></br><p>As an<br /> example, Zome sticks are both color and shape coded; a<br></br>fact<br /> that makes this an excellent tool for teaching math<br /> to<br></br>visually impaired students —<br></br>see <a href='http://emacspeak.sourceforge.net/raman/publications/thinking-of-math/'>Thinking<br></br>Of<br /> Mathematics</a>.</p><br></br><p>Redundancy can often be a<br /> consequence of the same basic fact<br></br>manifesting itself when<br /> viewed from different perspectives. As a<br></br>case in point, see<br /> the number of ways that the golden ratio<br></br>manifests itself,<br /> both within Zome, and more generally the wider<br></br>world we<br /> live in.</p><br></br><br></br> <br /><p><a rel='tag' href='http://technorati.com/tag/tv+raman'><img alt=' ' src='http://static.technorati.com/static/img/pub/icon-utag-16x13.png?tag=tv+raman' style='border:0;vertical-align:middle;margin-left:.4em'></img>tv raman</a></p><br /> </div><br></br> T. V. Ramanhttp://www.blogger.com/profile/03589687652590194428noreply@blogger.com0tag:blogger.com,1999:blog-4641716912361025364.post-31442342234022849342007-05-04T15:44:00.000-07:002007-05-04T15:50:58.513-07:00Some TidbitsSome Tidbits:<br /><br />Phi (the golden ratio) plays a very important role in Mathematics,<br />Sciences, Arts and Zome tool. Please take a look at<br /><a href="http://redsevenone.wordpress.com/"> http://redsevenone.wordpress.com/</a><br />(and click on the link on the right side ZOME - the Answers Before you ask for Zome tool),<br />and <a href="http://en.wikipedia.org/wiki/Golden_ratio">http://en.wikipedia.org/wiki/Golden_ratio</a> (for Phi and Golden Ratio).<br /><br />If Pi day is celebrated on March 14 (3/14), don't you think Phi day<br />should be celebrated on 6th of January (1/6)<br /><br />If Phi is 1.6180339887, then 1/Phi is 0.6180339887 - Do you know why? Because Phi satisfies the algebraic equation Phi^2 = Phi + 1.<br />If you divide this equation by Phi, you see the answer. This trick was<br />revealed to me by none other than my co-blogger <a href="http://emacspeak.sourceforge.net/raman">TV Raman<br /></a>mskmoorthyhttp://www.blogger.com/profile/04521212767155230436noreply@blogger.com0tag:blogger.com,1999:blog-4641716912361025364.post-54660808852918212802007-05-04T11:40:00.001-07:002007-05-04T11:57:43.639-07:00Math Fun With Zome<br /><br></br> <br /><div xmlns='http://www.w3.org/1999/xhtml'><br /><br></br><br /><p>Math can be fun, and it can be made even more enjoyable when<br /><br></br>elegant mathematical facts are made tangible via concrete models.<br /><br></br>Zome Systems --- a polyhedral building kit based on the geometry<br /><br></br>of the dodecahedron/icosahedron pair --- can provide many hours<br /><br></br>of fun-filled math edutainment.<br /><br></br>In this blog, we hope to help to share some of the insights we<br /><br></br>have gained over the last few years in the form of easy to<br /><br></br>consume byte-sized capsules.<br /><br></br>For an in-depth review of many of these facts, see <br /><br></br><br /><a href='http://emacspeak.sourceforge.net/raman/publications/polyhedra/'>Visual<br /><br></br>Techniques For Computing Polyhedral Volumes<br /></a>.<br /></p><br /><p>Author: <a href='http://emacspeak.sf.net/raman'>T. V. Raman</a></p><br /><br></br> <br /><br /></div><br /><br></br> <br />T. V. Ramanhttp://www.blogger.com/profile/03589687652590194428noreply@blogger.com0