Thoughts on the mathematical constant e

May 16th, 2010 admin

e for exponential

The mathematical constant shows up in strange places. Moreover, its significance is not as easy to grasp as that of the other famous constant, , because there is no easy physical object in whose context to imagine it. For example, is the ratio of the circumference to the diameter of a circle. Yes, it is irrational, but if you can get over that mystery (or ignore it for the time being), it is straightforward to imagine what is. Every circle does seem to have a certain circleness, which makes them all look the same. It is intuitively not hard to agree with the hunch that every circle has an unchanging ratio between the circumference and the diameter; and it makes sense to keep that ratio handy and give it a name.

The constant is considerably more elusive. It appears, at first, to be a number you would not go hunting after. You just happen to stumble upon in during one of your mathematical excursions; it seems interesting enough that you then pick it up and put in in your pocket for some potential use later. After stumbling upon the same thing along other mathematical excursions, in hindsight, it does seem to be something rather useful. Something you should have gone looking for. Read the rest of this entry »

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We, the people …

May 2nd, 2010 admin

Reference for the population figure: http://www.google.com/publicdata?ds=wb-wdi&met=sp_pop_totl&tdim=true&dl=en&hl=en&q=world+population

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We made a garden trellis with PVC piping

May 2nd, 2010 admin

Figure1: Basic plan showing the material required

We have two 4′x8′ (4 feet by 8 feet) raised beds, which we use for vegetable plants. Kavita has been asking me to either buy or build a trellis for her climbing plants (cucumbers, tomatoes and eventually some types of squash and gourds). I read several websites online and decided to build a simple trellis using PVC piping. It took one trip to the local Home Depot, and then about 2 hours of work. The cost for the material was under $10 (I already had all the tools needed).

We decided to build one and test it out before getting carried away and building more. We decided that we would roughly want the trellis to be 4 feet wide by 5 feet high. At the Home Depot we did some quick calculations based on the basic design we had in mind and came up with a total of about 29 feet of PVC tube. The calculation is shown in Figure 1.

Figure 2: Materials and Tools for the project (4-way, + shaped, PVC connector missing)

The PVC pipes are sold in 10′ pieces. We got 3 pieces. We also got some string (I tried polypropylene string since I did not know any better, we’ll see how that works out).

Figure 2 shows most of the material and tools. The one caveat is, since I took this picture after completing the project, the one 4-way 1/2″ PVC pipe connector used at the center of the frame is missing from the picture. I had extra connectors of the other type, so I could use them for the picture. Also, one other thing that is missing from the picture is a power drill and drill bits. I used a 3/16″ drill bit to drill evenly spaced holes in the pipes to draw the string through, to create a framework.

Figure 3: Trellis plan with the stringing shown

Figure 4: Framework ready, stringing is yet to be completed

The spacing between the holes and how the trellis is supposed to look eventually is shown in Figure 3.

The thing that took the most time was measuring and marking the PVC pipes, cutting them to the right size with the saw, then measuring and marking the locations for the holes for the string to go through and then drilling the holes with the power drill. Since the PVC pipe keeps rolling about, making it stable before drilling is important. I just used an old rag to wrap around the pipe in order to hold it somewhat still.

Once the pieces were all ready, putting the trellis together took less than 10 minutes. There was no need for glue, since the connectors fit quite snugly. Figure 4 shows the trellis laid out on the lawn (with only one piece of string drawn through, Kavita will work on getting the rest of the mesh this evening).

We are not sure how well this will hold up, how long it will last etc. I will update the post with some pictures on the trellis in action, later.

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Light Humor

May 2nd, 2010 admin

Light, at the end of the tunnel

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Out-of-the-box thinking

May 1st, 2010 admin

At the beginning of the previous post I had included a set of slides which propose the 4 squares problem and teach us that we should always be ready to think of a simple solution whenever possible. This theme caused a flurry of emails among some of my family members, and I would like to present some of the interesting ideas that arose in that discussion. Read the rest of this entry »

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The 4 squares problem – extended

May 1st, 2010 admin

A “4 Squares Puzzle” is doing the rounds over the internet nowadays. here are the original Microsoft Powerpoint slides which I received via email from my father. (Not sure who the original author of these slides is, but the slides report the author to be Nakit Yonetimi.) It may be useful to look over the pdf file once before proceeding. The question posted in the slides is different from the one I am going to pose, but going through the slides helps build context and helps get mentally warmed up.

The question I posed to myself after thinking through the puzzle was, “How can we divide a square into 7 equal parts with only a straight edge and a compass available?” Note that the question implies that we do not have a ruler or a scale. We have a straight edge, but without any markings on it to indicate inches or centimeters. Even if it did have the markings, such markings can only measure accurately up to a certain level. For example, say you have a scale with markings at the granularity of a millimeter. Say the square had a side equal to some irrational number, say or , or, even a simple integer which is not a multiple of 7, such as, 8 millimeters. There is no way to measure or or millimeters using such a scale.

There are at least 2 approaches to dividing this square into 7 parts. The first is a simpler approach and the second is slightly more involved. Let me talk about the second one first. The first one will then become easy to see. The main intuition behind the first idea is that a triangle’s area depends only on its base and height. If we can mark out 7 equidistant points along the square’s border, thus creating 7 equal bases, we can join the bases to the center of the square to create 7 regions with equal areas. The heights of these shapes will be equal, and the bases are equal by construction. Read the rest of this entry »

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