Interactive geometry programs such as GeoGebra and CaRMetal are intuitive tools for learning about geometry. You can use them to explore the basic principles of paper-folding (Origami). I learned about the mathematics of folding paper circles from: "Circle Origami Axioms", on Maria Droujkova's Math 2.0 Interest Group. I wondered if a rule for folding paper circles would apply to other shapes, and chose to explore triangles. An interesting configuration of folds is discovered when the triangle is equilateral. Using simple tricks, such as hidden line removal, the geometry diagram simulates a folded piece of paper. You can download my paper "Generalisation of a Paper-Folding Axiom and its Exploration using Interactive Geometry Software" and the data files for Geogebra, Dr Geo and CaRMetal from i2geo.net. You can also download the PDF file from Scribd.com. The paper is condensed into a slideshow on SlideShare.net.
07/23/2010
06/19/2010
Circles Inscribed In Triangles
I present some experiments with interactive geometry software on Slideshare.net at: Circles in Triangles using Geometric Construction. This follows on from my previous presentation: Inscribe Semicircle In Square by Geometric Construction. Both slide shows use Geometric Construction to explore basic geometric shapes. I used the free Dr Geo and Geogebra software to create constructions that explore Euclidean geometry in a plane. There are problems for you to investigate: (i) Inscribe four circles in a circle using geometric construction. (ii) Discover additional circles in the equilateral triangle shown in the diagram. This is a fresh look at some ancient mathematics, using readily available PC software. The geometry files are on http://i2geo.net/.
05/30/2010
Recursively Inscribed Semicircles
Dr. Geo is interactive geometry software for the GTK desktop on Linux. It saves the drawings in XML format. I had to edit the XML file of the square to set the corner points, to 0,0; 10,0; 10,10 and 0,10, as I use an older version of Dr. Geo that doesn't support coordinate entry. After exporting the drawing as a PNG picture, I used the open source Inkscape drawing software to colour the segments for clarity.
04/09/2010
Inscribe Semicircle In Square by Geometric Construction
Why would you use this technique? It is not trivial to solve this problem using algebra, as it involves factorisation of irrational numbers. Geometric Construction is an ancient mathematical technique that can be done without knowing algebra. The basic tools are a pair of compasses and a straight-edge or ruler. On a personal computer, you can use free geometry software such as Geogebra, as I did for these examples. The geometry files are on http://i2geo.net/.
12/02/2009
Mathematics Forum on LinkedIn.com Achieves 1000 Members
10/28/2009
Everything You Need to Know About Black Holes - One Week Late
A useful feature of eWeek is the list of “Most Read” stories, which is currently:
- Microsoft's Big Windows 7 Week. Microsoft's 2006-2007 release of Windows Vista was reported as disappointing; causing many PC users to stick with Windows XP, so the release of Windows 7 has been widely anticipated.
- NASA Space Funding Reaches Critical Point. NASA is planning the future of space travel against a background of controversial decisions, such as: whether to use the Moon as a stepping stone to Mars, whether to use humans or robots to explore space, and whether space travel should be a commercial or government funded venture.
- Why the Droid Can Challenge the iPhone. Should mobile phones be locked to applications and services that Apple Inc. permits, or should you have a free choice to customise mobile applications and services, as with (An)Droid?
- Windows 7 Comes on USB Drives for
Netbooks. Software and documents have traditionally been stored on spinning magnetic discs that are prone to mechanical failure, but we are now in the decade where solid-state memory (with no moving parts), has become a cost effective and more reliable alternative.
- Tilera Talks 100-Core Processor. Some calculations, such as simulating the collision between two black holes, would be too dangerous to try as experiments (1) and too complex to solve on an office PC, so scientists use dozens of computers on a single silicon chip, to complete their calculations in days rather than years. This new technology makes supercomputers available for business applications.
The business-like format of weekly publications made sense when they were printed on paper, but the Internet has enabled the era of reader generated content. The popular http://slashdot.org/, by contrast, has become the classic site where technology news is posted and rated by its readers. It is worth wading through the sometimes inane comments to find an occasional gem of alternative opinion.
Disclaimer: I have no interest in any of the above mentioned companies or organisations, and write this blog post for general interest purposes only.
08/17/2009
Rediscovering Straight-edge and Compass Construction
People who haven't had the right opportunity to learn math have
something in common with people who were born before math was invented.
Perhaps some of the techniques of Euclidean geometry could be revived in a format
that is more accessible today. Specifically, the ancient tools of straight-edge and compass, could be reinvented to suit today's teaching needs. Take this geometry problem: What is the largest area of the semi-circle that can be inscribed in a square of edge length 1 unit? That was the question posted by harpreet in the topic " maxima & minima" on a Mathematics forum on Orkut.com. Some mathematics enthusiasts took a team approach to this problem on the Mathematics24x7 social network, Christian drew a diagram of the solution and Steve calculated the radius of the semicircle. We calculated the radius as 2-sqrt(2) and the area as pi*(3-2*sqrt(2))
.(sqrt() represents the square root.) Danny proposed a less abstract expression of the problem, which would be more interesting to students. "I have been asked to help paint a mural on the outside wall of a
grocery store in my neighborhood. My task is to create the background
for the mural. The instructions are to create the largest possible
semicircle on the wall, with the semicircle touching all 4 sides. The
wall is square with 10 feet on each side. I need to find out how to
position the semicircle to satisfy the instructions. I also need to
know the radius and center of the semicircle. How can I figure this out
with the basic math that I know?" In the discussion Straight Edge and Compass Construction For Developmental Math, I described a method of drawing the mural using
a long plank, lengths of rope, a few pegs and some chalk. Use the rope to extend the base of the wall, to the right, by its
width, and mark the point with a peg. This defines an imaginary square
that is side by side with, and on the right of the square wall. Draw a
diagonal on the original square, because we know that the solution is
symmetrical about the diagonal. Mark a diagonal on the imaginary square
by stretching a rope from the peg to the top right corner of the square
wall. This is a way of calculating the square root of two. Using the
peg as the center point and the rope as a radius, follow an arc down to
the base of the wall, and mark the point where the arc intersects the
base. The distance of that point from the left wall is 2-sqrt(2), which
is the radius of the semicircle. Draw a perpendicular from that point,
and where it meets the diagonal, peg the center point of the semicircle.
Attaching a length of rope to the peg, stretch it to the furthest wall,
and with chalk held fixed on the rope, draw the semicircle. I also described the construction in more abstract terms, and used the GeoGebra geometry software to demonstrate.(See the diagram, where the biggest semicircle that fits in the square is positioned diagonally in the top left of the diagram.) Draw the square as a 4 sided regular polygon. Draw
an identical square to its right Draw the diagonals of the right hand
square and use them as radii of arcs that intersect the left hand
square. Create points where the arcs intersect the base and top side of
the left hand square. Join these two points with a vertical line. Its
intersection with the diagonal of the left hand square defines the
centre of the semicircle. Draw the semicircle through any one point on
the square and notice that it touches or intersects the square at 4 points. Draw the
base of the semicircle through the two intersection points. (I mistakenly drew the wall as 11 feet instead of 10, but that does not detract from the construction.) Constructive geometry involves no measurement of length, except use of
the compass as a tool to copy a length and duplicate it somewhere else
on the plane. It also involves no algebra. We multiply a length
by extending it with additional equal lengths. We draw the diagonal of a
square without realising that we are calculating the square root of
two. Constructive geometry is available to people who don't do algebra. They can solve ancient problems, rediscover the
history of mathematics and apply it to their own environment. Straight-edge and compass problems range in difficulty from simply drawing a hexagon to the more complex procedure for drawing a regular pentagon.
08/04/2009
Promoting Mathematics Via Social Networks
This is a story of the promotion of mathematics and science through social networks, digital repositories and other Web 2.0 technologies. It began in August 2008 when I was inspired by the 1999 Cluetrain Manifesto, and wrote a discussion paper "Use the Cloud to Get a Clue", which I published as a PDF file on Scribd.com and later as a slide show "Openness and Social Networking" on Slideshare.net. To demonstrate the Creative Commons licenses, I threw together a quick presentation "The Cool Physics of Heat", and was surprised that it notched up almost 1000 views on Slideshare.net and 1400 views on Scribd.com, much more than any other document that I have released. I then focussed on my interest in mathematics (not my main subject), in particular blogging about it here, at http://cmcallister.vox.com/ on the free blogging service, Vox.com. I discovered an active niche social network, Mathematics24x7, on the Ning.com platform, where teachers and other academics were expressing their enthusiasm for maths. I participated in a weekly #mathchat Twitter conference (2am GMT Thursdays). That exchange inspired me to create a public wiki "Online Mathematics Access", about math markup as a tool for discussing mathematics online. There were failures too; my "Mirimatics" forum on Friendster.com stimulated absolutely no discussion. I added a Math Problem Solving group to Mathematics24x7, which is more successful, and attracted a dozen participants in as many days. My online activity is neither scientific research nor publishing, in the formal sense. However, it is still worthwhile, and involves the exchange of academic ideas with a network of new online acquaintances. It is promising that this discourse has grown, without being published in an academic journal, or having any research focus. The discussion has drifted across such diverse topics as the solution of geometric problems and the significance of colour in cognition, which someone "Liked" on Facebook. I'm simply writing about a subject that I enjoy, not striving to maximise page hits. The only metrics are the view count on my uploaded files, and a few red dots on the visitor map on my blog. I uploaded snapshots, of both the math markup wiki and this Vox.com blog, to Scribd.com, to make them more available. The online discourse is dynamic, refreshing, and involves a broad cross section of maths enthusiasts. We exercised teamwork in solving maths problems and discussed some new ideas. The discussions are linked to other networks too, including the Math, Math Education, Math Culture group on LinkedIn.com and a Mathematics community on Orkut.com.
This story is dedicated to the many individuals who are persecuted for publishing their ideas, a few of whom I mention on my blog at http://cmcallister.blog.friendster.com/.
Links:
Social network: http://mathematics24x7.ning.com/ created by Rashmi Kathuria
#mathchat, hosted weekly by Maria Droujkova at http://twitter.com/mariadroujkova
Math Markup wiki: http://onlinemathematicsaccess.wikispaces.com/
Math, Math Education, Math Culture, managed by Opher Liba, at http://www.linkedin.com/groups?gid=33207
Math 2.0 Interest Group, managed by by Maria Droujkova, at http://mathfuture.wikispaces.com/
07/31/2009
Colourised Equations As An Aid To Mathematics
Yesterday, during a discussion about cognition, a colleague informed me that Physicist Richard Feynman perceived colours when he saw equations. "When I see equations," he once said "I see the letters in colors - I don't know why." (Ref 1). This ability is a form of synesthesia by which there is cross-over from one sense to another. Daniel Tammet, an amazing savant, also has this gift. He "sees numbers as shapes, colors, and textures, and performs extraordinary calculations in his head", as he describes in his book, Born on a Blue Day. This suggested to me that colourised equations would be a good aid to learning for some people. (This is just an idea. I have no evidence that it would be effective, and it could put a person who is colour blind at a disadvantage. Ref 2.)
I took the well known roots of the quadratic equation:
and used blue to identify irrational parts and red to identify negative terms. The mapping is not clear cut, because one term is plus-or-minus, and it is only the result of the square root that is irrational.
\frac{\uc{red}{-b}\pm{\uc{blue}{\sqrt[]{(b^2 \uc{red}{- 4*a*c})}}}}{2a}
into the equation renderer at: http://www.hamline.edu/~arundquist/equationeditor/. The LaTeX command \uc{red}{-b} defines that the term -b is to be coloured red.
The benefits of colouring may not be apparent for a single equation. I suggest that you try it the next time you need to present a long and tedious algebraic evaluation. The low-tech approach would be to use coloured highlighting pens to mark up a printed copy of the equations. This is not a new idea, accountants have been using red ink to represent negative numbers for generations.
For the benefit anyone who is unable to view the images, the equation shown by the above images is:
(-b +/- sqrt(b^2 - 4*a*c))/2a
The negative terms -b and -4*a*c are coloured red and the sqrt() function, which may be irrational is coloured blue. The square root may also be an imaginary number, if it is the square root of a negative number.
Another application of colouring is to present each digit in its standard resistor colour code. The pattern of repeating decimals for certain fractions becomes much more memorable. For example 264/999:
This was generated from the LaTeX code:
\pagecolor{gray}\uc{red}{2}\uc{blue}{6}\uc{yellow}{4}/\uc{white}9\uc{white}9\uc{white}9 = 0.\uc{red}{2}\uc{blue}{6}\uc{yellow}{4}\uc{red}{2}\uc{blue}{6}\uc{yellow}{4}\uc{red}{2}\uc{blue}{6}\uc{yellow}{4}\uc{red}{2}\uc{blue}{6}\uc{yellow}{4}...
using into the equation renderer at: http://www.hamline.edu/~arundquist/equationeditor/.
Ref 1: The book Sparks of Genius, By Robert Scott Root-Bernstein. The page containing Feynman's description may be viewed in Google Books.
Ref 2: In "Colourised Equations As An Aid To Mathematics" on the Mathematics24x7 network, Bob Mathews commented that we need to create presentations that are "color-blind friendly".
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