DaVinci Was Right

HPO MVI 0043 from U of T Engineering on Vimeo.

I subscribe to eGFI, a site that is there to encourage future students to go into Engineering. I recommend it to students who ask about or seem to want to go into Engineering. Today they had this on the site, with the video above..


Human-Powered Plane Flies Like A Bird

"For the past four years, Todd Reichert, an engineering student at the University of Toronto, has been working to perfect one of Leonardo Da Vinci’s greatest concepts – an ornithopter.

An ornithopter is a human-powered aircraft that flies by flapping its wings, and with the help of 30 other students, as well as $200,000, Reichert made history by building such a vehicle and piloting a sustained flight.

Called the Snowbird, the aircraft is made of carbon fiber, foam and balsa wood, and weighs less than 93 pounds. Its wingspan is 104 feet, which is comparable to that of a Boeing 737. The vehicle works by pumping a set of pedals attached to pulleys and lines that bring down the wings in an elegant flapping motion.

Leonardo Da Vinci conceived the ornithopter back in 1485. For much of his life, Da Vinci was fascinated by the phenomenon of flight and produced many sketches of flying machines; however, he never actually made one."

The Bead by Mamikon's Visual Caclulus

Over the last few days I came upon a way to explain the volume of a bead with an (almost) non-calculus approach. My solution was inspired by a comment posted by Arjen Dijksman. I admitted I didn't understand his suggestion to use Mamikon's visual calculus, so he did what good teachers do. He posted a very clear explanation that even I could not miss. It's a beautiful blog, pretty math... check it out..

and Thanks loads Arjen.

A Non-calculus Explanation for the Volume of a Bead

A few days ago I wrote about the "surprising constant" that a bead formed by drilling a hole through the center of a sphere had a volume that was dependent only on the height of the remaining bead. At the time I admitted I could not think of a good "non-calculus" explanation for why the volume for a given bead height was a constant.

Afterward Arjen Dijksman,(see his blog) who regularly provides helpful insights, suggested that the volume could be explained using the ideas of Mamikon Mnatsakanian that I discussed here. While I was not able to see Arjen's solution, it did lead me to realize that a "non-calculus" proof existed (with some allowances).

Thinking of Mamikon's annular sweep led me to realize that every cross-section through the bead perpendicular to the axis of the drilled hole would be an annulus. In fact, using R as the original sphere's radius, r as the radius of the hole, and h as the distance above the center of the sphere where the cross section was taken, the area of the annulus would be Pi times (R² - r²-h²), which is just the area of a circle.

Now by Cavalieri's Principal, "If, in two solids of equal altitude, the sections made by planes parallel to and at the same distance from their respective bases are always equal, then the volumes of the two solids are equal."

Several candidates exist with circular cross-sections; cylinders, cones, and spheres are examples. The cross-sections of the bead went from zero at the top, to a maximum at the center, so the cylinder is excluded. We can exclude the cone since their radii decrease linearly from top to bottom and our bead does not. And since we are already tempted with the idea that the volume of the bead is the same as a bead with a radius equal to half the total height of the bead, why not try to compare those?

To make the notation easier, I decided to call the height of the bead 2H (letting H² equal the value (R² - r²). So the cross sectional area of the sphere with raidus H at any height, h, above the center will be Pi(H² - h²). As we have shown above, the cross-sectional area of the bead at the same height h is Pi (R² - r²-h ²). and since H² is equal to (R² - r²) these two are equal.

So we have proven that the volume of the bead is the same as the volume of a sphere with the same height;... but did we use calculus? I fear that the proof of Cavalieri's Principal is founded in the calculus, and thus we still have a "calculus-based" explanation...but it's the best I have to offer to date.

Folium

I realized in Caculus class the other day that I didn't have a link for folium on my Mathwords webpage, so I decided to write one here and just put a link...
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Folium is the Latin word for leaf. Student’s frequently are introduced to them in learning to graph polar curves such as r=sin(3*theta) which is a trifoil, or three leaf, sometimes called a “rose” (shown above).

The one that came up in my Calculus class a few days ago was one of the more widely known mathematical folia, the “Folium of Des Cartes”, x³ + y³=3xy. A nice chance to use the newfound skill of implicit differentiation. This curve was created by Des Cartes to challenge the methods of Fermat in finding extrema in the early days of the emerging calculus. I should mention that both men found the extrema without the calculus of limits as we now know it. I mentioned Des Cartes method, as extended by Hudde, in an earlier posting here.



The Folium can be rendered in parametric form by using the substitution y=tx to give

x³ + (tx)³ = 3tx² and dividing out x² we get to

x + t³ x = 3t... This can be factored for x to get x(1+t³)= 3t.... or

x= (3t) / (1+t³) and thus by y=xt we get y= (3t²)/ (1+t³)

I recently came across this neat stamp image with a picture of Des Cartes and his Folium.