FEBRUARY 2013 VOL 1 NO 2

AB INITIO Because all Mathematics must have a beginning …

FROM WHENCE IT COMES … This reminds me of a saying my mother often used when I was a child: "Take it from whence it comes." Meaning, of course, that EVERYTHING has a context that's as important as the OBJECT / CONCEPT themselves.

KEY PHRASES CONIC SECTIONS

HYPERBOLA

ELLIPSE

PARABOLA

ECCENTRICITY

RECRIPOCAL FUNCTION

DISCONTINUNITY

LIMITS

In this edition of the newsletter, we will begin to explore a very fundamental model that governs the loci of the ripples as it skims the surface of the water towards an

intersection.

On a very still morning in MacRitchie Reservoir I was standing on the bridge overlooking the reservoir waiting for the fountain to start up – I’ve always enjoyed watching how water spouting from the fountain interrupts the stillness of the surface of the reservoir and how the ripples cascade towards the shore in a very distinctive pattern.

That was when it hit me – because the pattern is distinct and because it repeats every day, we must be able to generalize the effects of the ripples using a mathematical equation.

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AB INITIO – Vol 1 No 2 FEBRUARY 2013

What is the path of the intersection of 2 waves?

Diagram 1

Therefore, if we were to plot the point(s) of intersection over a period of time, we would note that the point(s) of intersection would look like Diagram 2

If we were to throw two stones into the reservoir before the fountain started, where would the 2 ripples from the 2 distinct impact points intersect?

Consider the diagram 1, the 2 stones would impact the water at 2 DISTINCT points of impact, !1 and !2.

Now, we know that the radius of the 2 circles, ! and !, will increase over time. This means that the function governing the increase in the radius of both circles can be expressed as

!1! = !(!) and !2! = !(!).

If we were to assume that the radius of both circles were to increase at a constant rate over time, this can be expressed in the form

, where k is a constant

value and where ! − ! = !"#$%&#%.

and this relationship is well know to us because the LOCI of the point(s) of intersection will follow a HYPERBOLIC path and this has applications in areas such as the scattering theory of subatomic particles.

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AB INITIO – Vol 1 No 2 FEBRUARY 2013

In greater detail …

1

If you were to trace the path through the points of intersection (see Diagram 2), you would see that the HYPERBOLIC curve is clearly visible.

Hence, by definition, if we are given 2 distinct points (the foci) !1 and !2, a hyperbola is the LOCUS of points such that the DIFFERENCE to each focus is CONSTANT.

This means that ! − ! = !"#$%&#%.

2

Hence, considering the representation in Diagram 3, the

!!! − !" = !"#$%&#%,

and because

!!! = !"

it would be more accurate to conclude that

!!! − !" = 2(!").

Diagram 2

Diagram 3

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AB INITIO – Vol 1 No 2 FEBRUARY 2013

Euclid, Archimedes, and Apollonius are the three mathematical giants of the third century B.C. upon whose work we obtain a lot of our geometrical interpretation of some of the fundamental theorems that surround the fields of Astronomy, Architecture and Mathematics.

His–STORY – Apollonius, who was younger than Archimedes by about twenty-five years and was born circa 262 B.C. in Perga, in southern Asia Minor. As a young man he went to Alexandria, studied under the successors of Euclid, and remained there for a long time. Later, he visited Pergamum, in western Asia Minor, where there was a recently founded university and library patterned after that at Alexandria. He later returned to Alexandria and died there sometime around 190 B.C.

His–WORK – Although Apollonius was an astronomer and although he wrote on a variety of mathematical subjects, his chief claim to fame rests on his extraordinary Conic Sections – a work that earned him the name, among his contemporaries, of "The Great Geometer."

Apollonius' Conic Sections, in eight books and containing about 400 propositions, is a thorough investigation of these curves, and completely superseded the earlier works on the subject by Menaechmus, Aristaeus, and Euclid.

Only the first seven of the eight books have survived to today – the first four written in Greek and the following three from a ninth-century Arabian translation.

The first four books, of which I, II and III are presumably founded on Euclid's previous work, dealing with the general elementary theory of conics, whereas the later books are devoted to more specialized investigations.

Some history … Prior to Apollonius, the Greeks derived the conic sections from three types of cones of revolution, according as the vertex angle of the cone is less than, equal to, or greater than a right angle – this

was done by cutting each of three such cones with a plane perpendicular to an element of the cone, an ellipse, parabola, and hyperbola will be produced. However, in the early Greek considerations, only one branch of a hyperbola was considered.

Apollonius, however, in Book I of his treatise, obtains all the conic sections in the now-familiar way from one right or oblique circular double cone. The names ellipse, parabola, and Hyperbola were supported by Apollonius and were borrowed from the early Pythagorean terminology of application of areas.

When the Pythagoreans applied a rectangle to a line segment (that is, placed the base of the rectangle along the line segment with line end if the base coinciding with one end of the segment), they said that had a case of "ellipsis," "parabole," or "hyperbole" according as the base if the applied rectangle fell short of the line segment, exactly coincided with it, or exceeded it.

AB INITIO – Vol 1 No 2 February 2013

In closing … During the Renaissance, Kepler's law of planetary motion, Descarte and Fermat's coordinate geometry, and the beginning of projective geometry started by Desargues, La Hire, Pascal pushed conics to a high level.

Many later mathematicians have also made contribution to conics, especially in the development of projective geometry where conics are fundamental objects as circles in Greek geometry. Among the contributors, we may find Newton, Dandelin, Gergonne, Brianchon, Dupin, Chasles, and Steiner.

Conic sections is a rich classic topic that has spurred many developments in the history of mathematics.

Menaechmus first studied a special case of the hyperbola. This special case was xy = ab where the asymptotes are at right angles and this particular form of the hyperbola is called a rectangle hyperbola.

Euclid and Aristaeus wrote about the general hyperbola but only studied one branch of it while Apollonius who was the first to study the two branches of the hyperbola gave the hyperbola. The focus and directrix of a hyperbola were considered by Pappus.

When we consider the double cone, we will need to INTERSECT the cone with a PERPENDICULAR plane and the angle in which it intersects the cones will determine the resultant conic section – parabola, circle, ellipse and hyperbola.

As this newsletter is to highlight the applicability of Mathematics in your daily lives, YOU are welcomed to write articles, thoughts and reflections for the future editions of the newsletter.

Find something that interests you and we’ll find the Mathematics that goes with it. Email ideas and submissions to

jason_ingham@sst.edu.sg

Now for you to do some work …

Look for examples from your daily life that resembles the shape

of the PARABOLA, ELLIPSE and HYPERBOLA and post the

photos on your Facebook wall. I’ll include the better examples

on the next edition.

REFERENCES – 1.Hollingdale, S. (1994). “Makers of Mathematics”. Penguin Books, UK. 2. Motz, L. and Weaver, JH. (1994). “The Story of Mathematics”. Avon Books, NY. 3. Merzbach, UC and Boyer, CB (2011). “A History of Mathematics”. John Wiley and Sons, New Jersey.

In the NEXT edition … Would you like to work for GOOGLE? This is an interview

question that is posed as part of their problem solving process –

“You work in a 100-story building and are given two identical

eggs. You have to determine the highest floor from which an

egg can be dropped without breaking it. You are allowed to

break both eggs in the process. How many (minimum) drops

would it take you to do it?