High School MathematicsPhysics SMILE Meeting
19972006 Academic Years
Mathematics: Geometry
13 October 1998: Porter Johnson [IIT]
He talked about a Geometry problem where twins drove 3 nails
at random into a
table and formed a triangle. If the nails happened to lie along a
straight line,
there would be no triangle [or else a very skinny triangle with no
interior!].
The probabilities of a new nail lying on the right or left of the line
are each 50%, and for a triangle there are three lines, and you
have to lie
"left" "left" and "left" as you circulate around
the boundary [counterclockwise]. By this simple argument, you might
expect the
probability of being inside as being 1/8 = 0.125. However, in
the historic words
of the 20th Century Physicist Wolfgang Pauli, "nicht Einfach; aber
Falsch"it isn't simple but it is wrong!
It is convenient to use vectors to decide whether a point is inside a triangle. Let us choose one vertex as special, call the vectors from it to the other vertices r_{1} and r_{2}, and call the vector from it to the point in question r. Then the criterion for being inside the triangle is
r = a r_{1} + b r_{2} .
where a and b are both positive, with a + b < 1.
To gain insight, he employed a pseudorandom number generator and applied the ***MonteCarlo technique. In a run of 10 million shots there was a computer crash because one of the sets of points was accidentally linear. He modified the program and then safely ran to 100,000,000 shots, without incident [it took several hours on the 80 MHz 486PC], obtaining 7,637,924 hits. The corresponding hit probability is thus .076379 ± .000100, which is consistent with the number 11/144 = .076388888888... , and lots of other more complicated numbers, as well. However, no clever soul has yet appeared with the solution. A copy of the FORTRAN program was passed out, along with a pseudorandom number generator touted by Liam Coffey [Physics Faculty Member and Computational Physics guru], just in case you also want to waste vast amounts of time on this problem or put your lazy computers to work For details see the website http://mypages.iit.edu/~johnsonpo/aapt0611.html.
Roy Coleman commented that some pseudorandom number generators are not very successful in producing "random" numbers, as you can see from plotting them in pairs. The "safest" random number sequences involve tabulations of radioactive decay times, but they are difficult to use. The two pseudorandom number generators used here gave similar results.
***Much of twentieth century science involves MonteCarlo simulations of actual experiments, theoretical models, hypothetical problems, and even "useless and insignificant puzzles". The inventor of the idea was the mathematician Stanislaw Ulam, who was involved in the development of the atomic and hydrogen bombs. Ulam became ill after the war, and spent a lot of time in hospital playing the card game Solitaire. He realized that it is more straightforward to play a few games to "estimate" the probability of winning the game, rather than to try to calculate the odds of winning directly. Ulam realized that many problems too difficult to be solved analytically could be resolved by this technique using fast computers. This story, along with many others, appears in his autobiography, Adventures of a Mathematician [ISBN 0520071549].
10 February 1998 James Chichester [Lincoln Way HS]
He brought in a product called Odd Balls, with ordering information
Orbix Corporation 6329 Mori Street McLean, VA 22101 (703) 3560695Here is a picture of them: Source: Shop; National Gallery of Art: [http://shop.nga.gov/]
Designer: Dr Ben F Sherman [a retired Nuclear Engineer]
Porter's comments: What happens when 2 cylinders intersect? They produce an intersection region that is rounded, but essentially becomes a square when looked at in a certain plane. The volume of that region may be computed as a single integral, and p does not appear in it.
10 October 2000 Fred Schaal (Lane Tech HS)
showed us "The Occurrence of Concurrence". He explained
that if 3
straight lines in a plane intersect in a single, common point, it is
called "concurrence."
With the aid of a meter stick, he constructed a large, nice looking
triangle on the white board. With a large compass having a marker
pen attached at its "chalk" end, Fred used the compass to
construct
the line which bisected one of the angles of the triangle. He
did
this in a contrasting color. Then he constructed the bisectors
of
the other two angles the triangle. If the board had not been so
slippery and the compass marker had made legible marks, the
bisectors of the three angles would have intersected at a single
point within the triangle: concurrence! Unfortunately, the
construction was not precise, and it didn't work out. But you made
your mark, Fred! Thanks for an interesting lesson.
27 February 2001 Fred Schaal (Lane Tech Park HS, Math)
made a presentation on finding the lateral surface area of pyramids
with a regular polygonal
base. He illustrated the point with a pyramid with a square base. The
lateral surface
consists of four equivalent triangles, of base b and height s.
The
area of each triangle is ½ b s, where the height s
is the slant
height of the pyramid, and the lateral surface area of the pyramid
is
06 November 2001: Fred Schaal (Lane Tech HS, Mathematics)
Congruences of Plane Figures
Fred passed out a statement of the following principles on
Euclidean
Isometries:
Fred sketched figures on the board, which provoked much discussion as to how to establish the third result. Porter Johnson suggested reflecting first about the perpendicular bisector to two identical vertices of congruence, so that the "old figure" and "new figure" will have that vertex (X=X') in common. Then, reflect about the the bisector to the angle YXY', where vertices Y and Y' are identical. The result will either be a congruence if the figures were not initially spacereflected, or else can easily be brought into congruence if the figures were initially space reflected.
We had the feeling that there may be many inequivalent reflections to bring about this congruence.
Relevant References:
Fred also called our attention to the fact that the planets Mercury and Venus are visible in the sky just before dawn.
23 April 2002: Walter McDonald (Bowen HS and Chicago Veterans
Administration)  Higher Dimensional Geometry
Walter described his efforts at tutoring students on
visualizations of
spaces of various dimensions. He presented the following table of
characteristic figures [called simplexes by mathematicians] in various
spatial
dimensions:
Number of Dimensions  Characteristic Figure 
0  Point 
1  Line 
2  Triangle 
3  Tetrahedron 
4  Figure with Tetrahedron Faces 
Walter asked what is the difference in 4 dimensional (Euclidean) space and what physicists call "spacetime"? Porter Johnson commented that the time interval Dt between two events and the spatial interval DL between the same two events can be regarded in terms of a unified "spacetime", and that because of the constancy of the speed of light, v, it is required to define the invariantintervalsquared between two events as [DL]^{2} [v Dt]^{2} . This space is called Minkowski space in Special Relativity, and which is a four dimensional (Euclidean) space, when expressed in terms of real space variables, with the time variable multiplied by i = Ö(1). Incidentally, the development of nonEuclidean geometry in mathematics and its applications in physics are an outgrowth of the examination of whether Euclid's fifth postulate [parallel lines never meet] is a consequence of the other four. NonEuclidean geometry is the mathematical framework for General Relativity, our (classical) theory of the Gravitational Field. For an interesting discussion of nonEuclidean geometry, see the St Andrews University web page: http://wwwgroups.dcs.stand.ac.uk/~history/HistTopics/NonEuclidean_geometry.html.:
Very good, Walter!
23 April 2002: Leticia Rodriguez (Ruben Salazar Bilingual Center)
 Tesselations; Mathematical
Applications; Scientific Method
Leticia passed around the following book, which contains various
tesselations
[which are regular periodic patterns, or periodic and quasiperiodic
"tilings"
of space]: Tesselations Teaching Masters; Dale Seymour
Publications, 1989; ISBN 0886614627. Leticia's primary
students color
these tesselations to make elaborate designs, and use them as a means
to learn
elementary ideas in mathematics [shapes, patterns, graphs, counting,
geometry,
etc] and elements of the scientific method [observing, estimating,
collecting
data, predicting, classifying, investigating, comparing, contrasting,
problem
solving, inferring, and drawing conclusions]. For further details
see her
website on the SMART home page: http://mypages.iit.edu/~smart/.
Porter Johnson mentioned that intricate, symmetric patterns are
employed
in many religions to convey a sense of spirituality in their
cathedrals,
chapels, churches, mosques, pagodas, shrines, and temples. One
beautiful example of these patterns is the
Baha'i Temple
in Wilmette Illinois; see the websites:
http://members.core.com/~fphayes/bahai.htm.
Leticia also pointed out that teachers are entitled to a 15% discount on educational and school supplies for classroom use (with proper identification) from April 15 to May 31, 2002 at Amazing Savings stores, located in Morton Grove (Harlem & Dempster) , Wheeling (Elmhurst & Dundee), Chicago (McCormick & Lincoln), Broadview (17th and Cermak), and Bloomingdale (Springbrook Shopping Center). Thanks, Leticia!
24 September 2002: Hoi Hyunh ( HS, Physics) Math Notation and
Visual Geometry
Hoi began by suggesting the mnemonic that D
is really a D, which stands for Difference.
[Do you remember "trouble that begins with T, which
rhymes with P, and stands for Pool".]
Hoi then showed us an ordinarylooking, boxlike configuration that she
had
constructed as follows:
Very nice work, Hoi!
08 October 2002: Maria Vinci [Evergreen
Park HS, Mathematics] Tiling and Tessellation
Maria passed around the book The Graphical Work by the
Dutch
graphical artist M C Escher (18981972) [Taschen GmBH
1989; ISBN 3828858641], which contained various patterns, tilings,
and
tessellations. [For more details on the life of Maurits
Cornelis Escher and his works see the website M C Escher by
Cordon Art BV
[http://www.mcescher.com/].
Maria
showed various tessellated figures that students made in her classes,
using
images of an elephant or a human face in making periodic tilings.
Although
Escher was primarily a graphical artist, he understood mathematics
rather well,
and his work has had a profound influence on mathematicians; for
details see the
website Mathematical Art of M C Escher:
http://www.mathacademy.com/pr/minitext/escher/index.asp.
PJ comment: The preparation of periodic microcrystalline samples
of protein
structures, such as DNA, is a crucial component in Xray
scattering to
determine the atomic structure of these materials. For example,
the double
helical structure was deduced by Watson and Crick upon the
basis of
analysis of Xray scattering of microcrystals of DNA.
Thus,
tessellations are also important throughout modern science. We
get the
picture, Maria!
08 October 2002: Walter McDonald [VA Hospital; Bowen
HS] Fractals: How Long is the Coastline of
Florida?
Walter explained that the length of certain intricate curves is
indeterminate, because the lengths depend upon the scale of
resolution.
For example, a tourist brochure may advertise that the coast of the
State of
Florida is 6000 km [4000 miles] in length, but even
this estimate is
imprecise, since it would be impossible to follow all the nooks and
crannies
that separate the land from the sea. As the scale of resolution
of the measurement decreases, the length increases. He showed
some "self similar curves",
for which the structure has the same form when viewed at various scales
 including
one on which we measured the following lengths with various
resolutions:
L: Length 
R: Resolution 

#1  3  2 
#2  7  1 
#3  20  0.5 
23 November 2004: Porter Johnson called attention to information on Fractals on the Yale University website [http://classes.yale.edu/fractals/]. Fractals occur frequently in everyday situations, such as Crumpled Paper [http://classes.yale.edu/fractals/FracAndDim/BoxDim/PowerLaw/CrumpledPaper.html] and Bean Bags [ http://classes.yale.edu/fractals/Labs/CrumpledPaperLab/BBProcedure.html]. Does this actually work??
05 November 2002: Fred Schaal [Lane Tech High School,
Mathematics] TI92 graphics calculator;
Railroads
Fred described an experiment that he had done, in which he could
lay out a
triangle on his calculator, and then have it determine all interior and
exterior
angles, as shown in the diagram below:
Note that, from the diagram, a + A = 180°; b + B = 180°; c + C = 180°. In addition, it is true that A + B + C = 360°, as can be seen by shrinking the triangle to a point, as shown. By adding the first three relations, and then subtracting the fourth, we obtain a + b + c = 180°, indicating that the sum of the angles in a triangle is equal to 180°. In addition, by drawing a line parallel to the base line at the top vertex, we obtain the figure:
Evidently, a + b = C, indicating that the exterior angle C is the sum of the two interior angles a and b. Similarly, we can show that b +c = A and c + a = B.
Fred noticed in his extensive travels by rail this summer that the tracks were noticeably bumpy  presumably because freight trains also travel on the same tracks. In addition, he observed a lateral [transverse] vibration of the train cars, and wondered why. The consensus of the group was that the slight "play" in the "knuckle couplers" that attach one car to another permits such transverse vibration, which can be amplified as the train travels along the track. Very interesting, Fred!
09 September 2003: Bill Colson [Morgan Park HS,
mathematics]
Flashy DooDah's
Bill showed us Tetratops, a set of flashing and colorful
polyhedron tops, with trading
cards. Bill bought them at Walgreens for about $5.00.Tetratops
http://www.tetratops.com/
are manufactured by Duncan Toys Company
http://www.theyoyostore.com/duntettop.html,
the famous YOYO manufacturer. This excerpt is taken
from the website
above:
"Tops have fascinated humans since they were first discovered. No one knows when or where they were first spun, but they have been found in nearly every culture on Earth. The variety of designs is endless, however there is one remarkable similarity to all of them. Traditional tops all have only ONE axis of spin! Unlike traditional tops, Duncan's new TETRATOPS™ all have multiple axes of spin!"These spinning tops include the five Platonic solids [ http://www.math.utah.edu/~alfeld/math/polyhedra/polyhedra.html], the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. Nifty, Bill!
27 January 2004: Fred Schaal [Lane Tech HS,
mathematics] Innies and Outies
Fred
freehandedly drew
a circle on the board, marked five points roughly equidistant on its
circumference, and connected
them to form a fivesided polygon, a pentagon or 5gon. He
then measured the
interior and exterior vertex angles with a large wooden
protractor, obtaining the following results:
Vertex  Interior angle  Exterior Angle 
A  104°  76° 
B  98°  82° 
C  116°  64° 
D  115°  65° 
E  115°  75° 
TOTAL  548°  352° 
Vertex  Interior angle  Exterior Angle 
A  104°  76° 
B  55°  135° 
C'  210°  150° 
D  70°  110° 
E  115°  75° 
TOTAL  554°  546° 
The expected relation no longer works, because the sum of the angles for vertex C' is now close to 540°, rather than 360°. If we had taken the exterior angle for C' to be 30°, we would have obtained 346°, which is more consistent with expectations.
Finally, Fred asked whether the rubber wheels on the Montreal and Paris METRO rail systems represent a practical means of noise reduction, or in fact are they too inefficient because of increased friction. Does anybody know the answer? [It was mentioned that rubber wheels were once used in Chicago years ago.] For additional information see the website Rubbertired Metro: http://en.wikipedia.org/wiki/Rubbertired_metro. Interesting, Fred!
09 March 2004: Fred Schaal [Lane
Tech, HS
Mathematics]
When is there more crust than "pie"?
Fred carefully drew a full circle on the blackboard, and marked
the
center of the circle, as well as two points on its circumference. He
drew radial lines
(r) from the center
of the circle to these two points A, B  enclosing a sector,
or a pieshaped
slice of circle. Then he drew a straight line (chord) connecting A and
B.
The area between the chord and the arc of the circle represents the crust,
or
outer region of the slice Here is a rough diagram.
The slice angle between the two radii is q. In terms of area, how much crust is there, and how much pie, for a given radius r and slice angle q? Fred explained that the slice area is A_{slice }=½ r^{2} q, with the slice angle q measured in radians, since the slice area is proportional to q, and for q = 2p (full circle) we get an area of pr^{2}. Fred then measured the radius r of his circle and the slice angle q, and calculated the slice area A_{slice}.
How do you determine the area A_{tri} of the central "edible" portion of the pie  the slice minus the crust? It is the triangular region of base b and altitude h; so that A_{tri} = ½ b h. Fred measured the chord length b and triangle altitude h directly on the diagram, and calculated A_{tri}. Alternatively, we can compute them using h = r sin q/2 and b = 2 r cos q/2, so that A_{tri} = r^{2} sin q/2 cos q/2 = ½ r^{2} sin q. Therefore, the fraction of the slice area that is crust area is given by
This result, depends only on the slice angle q, and is independent of the pie radius. Why? This table gives its value for various ways of slicing the pie into equal pieces:.
Number of Slices  q: degrees  q: radians  sin q / q  A_{crust} / A_{slice } 
12  30°  0.523  0.956  0.044 
8  45°  0.785  0.901  0.099 
6  60°  1.047  0.827  0.173 
4  90°  1.571  0.637  0.373 
3  120°  2.094  0.414  0.586 
2  180°  3.142  0.000  1.000 
Thanks for sharing the pie for us  figuratively speaking! Good, Fred!
23 March 2004: Fred Schaal [Lane
Tech, HS
Mathematics]
Of All the Crust!
Fred continued his discussion of slicing pie, which was begun at
the
last SMILE meeting.
He had shown that the ratio,
R, of crust area to total area is given by
R = 1  (sin q) / q. (q
in
radians)
He programmed this formula on his TI83 graphing calculator, and projected on the screen at the front of the room the graph of R versus q for various ranges. In particular, he noted that for q greater than 180° or p radians R became greater than 1. Here is a summary of results:
q: degrees  q: radians  R = 1  (sin q) / q 
60°  1.047  0.173 
120°  2.094  0.586 
180°  3.142  1.000 
270°  4.712  1.212 
360°  6.284  1.000 
450°  7.854  0.873 
630°  10.995  1.090 
810°  14.137  0.927 
It seems that the ratio R is approaching 1 at large slice angle q. Do we get more pie by circling many times?!
You can have your ppie and eat it, too! Good, Fred!
28 October 2004: Benson Uwumarogie [Dunbar HS,
mathematics]
Special Parts of a Triangle
Benson used patty papers to show us how to construct the
altitudes, median lines, angle bisectors, and perpendicular bisectors
of a triangle by drawing the triangle on the paper, and then
folding the paper appropriately. With these patty papers
(presumably named after the paper sheets placed between hamburger
patties by butchers), his students constructed these features of
triangles for themselves. Unfortunately, the constructions did
not show up well on the overhead projector, because the paper was not
transparent. By showing his work at various states, Benson
illustrated these geometrical concepts. Bill Colson
commented that the book Patty Paper Geometry by Michael
Serra has been published by Key Curriculum Press.
For details see their website http://www.keypress.com/.
A useful approach, Benson!
07 December 2004: F J Schaal [Lane Tech,
mathematics]
Spheres to Cubes
Fred reminded us that a cube of side b_{0}
has 6 square faces, a total surface area S_{0} = 6 b_{0}^{2},
and total volume V_{0} = b_{0}^{3}.
Let us enlarge the cube uniformly until its volume is doubled: V_{1}
= 2V_{0 }= b_{1}^{3}. The length of a side
is therefore equal to b_{1}= 2^{1/3 }b_{0}
~ 1.27 ^{ }b_{0}. Correspondingly, let
us double the surface area S_{2} = 2 S_{0} = 6 b_{2}^{2
}. We obtain b_{2}= 2^{1/2 }b_{0}
~ 1.41 ^{ }b_{0 }and V_{2 }= b_{2}^{3}
= 2.82 V_{0}. These results are the same as those
obtained by Fred at the last MP SMILE meeting for a solid sphere. Bill Shanks pointed out that, if an
inverted hollow cone filled halfway to the maximum height (with snow or
ice cream  pick your favorite), the volume of edible material is only
1/8 of that when it is filled to the top. What a
ripoff! Thanks for the ideas, Fred and Bill!
07 December 2004: Bill Shanks [Joliet Central, happily
retired] Various
Topics
Bill first held in his hand a hexagonal socket used with a
3/8" (8 mm) square drive to fit a 14 mm spark plug. He struck
the hexagonal end smartly against the palm of his hand several
times. Each time we heard a short "pop" sound with a
certain pitch. Bill then asked us what pitch of sound would
occur when he hit his palm with the other end. Our survey
consisted of votes in all three categories  lower pitch, same
pitch, higher pitch. Then he did it and we heard a "pop"
sound of obviously higher pitch. Bill then referred to the wine
jug instrument (Helmholtz Resonator) presentation made at the 25
February 2003 MP SMILE meeting by Don Kanner.
At the last MP SMILE meeting Bill measured the length of a little wooden cube (a giveaway), and obtained 1.27 cm (corresponding to a halfinch). He calculated the volume of the cube, obtaining (1.27 cm)^{3}, or a little more than 2 cm^{3}. We earlier had guessed that the cube was 1 cm on a side, with a volume of 1 cm^{3}. How can the volume of the cube more than double when its sides change only by a "small amount"? To explain this, Bill put x = 1.00 and Dx = 0.27 into the expansion formula for (x + Dx)^{3}:
Neat! Thanks, Bill!
22 February 2005: Fred Schaal [Lane Tech HS,
mathematics]
Pump Up the Volume
Fred brought out a metallic box (appropriately decorated for
holiday gifts of candy or
sweets), along with a ruler. Fred recruited an
assistant, Charlotte
WoodHarrington, to make measurements of its outside length L,
outside width W, and height H. Charlotte
obtained the
following values: L = 14.25 cm, W = 14.15 cm, and H = 4.50 cm.
We then treated
the box as a Rectangular Parallelopiped, and computed its
volume:
Porter Johnson mentioned that glasses in restaurants, cafes, and bars in Europe are commonly marked to indicate the volume of a given level of fluid. These markings, which were a legal requirement in Germany, might be one of the following:
0,3 L = 3 dL  3 deciLiters, or 300 cm^{3} 
0,5 L = 5 dL  5 deciLiters, or 500 cm^{3} 
1,0 L  1 Liter, or 1000 cm^{3} 
Watch out for those one liter beer steins! Thanks, Fred!
12 April 2005: Benson Uwumarogi [Dunbar HS,
mathematics]
Circle, Radius p
Benson used a cloth measuring tape to determine the circumference
and diameter of various round objects, and then calculated the ratio C
/ D, with these results:
Object  C: circumference  D: Diameter  Ratio 
metallic cookie tin  79 cm  25 cm  3.16 
plastic lid  32 cm  10 cm  3.20 
01 November 2005: Benson Uwumarogie (Dunbar HS,
mathematics) Patterns
Benson
presented an interesting geometric puzzle. He drew a series of
five circles marking either 2, 3, 4, 5, or 6 points on their
circumferences. He then connected the points in all possible ways by
straight lines. For example, for the circle with two points, only a
single straight line across the interior of the circle can be drawn;
for the circle with three points, a triangle can be drawn in the
interior of the circle, etc. Each circle was thus divided into a number
of
nonoverlapping regions within its interior. For example, for the
"twopoint" circle there are two regions; for the three point circle,
the are four regions; for the four point circle, there are eight
regions; for the five point circle there are 16 regions; for the six
point circle there are 32 regions. This pattern suggests the general
formula for n points, corresponding
to 2^{n1} regions. The general formula
works! How come?
What a neat way to discover the relationship between physical
patterns and mathematical formulas that describe the patterns.
Good work. Thanks,
Benson.
29 November 2005: Nneka Anigbogu (Jones College
Prep)
Random rectangles
Nneka handed out a sheet with 100 rectangles of various dimensions
displayed.
Each rectangle was composed one or more identical squares. Our
activity
was to obtain an estimate of the average number of squares. This
is an activity to illustrate use of statistics. Nneka
asked us to briefly look at the 100 rectangles and to shout out
estimates of the average area of all 100 in "standard units" (easy
because each had an area that was an integral number of the standard
squares mentioned above. Estimated averages were 8, 9, 11, 17, 50. Then
Nneka had us choose any 5 rectangles in a "continuous run"
(e.g., rectangles 59) and estimate the average of those 5. We got
averages of 112.8, for 10 samples of 5 rectangles, with an overall
average of 8.82. This completed the "subjective estimate" of the mean.
Nneke also handed out a table of 450 five digit random numbers. We used the table to pick 5 numbers and use any consecutive two digits from that five digit number, match these with the numbers of the corresponding rectangles, and get another group of 5rectangle averages. Ten averages ranged from 4.0 8.6, with an overall average of 5.81, the "random estimate" of the mean.
It turns out that the actual average of the 100 rectangles is 7.42. Surprisingly, the subjective estimate was a bit closer to the actual than the random estimate. Nneka noted that in her class the random method gave a number almost identical to the actual average. How come ours was so far off? Great stuff! Thanks, Nneka.
07 February 2006:
Fred Schaal (Lane Tech HS)
Constructing
Points on an Ellipse
Fred
showed us how to make
points on an ellipse using the blackboard and only a
(chalk) compass and a straight edge. An ellipse is defined as
a geometric shape with two focal points (foci), so that the sum of the
distances from
any point on the ellipse to each is always the same.
Fred drew the two foci (F) on the board and then a third
point (P), as shown.
P . / \ x / \ y / \ * * F F
07 March 2006: Fred Schaal (Lane Tech HS,
math)
Parabolic Points
In an extension of his presentation at the last meeting, Fred
used a
similar procedure to trace out the points on a parabola using
only his (chalk) compass, a meter stick and the blackboard. He chose a
focal
point (focus) at random above a horizontal line (directrix).
He used the compass to draw a portion of a circular arc with an
arbitrary radius, with the center at the focus . Two arcs are then made
with the compass held at
the same radius, with their centers on the line. A tangent to these two
arcs intersects
the first arc at two points, which lie on the parabola. The
process is repeated using the same focal point but different
radii, generating points to trace out a
parabola. For additional information see the interactive webpage The
Parabola by Alex Bogomolny: http://www.cuttheknot.org/ctk/Parabola.shtml,
Neat, Fred!
07 March 2006: Porter Johnson (IIT,
Physics) SangakuFollowup
Porter continued the discussion of the “Circle Inscribing Sangaku”,
which was introduced at the last class by Walter McDonald. This
problem is
discussed on the Mathworld Website on the web page
http://mathworld.wolfram.com/CircleInscribing.html.
However, that discussion is incomplete, in that it does not prove that
the
inscribed circle centered at O_{3} is tangent to the
isosceles
triangle ACB.
(1 + r) y = r Ö[2 (1  r)] .
Let the symbol j represent the angle ACD. Because the point C lies on the largest circle, its distance to the center O is 1/2. Furthermore, the right triangle ADC, has these side lengths:
Porter then told us about Morley’s Theorem. Start with any triangle and trisect all three angles. Pairs of the trisecting lines from adjacent angles will intersect to make three points inside the original triangle. Connection of these three points will always produce an equilateral triangle!! Fred then illustrated this by laying out a carefully drawn figure on the board. For more details see the website http://www.cuttheknot.org/Curriculum/Geometry/Morley.shtml, which contains an adjustable triangle showing the result. See also http://www.jimloy.com/geometry/morley.htm, which contains the following comment:
"One of the interesting side results of some of the proofs is that the side of the equilateral triangle is equal to 8R sin(A/3) sin(B/3) sin(C/3), where A, B, and C are the angles of the larger triangle, and R is the radius of the circumcircle."Fascinating, Porter.
04 April 2006:
Karlene Joseph (Lane
Tech)
Go Figure
Karlene brought a
children’s book called Go Figure by Johnny Ball:
.
It is a
book about numbers. One section asks the question, “What if
we had no numbers?” We couldn’t report sports scores, TV
listings, and a million other things! There is also a
section on the pyramids (Karlene loves ancient cultures!).
The dimensions of the pyramids are arranged so that several
interesting relationships occur. Another interesting item
is that about 100 years ago the State of Indiana tried to
pass a law decreeing that p would be
exactly
3.2! For details see The Indiana Pi Bill: http://www.agecon.purdue.edu/crd/Localgov/Second
Level pages/indiana_pi_bill.htm.
Here’s another one about p (we thank Archimedes for this one). A circle inscribed inside a square of side 1 has a diameter of 1 and a circumference of p. The perimeter of the square is 4, which means p must be less than 4. Repeat with a hexagon inscribed inside the circle (hexagon perimeter 3.0); now p must be greater than 3. Archimedes iterated this up to a 96sided polygon and found that 223/71 < p < 220/70 (or 3.14084507 < p < 3.142857143). For details see Archimedes Traps Pi: http://physics.weber.edu/carroll/archimedes/pi.htm.
Then Karlene gave us each a triangle cut from a piece of paper (random shapes and sizes). We then talked about the things we know about triangles. One is that the total of the three angles is 180 degrees (the same as a straight line). Karlene then had us tear off the three corners and use the little angles to tuck into one another and see if we got a straight line and we surely all did! One can do the same with a planar quadrilateral to obtain 360 degrees. Neat stuff! Thanks, Karlene.