Fred Schaal (Lane Tech HS, Mathematics) -- Hexcode Digital
Numbers
Fred presented an extension of the the lesson on Digital
Numbers by Monica
Seelman at the 23 April 2002 (last) SMILE
meeting. Monica
converted multiplication tables [from 1 ´
1 to
9 ´ 9] into tables of Digital
Numbers,
where digital decimal numbers are obtained by adding the digits of a
number sequentially until a number between 1 and 9 is obtained [for
example; 78
® 7 + 8 = 15 ®
1+5 = 6]. With unerring Mathematical Logic,
Fred decided that the same procedure would work with Hexcode
Numbers [that
is, hexadecimal numbers -- base 16]. First he reminded
us how to count in
base 10 as well as in base 16:
Base 10: | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | ... |
Base 16: | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | B | C | D | E | F | 10 | 11 | 12 | 13 | ... |
The new symbols A, B, C, D, E, and F represent the numbers between 9 and 16, whereas in base 16 the number 16 = 16^{1}+ 0 is written as [10]_{hex}, etc. He next made a multiplication table for base 16, a portion of which is given here:
one-sies | two-sies | three-sies | four-sies | five-sies | six-sies | seven-sies | eight-sies | nine-sies |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
2 | 4 | 6 | 8 | A | C | E | 10 | 12 |
3 | 6 | 9 | C | F | 12 | 15 | 18 | 1B |
4 | 8 | C | 10 | 14 | 18 | 1C | 20 | 24 |
5 | A | F | 14 | 19 | 1E | 23 | 28 | 2D |
6 | C | 12 | 18 | 1E | 24 | 2A | 30 | 36 |
7 | E | 15 | 1C | 23 | 2A | 31 | 38 | 3F |
8 | 10 | 18 | 20 | 28 | 30 | 38 | 40 | 48 |
9 | 12 | 1B | 24 | 2D | 36 | 3F | 48 | 51 |
A | 14 | 1E | 28 | 32 | 3C | 46 | 50 | 5A |
B | 16 | 21 | 2C | 37 | 42 | 4D | 58 | 63 |
C | 18 | 24 | 30 | 3C | 48 | 54 | 60 | 6C |
D | 1A | 27 | 34 | 41 | 4E | 5B | 68 | 75 |
E | 1C | 2A | 38 | 46 | 54 | 62 | 70 | 7E |
F | 1E | 2D | 3C | 4B | 5A | 69 | 78 | 87 |
Now, the plan is to convert these numbers to Hexa-decimal Digital Numbers [for example; 7E_{hex} ® 7_{hex} + E_{hex} = 15_{hex} ® 1_{hex} + 5_{hex} = 6_{hex}]. When we do so, we get the following table:
one-sies | two-sies | three-sies | four-sies | five-sies | six-sies | seven-sies | eight-sies | nine-sies |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
2 | 4 | 6 | 8 | A | C | E | 1 | 3 |
3 | 6 | 9 | C | F | 3 | 6 | 9 | C |
4 | 8 | C | 1 | 5 | 9 | D | 2 | 6 |
5 | A | F | 5 | A | F | 5 | A | F |
6 | C | 3 | 9 | F | 6 | C | 3 | 9 |
7 | E | 6 | D | 5 | C | 4 | B | 3 |
8 | 1 | 9 | 2 | A | 3 | B | 4 | C |
9 | 3 | C | 6 | F | 9 | 3 | C | 6 |
A | 5 | F | A | 5 | F | A | 5 | F |
B | 7 | 3 | E | A | 6 | 2 | D | 9 |
C | 9 | 6 | 3 | F | C | 9 | 6 | 3 |
D | B | 9 | 7 | 5 | 3 | 1 | E | C |
E | D | C | B | A | 9 | 8 | 7 | 6 |
F | F | F | F | F | F | F | F | F |
Now, divide a circle into 15 equal arcs, number the vertices using base 16 symbols [1 ... E], and connect the vertices in the order given in one of the columns. You will produce a polygonal figure that eventually closes, as before. The structure of the figures [quite different in appearance from last time in the base 10 case] is summarized here:
Sequence Name |
Sequence |
Figure Name |
One-sies: | 1 2 3 4 5 6 7 8 9 A B C D E F | Regular 15-agon |
Two-sies | 2 4 6 8 A C E 1 3 5 7 9 B D F | 2-star 15-agon |
Three-sies | 3 6 9 C F 3 6 9 C F 3 6 9 C F | Pentagon |
Four-sies | 4 8 C 1 5 9 D 2 6 A E 3 7 B F | 4-star 15-agon |
Five-sies | 5 A F 5 A F 5 A F 5 A F 5 A F | Triangle |
Six-sies | 6 C 3 9 F 6 C 3 9 F 6 C 3 9 F | 2-star Pentagon |
Seven-sies | 7 E 6 D 5 C 4 B 3 A 2 9 1 8 F | 7-star 15-agon |
Eight-sies | 8 1 9 2 A 3 B 4 C 5 D 6 E 7 F | 7-star 15-agon |
Nine-sies | 9 3 C 6 F 9 3 C 6 F 9 3 C 6 F | 2-star Pentagon |
Ten-sies | A 5 F A 5 F A 5 F A 5 F A 5 F | Triangle |
Eleven-sies | B 7 3 E A 6 2 D 9 5 1 C 8 4 F | 4-star 15-agon |
Twelve-sies | C 9 6 3 F C 9 6 3 F C 9 6 3 F | Pentagon |
Thirteen-sies | D B 9 7 5 3 1 E C A 8 6 4 2 F | 2-star 15-agon |
Fourteen-sies | E D C B A 9 8 7 65 4 3 2 1 F | Regular 15-agon |
Fifteen-sies | F F F F F F F F F F F F F F F | Point |
Interestingly, you get a 15-sided polygonal figure only
for rows 1, 2, 4, 7, 8,
11, 13, and 14. It is significant that these numbers, and
only these
numbers, have no common factors with the number 15.
The regular
15-agon is called a pentadecagon; for
additional information see the websites
http://mathworld.wolfram.com/Pentadecagon.html
and http://www.kole-slaw.com/gauss/pentadecagon.html.
The latter site contains the following note of historical interest:
"In Book IV, Euclid’s main achievement was to construct a regular pentagon in a circle. By combining this construction with that of an equilateral triangle, he was then able to construct a regular 15-sided polygon. In the 1790s, Gauss extended this idea to determine exactly which regular polygons can be constructed – they are ones based on the numbers 3, 5, 17, 257 and 65537 – the so-called Fermat primes."
Beautiful work, Fred!
F Lee Slick (Morgan Park HS, Physics) -- Anti-gravity Fluid No
41,086
Lee brought in a friction-reduction additive that he had obtained
from Edmund Scientific
[http://www.edsci.com/]
some time ago. Unfortunately, the product no longer appears in
their
catalog, and seems to be unavailable. This additive may be identical or
similar
to the substance Polyethylene
oxide [Polyox], which is widely used in commercial
applications
such as water jets and foams. [http://www.dow.com/dowwolff/en/industrial_solutions/]
for friction reduction in fluids. Here is an excerpt from the second
reference:
"Foam systems could contain friction reducing agents, like polyethylene oxide, which could render roadway slopes impassable – inclined ramps to highways, for instance."
Our additive is to be mixed thoroughly and completely with water to form a clear liquid. The long polymeric molecules serve reduce friction, to the extent that, under proper conditions, one could pour the liquid from one cup to another, and set up a "pipeless siphon", with the fluid continuing to creep up the side of the top cup, even after the cup is gradually tilted back so that the fluid can no longer simply spill out. This matter deserves further exploration. Come next fall ... !
Believe it or not; this is amazing! Thanks Lee!
Don Kanner (Lane Tech HS, Physics) Liquid Nitrogen Bath
Don described an experiment in which he poured Liquid Nitrogen into
a bin to
form a layer about 2 cm deep, and then placed a beaker
containing about 20 ml of
water slowly into the bin. When the water was solidly frozen, he
and the
class observed that the water had frozen into ice that had formed a
peak at the center of the
beaker. How come? The effect apparently occurs
because water expands when
freezing, since ice is less dense than water. The ice forms
first near the outside of the glass, causing the liquid inside to be
pushed
upwards. One can also see "central peaks" in ice cubes formed in
refrigerators, for the same reason. We then discussed the
inherent dangers
in handling Liquid Nitrogen LIN (or LN). In
addition, taking precautions
because of its low temperature [71 K], one must be careful NEVER
NEVER
to close or clog the vents on a LIN container. The
liquid
continually vaporizes in a room temperature environment, and in the
process of
evaporation its density is reduced by a factor of about 1000.
Unless the vapor is allowed to exit the LIN container, a
dangerously high
pressure [up to about 1000 atmospheres] will arise fairly
quickly. Good,
Don!
John Bozovsky (Bowen HS, Physics) Laser Light Show
John put on a Laser Light Show using a laser pointer [a
HeNe Laser]
and mirrors, which were bonded obliquely to shafts that are rotated
independently at
continuously adjustable rates by motors. The laser beam strikes
the surface
of the first rotating mirror, and then it is reflected onto the
second
rotating mirror -- and then perhaps to a third rotating mirror --
before being
reflected into the room. Because of the independent rotations of
the
reflecting mirrors, the laser beam traces an interesting pattern on the
wall or
ceiling, which is reminiscent of Lissajous Figures on
oscilloscopes
[http://www.jmargolin.com/mtest/LJfigs.htm]
or mechanically, as done originally by Jules Antoine Lissajous
[
http://www-groups.dcs.st-and.ac.uk/history/Mathematicians/Lissajous.html].
By varying the frequencies of rotation, we can alter the pattern of the
beam,
and set up closed figures in certain cases. At last, we are
stirred by the
light! Thank you, John.
Hoi Huynh (Clemente HS) Division with Mirrors [*** but
without
smoke!]
Hoi taped together two big mirror tiles [30 cm ´ 30 cm] along the edges, and stood
them on the table. The tape acted
like a hinge, so that the angle between the mirror faces could be
changed as one
wished. By placing
a ubiquitous cola can between the mirrors and varying the opening angle
between
the mirrors, we could produce various patterns of cola can
reflections. In
particular, we noticed that at certain angles the reflected images of
the cola
can formed regular patterns. Specifically, when the angle
between
the mirrors was 360º/N, these images lay at the vertices
of a regular
polygon with N sides. The cases N = 3, 4, 5, 6, and 7
are
shown below:
Hoi described what happened as the opening angle q = 360º/N becomes small, as N becomes large. In the limit as the opening angle q goes to zero, the number of images, N, becomes infinite. It must be so! Incidentally, the choice of 360º for one revolution was made by the Babylonians; see http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Babylonian_mathematics.html. The website http://mathforum.org/library/drmath/sets/select/dm_circle360.html contains several explainations while another contains the following information:
"Why, when, where and by whom was it decreed that there should be 360 degrees of arc in a complete circle?Hoi separated the mirrors, and placed them parallel to one another with the reflecting sides facing one another --- allegedly like in the hairdresser's shop! We could see an endless progression of images of an object placed between the mirrors. Very impressive, Hoi! At last, we can visualize/see ¥ -- the infinite!It was during the reign of Nebuchadnezzar (605-562 BC) in the Chaldean dynasty in Babylon that the circle was divided into 360 degrees. This was because the Chaldeans had calculated by observation and inference that a complete year numbered 360 days. The basis of angular measure for the mathematicians of Babylon was the angle at each of the corners of an equilateral triangle. They did not have decimal fractions and thus found it difficult to deal with remainders when doing division. So they agreed to divide the corner of an equilateral triangle into 60 degrees, because 60 could be divided by 2, 3, 4, 5 and 6 without remainder. Each degree was divided into 60 minutes and each minute into 60 seconds. If the angles at the corners of six equilateral triangles are placed together they form the angle formed by a complete circle. It is for this reason that there are six times 60 degrees of arc in the complete circle."
We then discussed the operation of reflecting mirrors for drivers of motor vehicles, which usually have an "anti-glare" setting, in addition to the regular one. These mirrors are actually wedge-shaped, and most of the reflection occurs from the silvered "back surface" of the mirror. However, a small amount of reflection occurs from the front surface, which is slightly tilted relative to the back surface. When a bright light falls on the mirror, the driver may switch to the "anti-glare" front surface reflection, in order to avoid being distracted by the light. Of course, the intensity of all images is reduced in this process, and the driver should switch back to the regular setting when the bright light is removed, to maintain visibility.
Arlyn Van Ek (Illiana Christian HS, Physics) Illustration of
Polarization
Arlyn brought in some polarizing filters which he obtained as part
of a
Microwave Optics Kit, for doing microwave experiments such as the one
described in the website http://www.scienceinschool.org/2009/issue12/microwaves.
He remarked that he seldom employed the Microwave Optics Kit, but very
often
used these polarizing filters [metal sheets about 20 cm ´
20 cm, with parallel
slots in them], as shown:
_______________________When microwave radiation falls on this sheet, only the component that has the electric field parallel to the slits will be transmitted through the filter. When two filters are in the path with their slits perpendicular to one another, no radiation can get through. Arlyn pointed out that, if you just hold the filters up and look through them, you see a good image when they are parallel, and essentially nothing when they are perpendicular. Also, by using the scroll saw to drive the transverse oscillations of a rope, as described at the 23 April 2002 (last) SMILE meeting, these sheets serve the same purpose.
| _________________ |
| '-----------------' |
| _________________ |
| '-----------------' | Polarizing Microwave Filter
| _________________ |
| '-----------------' |
| _________________ |
| '-----------------' |
|_______________________|
We began to plan experiments with microwave ovens, as described in the article in the April 2002 issue of The Physics Teacher, http://www.aapt.org/tpt/: by Michael Gallis, entitled Automating Microwave Optics Experiments [p 217ff].
Arlyn began discussing why AM radio waves bounce off the
earth's ionosphere and can be
detected at great distances from the
source, whereas FM radio waves [and TV signals] are
generally limited by "line of site" from the antenna.
Porter Johnson
said that the free electrons in the ionosphere, which constitute a
plasma.,
oscillate naturally when disturbed with a characteristic frequency
known as the
plasma frequency w_{P}.
In terms of
n, the number of electrons per unit
volume, the plasma frequency is determined by the electron mass m,
the magnitude of
its charge e, and the dielectric constant
e_{0} as
The same physical principles can be used to explain why a conducting metallic sheet reflects light in, say, the silvered surface of a mirror. The density of the plasma of free conduction electrons in the metal is about 10^{23} electrons/cm^{3}, so that the plasma frequency is about 10^{16} Hz. Consequently, the frequency of visible light lies below the plasma frequency, and it is reflected by the metallic surface. Intriguing Physics, Arlyn!
See you in September!
Notes taken by Porter Johnson