A course
“ Mathematics and Technology ”
Mathematics is a living science, everywhere
present in science and technology
The teacher should have experienced how science
develops in the real world. He(she) can then show
it.
The scientist :
•He(she) asks questions;
•He(she) dares to say: “I do not know.”
•He(she) has an open and critical mind.
Helping discovering the power of the
mathematical method:
•Modelling
•Problem solving
•Mathematical sophistication
•Use of computer
Some messages through the course :
•Mathematics are useful and constantly developing
around us.
•The questions “Why?” and “What is it useful for?”
should be encouraged and deserve an answer.
•The beauty of mathematical constructions
•Mathematics are much more than numbers
Most subjects treated are too advanced so that
preservice teachers following the course can hope
bring them directly to classroom. The purpose of
the course is rather to teach them how to prepare
such kind of material.
Mathematics and technology
• New course since winter 2001. Most students
are preservice secondary school teachers.
• Purpose: Discover mathematics present in
everyday technologies
Joint creation with my colleague:
Yvan Saint-Aubin
• Yvan is a physicist and I am a mathematician.
• We knew very little of the material of the course when it was
created. We now have enough material for at least two
courses.
• The game is to take some technologies into pieces, to
dismantle them in order to discover and explain the
mathematics that make them work.
• We play the game to prepare the course. We try to
teach the students to do the same.
Description
Two formats:
• Flashs-science (1 hour)
• More elaborate subject
- 1 week = 3 hours plus 2 hours of exercices
- or two weeks on one subject
• The lectures are of two different types
– elementary parts (subject matter for exams)
– conference type lectures on advanced parts
Evaluation
• Two exams with open book and personal notes. Non
cumulative contents
• A session project on an application of mathematics
(by teams of two, if possible; by larger teams (4-6)
otherwise )
• A half-hour oral presentation of the project
The exercices
•We have spent a lot of time writing
interesting exercices that make the
students practice modelling and review
their elementary maths
•Finding appropriate exam questions is not
a trivial task
•A few examples below
A book for the course
Mathematics and technology (september 2008)
C. Rousseau and Y. Saint-Aubin, Springer-Verlag
Mathématiques et technologie (october 2008)
C. Rousseau and Y. Saint-Aubin, Springer-Verlag
Flashs-science
• Antennas and radars are parabolic. Why? (Geometric
definition of conics)
• Computer vision: calculating the position of one
object from its position on two photos (The
parametric equations of lines in 3-dimensional space)
• Covering a territory with antennas for a mobile phone
network (Euclidean geometry)
The corresponding exercice at the exam
We fill a large planar region with nonoverlapping disks of radius r. We use two methods:
in the first method we place the centers of the disks on a square network and in the second
method we place them on a regular triangular network of equilateral triangles.
(b)
(a)
Which method gives the denser filling? Suggestion: compute the proportion
of each square covered by portions of disks in case (a) and the proportion of
each triangle covered by portions of disks in case (b).
• Physics : unifying the laws of reflection and refraction. The
laws of nature follow optimization principles. Applications :
short waves, optical fiber
• A short look in the architecture of computers describing logic
circuits
• The regular tiling of the sphere with twelve spherical
pentagons
• Voronoï diagrams (Euclidean geometry)
More elaborate subjects
• Positioning in space : GPS, GPS signal, cartography,
localization of thunderstorms (Geometric locus,
differential geometry, theory of finite fields)
• How is a musical CD engraved: why 44100 numbers
per second? (Elementary Fourier analysis)
• Public key cryptography (Elementary number theory:
congruences)
• Error correcting codes : Hamming codes and ReedSolomon codes (Linear algebra, finite fields)
• Image compression: iterated function systems (Affine
transformations of the plane)
• The JPEG format (.jpg) (Elementary Fourier analysis)
• Robots (Rotations in 3-dimensional space, change of
reference frame)
• Friezes and tilings (Symmetries – linear algebra)
• Google and the Pagerank algorithm (Markov chains –
linear algebra)
• The skeleton and the gamma-knife surgery (Geometry)
• Turing machines and DNA computers (The hierarchy
of functions starting from the basic ones)
• Random number generators (Finite fields)
• Calculus of variations (Multi-variable calculus)
• Sparing and borrowing money
Some students projects
(a list on my webpage)
• Rollercoasters
• The search of boundaries in a photo
• Morphing IMAGES
• Text compression
• Mathematical morphology in treating images
• Benford law of significant digit
• How to complete the hole in Escher’s painting
“ Print Gallery”
• Polyhedra and fullerenes
Carbone 60
Truncated icosahedron
• Voronoï diagrams and Delaunay triangulation in
image analysis
• Sphere packings and honeycombs
• The best skateboard track
• Other cryptographic methods
• Reed-Müller error-correcting codes
• Knots and the action of enzymes on DNA
• Digital fingerprintING
• Image compression : from fractals to practical
applications
• Penrose tilings
• The seasons, the locus of the sunrise and sunset at a
given date, the length of day at a given date, …
• Calculation of astronomic distances, from the ancient
Greeks to now
• The eclipses
• The shape of sand dunes
• Phyllotaxy (how to explain spirals in sunflowers, etc.)
• Population growth under constraints
• Mathematical modelling of epidemics
• Chaos
A remarkable property of the parabola
All rays parallel to the axis are reflected to a
single point.
Applications: the shape of many objects
among which
•Telescope mirrors
• Solar furnaces
• Parabolic antennas
• Radars
The corresponding property of the ellipse
Any ray issued from one focus is reflected to the other focus.
Applications: mirrors, accoustic phenomena
• Elliptic mirrors for instance behind the lamp of a
cinema projector
• Accoustic phenomena: for instance Paris’ subway
Google and the PageRank algorithm
A search engine that does
not order entries properly
is useless.

0


1


0

0


 0


0

1

3
1
3

1
0 0 0
3

1
0
0 0 
3

1 1
1
2 3
1
0
0
3
1
0 0
2
Where are we after two clicks?
1

2

0

1
2

0


 0

1 1
6 6
2 4
3 9
5
0
18
1
0
9
1
0
6
0
1
0
0
0
11 

18 
1 

9 
1
6

0 

1 

9 
Where are we after n clicks?
Why?
 0.29

 0.39
 0.22

 0.02
 0.07

0.29 0.29 0.29 0.29 

0.39 0.39 0.39 0.39 
0.22 0.22 0.22 0.22 

0.02 0.02 0.02 0.22 
0.07 0.07 0.07 0.07 
Order of pages
 0.29

 0.39
 0.22

 0.02
 0.07

B, A, C, E, D
0.29 0.29 0.29 0.29 

0.39 0.39 0.39 0.39 
0.22 0.22 0.22 0.22 

0.02 0.02 0.02 0.22 
0.07 0.07 0.07 0.07 
Image compression
The easiest way to store an image inside the
memory of a computer is to store the color of each
pixel.
This requires an enormous quantity of memory!
Can we do better?
Let’s suppose we have drawn a city:
We store in memory the line segments, circle arcs,
etc…, which approximate our image.
We approximate our image by known
geometric objects
To store a line segment in memory it is
sufficient to store:
• the two endpoints of the line segment
• a program explaining to the computer how to
draw a line segment with given endpoints.
The geometric objects are our
alphabet.
How to store more complex images, for
instance landscapes?
• We use the same
principle but we
enlarge our alphabet:
• We approximate our
landscape by fractals,
for instance the fern.
We store in memory a program to draw the fern.
Such a program on Mathematica
m=15000
L[n_]:=If[1<n<87,2,n]
H[n_]:=If[86<n<94,3,L[n]]
K[n_]:=If[n>93,4,H[n]]
R=Table[K[Random[Integer,{1,100}]],{m}];
F[1,x_,y_]:=0
G[1,x_,y_]:=0.16*y
F[2,x_,y_]:=x*0.85+y*0.04
G[2,x_,y_]:=-x*0.04+y*0.85+1.6
F[3,x_,y_]:=x*0.2-y*0.26
G[3,x_,y_]:=0.23*x+0.22*y+1.6
F[4,x_,y_]:=-x*0.15+y*0.28
G[4,x_,y_]:=x*0.26+y*0.24+0.44
x[1]:=0
y[1]:=0
Do[{x[n+1],y[n+1]}={F[R[[n]],x[n],y[n]],G[R[[n]],x[n],y[n]]},{n,1,m}]
T=Table[{x[n],y[n]},{n,m}];
ListPlot[T, AspectRatio->1, Axes-> False]
Principle for drawing the fern
The fern is the
union:
•of a stalk
•of three copies of the
initial fern
We can reconstruct the fern from 4
affine transformations:
• the transformation T1 which
sends the large fern to the fern
minus two branches,
• the transformation T2 which
sends the large fern on the left
branch,
• the transformation T3 which
sends the large fern on the right
branch,
• the transformation T4 which
sends the large fern on the stalk.
In order to reconstruct the fern, it suffices to store
in memory this information!
Algorithm:
•We take a point P on the fern.
•We choose at random i in {1,2,3,4} and we plot
P1 = Ti(P).
•We choose at random i in {1,2,3,4} and we plot
P2 = Ti(P).
•Etc...
This method is called « Iterated function systems ». It
works because the fern is self similar.
Why does it work?
Let’s look at the Sierpinski
carpet:
It is a union of three
Sierpinski carpets.
Let us start with a square
and iterate a construction
algorithm
This works with any initial set! Let’s try
another one:
In practice
• Coding:
We replace any small square by
the image of a similar larger
square under a homothety of ratio
½ composed with one of 8
transformations:
• Identity plus 3 rotations
• 4 symetries
We adjust contrast.
We make a translation of the level
of grey.
Example
First iterate
Sixth iterate
Some exercices
The GPS (Global positioning system)
fully operational since 1995
• Network of orbiting
satellites whose
position is known
• The receptor measures the travelling
time t of a signal from one satellite
to the receptor.
• The distance from the satellite to the
receptor is d = ct
c: speed of light
• The points located at a distance d
from a satellite are on a sphere of
radius d, with center at the satellite.
• The intersection of two
spheres is a circle:
• The intersection of
three spheres is
two points. One of
them is excluded
because it is non
realistic.
• Hence, if we know the travelling time of the
signals of three satellites to the receptor we
know the position of the receptor.
This is the theory…
In practice … the satellites have atomic clocks perfectly
synchronized.
The receptor has a cheap clock.
We have a fourth unknown: the shift between the clock
of the receptor and the clocks of the satellites.
We then need to “measure” the travelling time of a
signal from a fourth satellite.
4 measured times

4 unknowns
• The shift between
clocks
• The three
coordinates of
position
With this method we get a precision of 20 meters.
Applications of the GPS
• Finding one’s way in nature
• Drawing a map
• Managing a fleet of vehicles
• Measuring Mount Everest and observing its growth
• Helping blind people
• Find one’s way on the road
• Landing a plane in the fog
GPS are a reference of time!
• Electronic equipments can be synchronized with the
help of GPS.
• Hydro-Québec uses this method to synchronize its
lightnings detectors. Once thunderstorms are
localized, one can reduce the current through lines
passing through zones of thunderstorms so as to
minimize the risk of breakdown of the electrical
network, in case one transit line receives a lightning.
A related exam question
Meteorites regularly enter the atmosphere, rapidly heat up,
disintegrate, and finally explode before hitting the surface of the
Earth. This explosion generates a shock wave that travels in all
directions at the speed of sound v. The shock wave is detected
by seismographs installed at various locations on the surface of
the Earth.
If four stations (equipped with perfectly synchronized clocks)
note the moment that the shock wave arrives, explain how to
calculate both the position and time of the explosion.
Signal of the GPS
Shift-register:
Example: we take (q0, q1, q2, q3 )=(1,1,0,0)
000100110101111
001001101011110
010011010111100
100110101111000
001101011110001
011010111100010
110101111000100
101011110001001
010111100010011
101111000100110
011110001001101
111100010011010
111000100110101
110001001101011
100010011010111
Why?
Random number generators
• Consider sequences of 0 and 1:
• 0 and 1 must each appear with probabilty ½.
• All sequences of length 2 must each appear with
probability ¼.
• All sequences of length n must each appear with
probability 1/2n.
Theorem: in the sequences of period 2n – 1 generated by
the shift-register:
• 1 appears 2n-1 times and 0 appears 2n-1 – 1
times,
• Each sequence of length 2 appears 2n-2 times
except 00 which appears 2n-2 – 1 times
• Each sequence of length r appears 2n-r times
except 0…0 which appears 2n-r – 1 times, for
r <= n.
Error correcting codes
Principle: we lengthen the message in a redundant way. This
allows to correct some errors.
Example: We repeat each bit three times. We want to send 0.
We send 000.
If we receive 000 we decode
0
100 we decode
0
010 we decode
0
001 we decode
0
We have corrected 0 or 1 error.
However
If we receive
110
we decode
101
we decode
011
we decode
111
we decode
And the transmission is erroneous.
1
1
1
1
An error correcting code is efficient if there are few
errors.
This code is not economical: a word of 4 bits is
lengthened to 12 bits and we may only be able to correct
one error.
We can do much better…
Hamming code:
We want to send a 4 bits word: u1, u2, u3, u4
We send a 7 bits word. We add (mod 2):
u5 = u1+ u2 + u3
u6 = u2+ u3 + u4
u7 = u1+ u2 + u4
This code can correct one error.
u1 erroneous: u5 and u7 incompatibles
u2 erroneous : u5, u6 and u7 incompatibles
u3 erroneous : u5 and u6 incompatibles
u4 erroneous : u6 and u7 incompatibles
u5 erroneous : u5 incompatible
u6 erroneous : u6 incompatible
u7 erroneous : u7 incompatible