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While you are waiting
• Which triangle is larger in area, one with sides of
length, 5,5,6 or one with sides of length 5,5,8?
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28/07/2014
MEI Conference 2014
Maths in Popular Culture
Maths in Popular Culture
• Idea for this session came from thinking about
an FMSP podcast about enrichment
• The idea was to have a panel of people
reviewing/chatting about something from
popular culture that would provide a link into
some mathematics.
• Popular culture included: TV (and Youtube),
Books, Films, Radio (and Podcasts), Games
(and Apps), Music, Sport, Social Networks,
Websites and Blogs
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This talk
This talk is designed to demonstrate how to
piggy-back on popular culture to prompt
discussion and engagement in mathematics.
Hopefully it is interesting and will provide you
with ideas for the classroom.
It’s not intended to be comprehensive in any way
and your view of ‘popular’ and ‘culture’ might be
different to mine!
Hollywood hates Math(s)
• This blog post from Dan Meyer is interesting
giving an insight into how mathematics is
typically portrayed in movies
http://blog.mrmeyer.com/2013/hollywoodhates-math/
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Stanislaw Lem, Return from the Stars
“These are metagen expansions in an
n-dimensional, configuration, degenerative series”
“What are you saying? Didn’t Skriabin prove that
there are no metagens other than the variational?”
“Yes, a very elegant proof. But this, you
see, is transcontinuous.”
“Impossible! That would…but it
must have opened up a
whole new world”
Parody of Maths Education TV
• Look Around You BBC:
http://vimeo.com/13497928
Amuse your maths class with this!
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Dara O’Briain’s School of Hard Sums
• Some great starter problems can be found
here
http://dave.uktv.co.uk/shows/dara-o-briainschool-hard-sums/
• Resources for the series made available
through TES
http://www.tes.co.uk/teaching-resource/DaraO-and-39-Braian-and-39-s-School-of-Hard-SumsProblems-6351740/
The Office
• Here is a clip from the US version of The Office
which leads nicely into a rich mathematics
problem
http://www.math.harvard.edu/~knill/mathmovie
s/swf/office_surplus.html
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A maths problem from this
90cm
60cm
Generalisation
10m
cm
10n cm
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Twitter and Facebook – Maths Rich
Trending Twitter Topics
• Early detection of trending in Twitter occupies
many mathematicians
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Facebook patents
Degrees of separation
• Oldest model for a random graph is that of
Erdos and Renyi
• G(n,p) is a graph with n – nodes where the
probability of the existence of a possible edge
is 1/p for all edges
• Erdos and Renyi showed that if p > 1/n then
the probability of there being a giant
connected component is greater than 0.5.
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Giant Connected Component
Triangles of friends in social networks
• In social networks triangles tend to form in the
friends graph.
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Triangles of friends in social networks
• In social networks triangles tend to form in the
friends graph.
Triangles of friends in social networks
• In social networks triangles tend to form in the
friends graph.
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Triangles of friends in social networks
• In social networks triangles tend to form in the
friends graph. This is called transitivity.
This
friendship
now forms
Strength of ties
• Most social network algorithms for friend
recommendation are based on weighted
graphs, where the weight represents the
strength of the relationship.
• An example of a function to determine the
strength of a relationship between two people
might be
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•
Ways of measuring the
strength
of a tieA and B might be
Strength of tie between
two individuals,
measured as something like:
•
•
•
•
•
Iout-set of outgoing e-mails, A to B
Iin-set of incoming e-mails, B to A
tnow-is the current time
t(i) -timestamp of an interaction
λ –half life, determines speed at which an interaction's
importance decays
• ωout-weight that determines relative importance of outgoing
interaction vs. incoming interaction
Why weak ties are important
• Suppose we have only strong ties and weak
ties.
• Assume that if A is strongly tied to B and B is
strongly tied to C then A will be strongly tied
to C.
• A bridge is an edge that is the only path
between two points.
• Assuming that everyone has a least one strong
tie, then a bridge must be a weak tie
• Prove it
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Why weak ties are important
• The two strongly tied communities shown
would have no way of contacting one another if
it were not for this weak link
Why your friends have more friends
than you do!
• This from a paper with this title in the
American Journal of Sociology by Scott L Feld
• People might ask themselves what is a
reasonable number of friends to have
• It is reasonable to assume that individuals use
the number of friends their friends have as a
basis for this comparison
• If you do this you may feel inadequate…
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Papers related to this
• http://www.cs.umd.edu/~srin/PDF/2012/rwconf.pdf
• http://en.wikipedia.org/wiki/Friendship_para
dox
Feld, Scott L. (1991), "Why your friends have
more friends than you do", American Journal of
Sociology 96 (6): 1464–1477
This is always true
It can be shown that if xi are the degrees of the
nodes in a friends network then
• The average number of friends is mean(xi)
• The average of the number of friends of friends
is mean(xi) + variance(xi)
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A
D
E
B
C
Person
No of
friends
Average
number of
friends their
friends have
A
1
4
B
1
4
C
1
4
D
1
4
E
4
1
Average number of friends = 8/5 = 1.6
Average number of friends of friends = 17/5 = 3.4
Variance of number of friends =
( (1 – 1.6)2 + (1 – 1.6)2 + (1 – 1.6)2 + (1 – 1.6)2 + (4 – 1.6)2)/4 = 1.8
Dislikes –
• You have a group of facebook users and you
know who dislikes who. This can be
represented as a graph.
• Is it possible to arrange three facebook groups
and in such a way that no two people who
dislike one another are in the same facebook
group?
• This is the same as asking the three colouring
question for a map.
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•
This problem is known to be NP
The only way to solve
this problem in general
complete
is to try all possible colourings and see if there
is one that works. Each possible colouring can
be checked quickly.
• If n is the number of nodes/countries/facebook
users this means checking 3n possibilities.
• This is an example of an NP problem (nondeterministic polynomial time)
• NP – complete problems are a subset that are
equivalent to one another
What is P? What is NP?
What is P = NP?
• Adding or multiplying two numbers have P
complexity
• The travelling salesman problem is NP complete
• The boolean satisfiability problem is NP complete
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Perlin Noise
• This was invented by Ken Perlin as a way to
make metallic surfaces in Tron (1982) look
more organic
Perlin Noise
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Perlin Noise
w
x
a
d
b
c
z
y
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w
w.a/|w||a|
a
b
x
x.b/|x||b|
c
d
z
y
y.c/|y||c|
z.d/|z||d|
w
k = w.a/|w||a|
a
d
b
x
l = x.b/|x||b|
c
z
y
n = z.d/|z||d|
m = y.c/|y||c|
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k
αk+ (1 – α)l
α
l
β
1–α
β [αk+ (1 – α)l] + (1 – β)[ αn+ (1 – α)m]
1–β
n
αn+ (1 – α)m
m
Perlin Noise Demo
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28/07/2014
Elite (1984) - A huge world to explore
• 8 galaxies each with 256 planets
• For each planet a data page containing details
of its
◦ government type
◦ economy type
◦ tech level
◦ commodity prices
◦ …..
in fact….
Planet Data Page
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Elite
Not only that but each planet has a one sentence
description of what it is most notable for:
This planet is most notable for Tibediedian Arnu
brandy but ravaged by unpredictable solar
activity.
Elite
• So how was this done inside approx 20K?
• Start with the hexadecimal system, base 16:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E
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Elite
• This is how Galaxy 1 was generated.
• Start with a seed value of 5A4A0248B753. View
this as
5A4A|0248|B753
• Calculate the next value as
0248|B753| _ _ _ _
Elite
• The end four digits are the three sections
added together module 164
5A4A + 0248 + B753 (modulo 164)
• This is 13E5
• So the next term is
0248|B753|13E5
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Elite
• We can generate a sequence in this way
Elite
• Then we can convert each section into binary
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Elite
Elite
x coordinate of
planet 1 = 90
y coordinate of
planet 1 = 2
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Elite
10111 = TI 10010 = BE 01101 = DI 11000 = ED
Name of first planet in Galaxy 0 is
TIBEDIED
Elite
Adding 10 to this number then multiplying by 256
gives the radius of planet 1 in Galaxy 0.
So the radius of this planet is (10 + 7) * 256 = 4352
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28/07/2014
From a forum about Minecraft
Spheres and other shapes in Minecraft
• Here is a nice website showing how to
construct various shapes from cubes in
Minecraft
http://www.plotz.co.uk/
• It’s interesting to get students to think about
how these constructions are generated (what
determines whether a given block is included
or not)
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Professor Layton
• Here are couple of nice problems from the
Professor Layton series of video games:
http://professorlaytonwalkthrough.blogspot.co.u
k/2008/02/puzzle102.html
http://professorlaytonwalkthrough.blogspot.co.u
k/2008/02/puzzle086.html
• Here are all the problems
http://professorlaytonwalkthrough.blogspot.co.u
k/
Other things on the web
• Webcomics - http://xkcd.com/ (you’ll find references
to various mathematical ideas and theorems here,
these can make good starters, try searching the site)
• Maths Gifs
http://www.presentandcorrect.com/blog/a-collectionof-maths-gifs-posted-purely-for-aesthetic-reasons
(these are excellent, get students to try to work out
what they are showing), also
http://www.tumblr.com/tagged/math-gifs
http://www.reddit.com/r/mathgifs
• Podcasts
My favourites are http://mathfactor.uark.edu/
(particularly the interview with John Conway) and
http://www.furthermaths.org.uk/podcasts
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