Mathematical

1
Mathematical Patterns
By

Korie Fudge
2
Intro
Mathematical patterns are found all around us.
There is math patterns in nature from the design
of a sunflower to how rabbits reproduce. There
is math patterns in the braiding hair, in the
leaves of a plant and in a beehive. This
presentation will teach us about mathematical
patterns, tessellations, math in nature and tell
about some of Fibonaccis work with mathematical
patterns in nature.
3
Fibonacci
4
Biography of Fibonacci
Leonardo Fibonacci or as he is also known,
Leonardo of Pisa, was the first great
mathematician of medieval Christian Europe. He
played an important part in the revival of
ancient mathematics and made significant
contributions of his own. His most important work
was in indeterminate analysis and number theory.
The Fibonacci Sequence is named for him.
5

• Fibonacci was the son of Guilielmo and a member
  of the Bonacci family. His possible date of birth
  was 1170 in Pisa (now Italy), and his possible
  date of death was 1250 in Pisa (now Italy).
  Although born in Italy he was educated in North
  Africa where is father held a diplomatic post.
• Fibonacci lived in the days before printing, so
  his books were hand written. Of his books, four
  are around today. His most impressive piece of
  work is his book Liber quadratorum, written in
  1225.
• Fibonacci proved many interesting number theory
  results. Two of these theories are
• there is no x, y such that x2 y2 and x2 y2
  are both squares
• and x4 y4 cannot be a square

6
Fibonacci Numbers
7
What are Fibonacci numbers?
Fibonacci numbers is a sequence of numbers where
the sum of two consecutive numbers is equal to
the next consecutive number. The first ten
numbers are as follows 1, 1, 2, 3, 5, 8 ,13, 21,
34, 55 (1 1 2, 1 2 3, 2 3
5.) These numbers are named for the medieval
Italian mathematician, Leonardo Fibonacci, who
first noted them.
8
Fibonacci numbers in Nature
9
Fibonacci numbers are all around us in nature.
Cutting a bell pepper cross wise reveals 3
chambers. An apple has a 5 point start
cross-section and a lemon has an 8 chambered
cross section. A daisy generally has 13, 21 or 34
petals. Each set of spirals in a sunflower
contain two consecutive Fibonacci numbers. Their
seeds spiral out from the center with 21 spirals
in one direction and 34 in another. The giant
sunflower contains 89 and 144 spirals and the
whopper sunflower has 144 and 233 spirals.
These numbers are also found in the way rabbits
reproduce in the ancestry of honeybees, and in
the number of turns on a flower
10
Fibonacci numbers

Rabbits
11
One way in which mathematical patterns are found
in nature is in the way rabbits reproduce. This
was the original problem that Fibonacci
investigated. This uses the Fibonacci sequence
(add any 2 consecutive numbers to obtain the next
number.) This isnt very realistic however
because it implies that the rabbits always have 1
male and 1 female, that the brothers and sister
rabbits mate with each other and that the rabbits
dont die.
12
The Reproduction of
Rabbits
An example of Fibonacci Sequence found in nature
13
Here is an example
14
Fibonnaci numbers

Honeybees
15

• Fibonacci sequence can also be used to count a
  honeybees ancestors. First of all we have to
  recognize some things about honeybees
• Not all honeybees have 2 parents
• Female honeybees have 2 parents (a male and a
  female)
• Male honeybees only have 1 parent (a female)

16
The Ancestors of Honeybees
A second example of Fibonacci Sequence in Nature
17
Fibonacci numbers
in
Plants
18
Fibonacci numbers are found on plants in the
arrangements of the leaves around their stems. On
a plant, the leaves are often arranged so that
leaves above do not hide leaves below. The
Fibonacci numbers also occur when counting the
number of times we go around the stem, going form
leaf to leaf, as well as counting the leaves we
meet until we encounter a leaf directly above the
starting one. If we count in the other direction
we will get a different number of turns for the
same leaf number. The number of turns in each
direction, as well as the number of leaves met
are three consecutive Fibonacci numbers.
19
In the top plant on the picture to the right, we
have 3 clockwise rotations before we come to a
leaf directly above the first one, passing 5
leaves on the way. However, if we go counter
clockwise, we only need to turns. (notice how 2,
3 and 5 are consecutive Fibonacci numbers). For
the lower plant we have 5 clockwise rotations and
have to pass 8 leaves. In the counter clockwise
direction we only need 3 turns. (3, 5 and 8 are
consecutive Fibonacci numbers).
20
Sequences Forumulas
21
What is a sequence?
A sequence is an ordered set of mathematical
quantities called terms. A sequence is known if a
formula can be given for any particular term
using preceding terms or using its position in
the sequence. For example, the sequence 1, 1, 2,
3, 5, 8, 13 is found by adding any two
consecutive terms to obtain the next term. The
sequence 0, 3, 8, 15, 24, 35 is determined by
the formula n2-1. Where n is the term.
22
How to determine the general term for the
sequence 0, 1, 26, 63, 124. Set up a table of
values relating the domain values (1, 2, 3, 4,
5,) to the range values (0, 7, 26, 63, 124)
23
6
6
6
6
Level 3
(constant difference
of 6)
Level 2
Level 1
24
Level 1 7 0 7 26 7 19
63 26 37 124 63 61 Level 2
19 7 12 37 19 18 61
37 24 Level 3 18 12 6 24
18 6
The formula is tn n3 1
25


Here is another example
0, 15, 80, 255, 624, 1295
26
24
24
Level 4
Level 3
(constant difference of 24)
Level 2
Level 1
27
The formula is tn n4 1
Now we can find the next three numbers in the
pattern.
0, 15, 80, 255, 624, 1295, 2400, 4095, 6560
28
If you already have a formula, and you would
like to find n term, you would place n in the
appropriate place in the formula. For example if
you wanted to find the 5th number a sequence and
the formula is n2 3, you would place 5 where it
says n. t5 n2 3 52 3 25 3
28 The fifth term is 28
4, 7, 12, 19, 28, 39, 52
29
Tessellations
30
A tessellation is a filling up of a
two-dimensional space by congruent copies of a
figure that do not overlap. Only 3 regular
polygons can tessellate the square, the
equilateral triangle and the regular hexagon.
Only these three regular polygons will work
because the corners of each of their angles is a
denominator of 360o. (Each angle of a square is
90o, each angle of a equilateral triangle is 60o,
and each angle of a regular hexagon is 120o.
As well, variations of these regular polygons
can tessellate. This can be done by modifying one
side of the polygon and the modifying the
opposite side the exact same way.
31
Regular Rectangle Tessellation
32
Equilateral Triangle Tessellation
33
Regular Hexagon Tessellation
34
A Non-Standard Figure Tessellation
35
Tesselations
in
Hair Braiding
36
There are different ways of braiding hair and in
some ways, you tessellate certain shapes to get
the design. For a box braid, you tessellate
boxes shaped like rectangles and in the end the
pattern will resemble a brick wall. You start
with two boxes at the nape of the neck and at
each level you increase with one box. To make the
box you take the hair inside the box and braid it
at the point of intersection of the diagonals of
the box. For triangular braids, the hair inside
the triangle is drawn to the point of
intersection of the bisectors of the the angles
of the triangle.
37
It is also possible to braid with hexagons as
shown in the picture above. These braids resemble
the design on a pineapple or in a beehive.