Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity....
Transcript of Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity....
![Page 1: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/1.jpg)
!1
Is Mathematics Beautiful?
Tex
If It Is,
Is It Because God Is A Mathematician?
![Page 2: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/2.jpg)
!2
When the famous mathematician Gauss was a child, his teacher, trying to keep him busy, set him on the task of finding the sum of the whole numbers 1 to 100.
His teacher was shocked when he completed the problem in just a few seconds.
1 + 2 + 3... + 50
50 x 101 = 5050
100 + 99 + 98 ...+ 51
101 + 101 + 101 + 101
![Page 3: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/3.jpg)
!3
Paul Erdos
![Page 4: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/4.jpg)
!4
Paul Dirac
Dirac’s Equation
![Page 5: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/5.jpg)
!5
Euler’s Identity
![Page 6: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/6.jpg)
SOME BEAUTY FROM NUMBER THEORY
• What are the next 3 terms of the Fibonacci Sequence?
1, 1, 2, 3, 5, 8, __ , __ , __ , ... 13 21 34
• Who was Fibonacci?
Leonardo Pisano (Leonardo of Pisa) (right), better known as Fibonacci, was an Italian mathematician who is most famous for his Fibonacci Sequence and for popularizing the Hindu-Arabic numeral system in Europe.
Fibonacci
!6
![Page 7: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/7.jpg)
!7
The Fibonacci Sequence in Nature- Humans(cm)
1,1,2,3,5,8,13,21,34,55
![Page 8: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/8.jpg)
!8
The Fibonacci Sequence in Nature- Sunflowers
1,1,2,3,5,8,13,21,34,55
![Page 9: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/9.jpg)
!91,1,2,3,5,8,13,21,34,55
![Page 10: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/10.jpg)
!101,1,2,3,5,8,13,21,34,55
![Page 11: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/11.jpg)
!111,1,2,3,5,8,13,21,34,55
![Page 12: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/12.jpg)
!12
The Fibonacci Sequence in Nature- Pine Cones
1,1,2,3,5,8,13,21,34,55
![Page 13: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/13.jpg)
!13
The Fibonacci Sequence in Nature- Flower Petals
1,1,2,3,5,8,13,21,34,55,89
![Page 14: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/14.jpg)
!1455?
![Page 15: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/15.jpg)
!15
The Fibonacci Sequence in Nature- Trees & Bees
1
1
2
3
5
Female Bees can produce with or without a male!
Funfertilized Male
Ffertilized by male Female
Ed
1 Parent
2 Grandparents(Gps)
3 greatGps (gGps)
5 ggGps
![Page 16: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/16.jpg)
!16
The Fibonacci Sequence/Spiral in Nature- Shells
![Page 17: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/17.jpg)
!17
The Fibonacci Spiral in Nature
Aerial View of a Hurricane Telescopic View of a
Galaxy
![Page 18: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/18.jpg)
!18
The Fibonacci Spiral in Nature- Where You Least Expect It!
![Page 19: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/19.jpg)
!19
The Fibonacci Spiral in Life /Living Things
Double Helix DNA Spiral
Proboscis of a Sap-Feeding Butterfly
![Page 20: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/20.jpg)
!20
The Fibonacci Spiral in Art
1,1,2,3,5,8,13,21,34,55,89,144,...
![Page 21: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/21.jpg)
SOME BEAUTY FROM GEOMETRY
A GOLDEN RECTANGLE IS A SPECIAL, AESTHETICALLY PLEASING, RECTANGLE IN WHICH THE LENGTH L DIVIDED BY THE WIDTH W IS APPROXIMATELY 1.618.
GOLDEN RECTANGLEW
L
L / W = ≈ 1.618Phi
(Golden ratio)
![Page 22: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/22.jpg)
CONSTRUCTING A GOLDEN RECTANGLE
c = (a/2) + a = 5a /4 c = (√5 a)/22 22 2
Length = a/2 + (√5 a)/2 = a(√5 + 1)/2 ≈ 1.618aLength / Width = 1.618a/a ≈ 1.618 = Golden Ratio
![Page 23: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/23.jpg)
!23
The GOLDEN RATIO IN LIFE
The ratio of the Major groove to the Minor groove in the DNA Double Helix Spiral is the Golden Ratio
![Page 24: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/24.jpg)
!24
THE GOLDEN RATIO IN THE HUMAN BODY
HEIGHT/ HEIGHT OF NAVEL
![Page 25: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/25.jpg)
!25
“Beauty is in the Phi of the beholder.”
Florence Colgate, England
Nose to Chin/Lips to Chin
Bottom of Nose to Top of Lips/Top of lips to bottom of lips
Eyes to lip center/Lips to Chin
Height of head/Width of Head
The above, and at least 16 others. All are the Golden Ratio (Phi) ≈ 1.618
![Page 26: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/26.jpg)
![Page 27: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/27.jpg)
!27
More About “God’s Fingerprint” in the Human Body
From measurement of 5000 Uteruses per year by ultrasound, Dr Jasper Verguts, a Gynecological Specialist at the University Hospital Lueven in Belgium looked at the ratio of uterus length to width (uterus ratio) and found:
For newborn girls, the uterus ratio is about 2
In old age, the uterus ratio shrinks to 1.46
In the most fertile time of a woman’s life (Between ages 16 and 20) the uterus ratio is 1.6, strikingly close to the Golden Ratio!
![Page 28: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/28.jpg)
!28
THE GOLDEN RATIO IN THE GREAT PYRAMID
![Page 29: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/29.jpg)
!29
THE GOLDEN RATIO IN ANCIENT BUILDINGS
The Taj Mahal, IndiaThe Parthenon, Greece
![Page 30: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/30.jpg)
!30
THE GOLDEN RATIO IN MODERN BUILDINGS
The CN Bulding in Toronto.The ratio of its height (553 m) to the height ofit’s observation tower is 1.618
The UN Building in NYC. It’s width compared to the height of every 10 floors is 1.618
![Page 31: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/31.jpg)
!31
THE GOLDEN RATIO IN VIOLIN CONSTRUCTION
![Page 32: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/32.jpg)
Applications in Music
!32
![Page 33: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/33.jpg)
!33
PURSUING THE BEAUTY FURTHER-AN AMAZING RELATIONSHIP!
The Golden Ratio is “Married to” the Fibonacci Sequence!
1 , 1 , 2 , 3 , 5 , 8 , 13 , 21, 34
A term of the Fibonacci Sequence, say 21, divided by the term before it, 13, is the Golden Ratio, approx 1.618, as it’s limiting value.
![Page 34: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/34.jpg)
!34
SOME BEAUTY FROM TOPOLOGY Topology is the area of mathematics dealing with properties of space that are preserved under continuous deformations, such as stretching, crumpling, and bending, but not tearing or gluing.
![Page 35: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/35.jpg)
SOME BEAUTY FROM TOPOLOGY
!35
IS IT POSSIBLE TO TAKE OFF YOUR VEST WITHOUT TAKING OFF YOUR COAT?
TOPOLOGISTS KNOW!
![Page 36: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/36.jpg)
![Page 37: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/37.jpg)
!37
“Mathematics is the language in which God has written the universe”
“If I were again beginning my studies, I would follow the advice of Plato and start with mathematics”
Galileo Galilei
![Page 38: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/38.jpg)
!38
![Page 39: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/39.jpg)
!39
![Page 40: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/40.jpg)
!40
![Page 41: Is Mathematics Beautiful?€¦ · Paul Erdos!4 Paul Dirac Dirac’s Equation!5 Euler’s Identity. SOME BEAUTY FROM NUMBER THEORY ... Topology is the area of mathematics dealing with](https://reader033.fdocuments.net/reader033/viewer/2022052611/5f08a1a97e708231d422f67c/html5/thumbnails/41.jpg)
!41