SlideShare ist ein Scribd-Unternehmen logo
1 von 9
Logarithmic
Spirals
Francesca Farris
What are they?
Logarithmic spirals are spirals found in nature, unique
because they are self-similar. Self-similarity means that
a part of an object or image is the same as the whole.
Self-similarity in a fern
plant
Fractals, which we learned about in class,
are self-similar. The link here is to an
animated Mandelbrot sequence zoom. You
can see that as it zooms deeper and
deeper into the fractal set, the image stays
the same. Logarithmic spirals are also
seen in the animation.
Logarithmic
spiral
The Basics
The basic spiral is the
Archimedean spiral, in
which the distance
between the curves of the
spiral is constant, as seen
to the right.
In logarithmic spirals, the
distance between the
curves increases in
geometric size by a scale
factor, but the angle at
which each curve is
formed is constant and the
spiral retains its original
shape.
Archimedean spiral
Logarithmic spiral in nature
Spira Mirabilis
This fact, that logarithmic spirals have the unique quality of
increasing in size while retaining an unaltered shape, caused
Jacob Bernoulli, in his studies, to call them spira mirabilis
(“miraculous spiral”, in Latin).
Interestingly, Jacob Bernoulli was so fascinated by logarithmic
spirals that he wanted to have one put on his headstone, along
with the Latin quote “Eadem mutata resurgo” (“Although
changed, I shall arise the same”), which describes logarithmic
spirals very well. Ironically, an Archimedean spiral was placed on
his headstone by mistake.
Spira
mirabilis, as
seen in a
shell
Spira
mirabilis, as
seen in a
head of
Romanesco
broccoli
Polar Coordinates
Logarithmic spirals can be created on a polar coordinate
graphing system, rather than the Cartesian coordinate
system of graphing which we would use to graph normal
functions.
To graph polar functions, you would use a number that lies
along the x-axis, just like with the Cartesian system, as your
first point. But rather than using a number that lies along the
y-axis as your second point, you would use an angle to
determine where that point was.
Logarithmic Formula
In order to graph a logarithmic spiral (or any polar coordinates),
you must find the values of r and theta (r,θ), just like how you
would find the values for x and y (x,y) to graph a normal function.
Logarithmic curves are expressed using the formula r=a . ebθ,
where r is the radius, or distance from the center point (called the
pole), e is the base for the logarithm, a and b are constants, and θ
is the angle of the curve. You can use this formula, substituted with
values on a graph for a and b, to create a logarithmic spiral.
By increasing a, the distance of the curve from the pole on the
graph, you are widening the spiral, but by leaving θ at a constant,
you are keeping the angle the same; therefore, the spiral does not
change shape.
The Golden Spiral
In class we learned about the golden ratio and how it can
form a golden spiral, using the growth factor phi (ϕ). This
sort of spiral increases in size by a rate that follows the
Fibonacci sequence (1+0=1, 1+1=2, 2+1=3, 3+2=5, 5+3=8,
8+5=13, …). This spiral forms a golden rectangle, which is
an example of the golden ratio at work, as well as the
Fibonacci sequence; each square in the golden rectangle
increases in size based on the next number in the Fibonacci
sequence.
Logarithmic Spirals in Nature
The logarithmic spiral is a prime example of nature’s
perfection in its fundamental structure. These spirals can be
seen in many plants, animal shells, the path birds fly on to
spiral in on prey, the formation of hurricanes and whirlpools,
spiral galaxies (like the Milky Way), and many other things.
Logarithmic spiral as seen
in a whirlpool Logarithmic spiral as seen
in the galaxy
In Conclusion
The prevalence of so many logarithmic and other
similar spirals in nature can be taken as a philosophical
statement on the similarity of all things, and teaches us
that despite variations, there are some things that we
all share. This, among other things, is one example of
the link between mathematics and our tangible
existence.
Image designed by Alex Grey

Weitere ähnliche Inhalte

Was ist angesagt?

Ray optics and optical
Ray optics and opticalRay optics and optical
Ray optics and opticalfathimakk
 
simple harmonic motion
simple harmonic motionsimple harmonic motion
simple harmonic motionsaba majeed
 
Quantum mechanics
Quantum mechanics Quantum mechanics
Quantum mechanics Kumar
 
Special Theory Of Relativity
Special Theory Of RelativitySpecial Theory Of Relativity
Special Theory Of RelativityGreenwich Council
 
Optics basics concepts
Optics basics conceptsOptics basics concepts
Optics basics conceptsAnisur Rahman
 
POLARIZATION - BIREFRINGENCE AND HUYGEN'S THEORY OF DOUBLE REFRACTION
POLARIZATION - BIREFRINGENCE AND HUYGEN'S THEORY OF DOUBLE REFRACTION POLARIZATION - BIREFRINGENCE AND HUYGEN'S THEORY OF DOUBLE REFRACTION
POLARIZATION - BIREFRINGENCE AND HUYGEN'S THEORY OF DOUBLE REFRACTION Anuroop Ashok
 
Topic 7 wave_interference(latest)
Topic 7 wave_interference(latest)Topic 7 wave_interference(latest)
Topic 7 wave_interference(latest)Gabriel O'Brien
 
Young’s double slit experiment
Young’s double slit experimentYoung’s double slit experiment
Young’s double slit experimentTrisom Sahu
 
Total internal reflection (3)
Total internal reflection (3)Total internal reflection (3)
Total internal reflection (3)Chithra VM
 
Project math in nature
Project math in natureProject math in nature
Project math in nature9562
 
Fibonacci sequence
Fibonacci sequenceFibonacci sequence
Fibonacci sequencelmrio
 
Projectile motion of a particle
Projectile motion of a particleProjectile motion of a particle
Projectile motion of a particleKhanSaif2
 
Periodic Motion P2
Periodic  Motion   P2Periodic  Motion   P2
Periodic Motion P2guesta71bdd
 
Fibonacci Sequence and Golden Ratio
Fibonacci Sequence and Golden RatioFibonacci Sequence and Golden Ratio
Fibonacci Sequence and Golden Ratiovayappurathu
 

Was ist angesagt? (20)

Ray optics and optical
Ray optics and opticalRay optics and optical
Ray optics and optical
 
simple harmonic motion
simple harmonic motionsimple harmonic motion
simple harmonic motion
 
Quantum mechanics
Quantum mechanics Quantum mechanics
Quantum mechanics
 
Special Theory Of Relativity
Special Theory Of RelativitySpecial Theory Of Relativity
Special Theory Of Relativity
 
Nature of light (2)
Nature of light (2)Nature of light (2)
Nature of light (2)
 
Optics basics concepts
Optics basics conceptsOptics basics concepts
Optics basics concepts
 
POLARIZATION - BIREFRINGENCE AND HUYGEN'S THEORY OF DOUBLE REFRACTION
POLARIZATION - BIREFRINGENCE AND HUYGEN'S THEORY OF DOUBLE REFRACTION POLARIZATION - BIREFRINGENCE AND HUYGEN'S THEORY OF DOUBLE REFRACTION
POLARIZATION - BIREFRINGENCE AND HUYGEN'S THEORY OF DOUBLE REFRACTION
 
Topic 7 wave_interference(latest)
Topic 7 wave_interference(latest)Topic 7 wave_interference(latest)
Topic 7 wave_interference(latest)
 
Quantum theory ppt
Quantum theory ppt Quantum theory ppt
Quantum theory ppt
 
Young’s double slit experiment
Young’s double slit experimentYoung’s double slit experiment
Young’s double slit experiment
 
Total internal reflection (3)
Total internal reflection (3)Total internal reflection (3)
Total internal reflection (3)
 
Project math in nature
Project math in natureProject math in nature
Project math in nature
 
Helium Neon Laser
Helium Neon LaserHelium Neon Laser
Helium Neon Laser
 
Fibonacci sequence
Fibonacci sequenceFibonacci sequence
Fibonacci sequence
 
Blackbody ppt
Blackbody pptBlackbody ppt
Blackbody ppt
 
Projectile motion of a particle
Projectile motion of a particleProjectile motion of a particle
Projectile motion of a particle
 
Quantum
QuantumQuantum
Quantum
 
Periodic Motion P2
Periodic  Motion   P2Periodic  Motion   P2
Periodic Motion P2
 
Fibonacci Sequence and Golden Ratio
Fibonacci Sequence and Golden RatioFibonacci Sequence and Golden Ratio
Fibonacci Sequence and Golden Ratio
 
Wave Motion
Wave Motion Wave Motion
Wave Motion
 

Ähnlich wie Logarithmic Spirals

Mathematics in nature
Mathematics in natureMathematics in nature
Mathematics in natureJovin John
 
Mathematical patterns in nature
Mathematical patterns in natureMathematical patterns in nature
Mathematical patterns in natureanshuman264054
 
Fractals and symmetry group 3
Fractals and symmetry   group 3Fractals and symmetry   group 3
Fractals and symmetry group 3Leiko Ravelo
 
How can I tell when it is going to be a Archimedean Spiral. What is .pdf
How can I tell when it is going to be a Archimedean Spiral. What is .pdfHow can I tell when it is going to be a Archimedean Spiral. What is .pdf
How can I tell when it is going to be a Archimedean Spiral. What is .pdfformicreation
 
Maths in nature (complete)
Maths in nature (complete)Maths in nature (complete)
Maths in nature (complete)Abhay Goyal
 
Fractals and symmetry by group 3
Fractals and symmetry by group 3Fractals and symmetry by group 3
Fractals and symmetry by group 3Leiko Ravelo
 
GE 4 Nature of Mathematics The first module
GE 4 Nature of Mathematics The first moduleGE 4 Nature of Mathematics The first module
GE 4 Nature of Mathematics The first modulee5141nunezascotbagui
 
MATH ONLINE ASSIGNMENT
MATH ONLINE ASSIGNMENTMATH ONLINE ASSIGNMENT
MATH ONLINE ASSIGNMENTFathima Fatah
 
Question 1
Question 1Question 1
Question 1inner4zn
 
Line symmetry for 7th std
Line symmetry for 7th stdLine symmetry for 7th std
Line symmetry for 7th stdMalini Sharma
 
PATTERNS-AND-NUMBERS-IN-NATURE.pdf
PATTERNS-AND-NUMBERS-IN-NATURE.pdfPATTERNS-AND-NUMBERS-IN-NATURE.pdf
PATTERNS-AND-NUMBERS-IN-NATURE.pdfjaymarkawra
 
Fi̇bonacci̇ sequence
Fi̇bonacci̇ sequenceFi̇bonacci̇ sequence
Fi̇bonacci̇ sequenceShohrat Ovezov
 
The Nature of Mathematics
The Nature of Mathematics The Nature of Mathematics
The Nature of Mathematics SergsMacuja
 
Fibonacci sequence
Fibonacci sequenceFibonacci sequence
Fibonacci sequenceAnushkaSahu
 
Chapter 1 - Nature of Mathematics.pptx
Chapter 1 - Nature of Mathematics.pptxChapter 1 - Nature of Mathematics.pptx
Chapter 1 - Nature of Mathematics.pptxMinaSaflor
 

Ähnlich wie Logarithmic Spirals (20)

Mathematics in nature
Mathematics in natureMathematics in nature
Mathematics in nature
 
CHAP1.pdf
CHAP1.pdfCHAP1.pdf
CHAP1.pdf
 
Mathematical patterns in nature
Mathematical patterns in natureMathematical patterns in nature
Mathematical patterns in nature
 
Fractals and symmetry group 3
Fractals and symmetry   group 3Fractals and symmetry   group 3
Fractals and symmetry group 3
 
How can I tell when it is going to be a Archimedean Spiral. What is .pdf
How can I tell when it is going to be a Archimedean Spiral. What is .pdfHow can I tell when it is going to be a Archimedean Spiral. What is .pdf
How can I tell when it is going to be a Archimedean Spiral. What is .pdf
 
Maths in nature (complete)
Maths in nature (complete)Maths in nature (complete)
Maths in nature (complete)
 
Mathematics in nature
Mathematics in natureMathematics in nature
Mathematics in nature
 
The Golden Ratio
The Golden RatioThe Golden Ratio
The Golden Ratio
 
Fractals and symmetry by group 3
Fractals and symmetry by group 3Fractals and symmetry by group 3
Fractals and symmetry by group 3
 
GE 4 Nature of Mathematics The first module
GE 4 Nature of Mathematics The first moduleGE 4 Nature of Mathematics The first module
GE 4 Nature of Mathematics The first module
 
MATH ONLINE ASSIGNMENT
MATH ONLINE ASSIGNMENTMATH ONLINE ASSIGNMENT
MATH ONLINE ASSIGNMENT
 
Question 1
Question 1Question 1
Question 1
 
Line symmetry for 7th std
Line symmetry for 7th stdLine symmetry for 7th std
Line symmetry for 7th std
 
PATTERNS-AND-NUMBERS-IN-NATURE.pdf
PATTERNS-AND-NUMBERS-IN-NATURE.pdfPATTERNS-AND-NUMBERS-IN-NATURE.pdf
PATTERNS-AND-NUMBERS-IN-NATURE.pdf
 
Danny Carey Essay
Danny Carey EssayDanny Carey Essay
Danny Carey Essay
 
Fi̇bonacci̇ sequence
Fi̇bonacci̇ sequenceFi̇bonacci̇ sequence
Fi̇bonacci̇ sequence
 
The Nature of Mathematics
The Nature of Mathematics The Nature of Mathematics
The Nature of Mathematics
 
Symmetry
SymmetrySymmetry
Symmetry
 
Fibonacci sequence
Fibonacci sequenceFibonacci sequence
Fibonacci sequence
 
Chapter 1 - Nature of Mathematics.pptx
Chapter 1 - Nature of Mathematics.pptxChapter 1 - Nature of Mathematics.pptx
Chapter 1 - Nature of Mathematics.pptx
 

Kürzlich hochgeladen

General Principles of Intellectual Property: Concepts of Intellectual Proper...
General Principles of Intellectual Property: Concepts of Intellectual  Proper...General Principles of Intellectual Property: Concepts of Intellectual  Proper...
General Principles of Intellectual Property: Concepts of Intellectual Proper...Poonam Aher Patil
 
Micro-Scholarship, What it is, How can it help me.pdf
Micro-Scholarship, What it is, How can it help me.pdfMicro-Scholarship, What it is, How can it help me.pdf
Micro-Scholarship, What it is, How can it help me.pdfPoh-Sun Goh
 
ComPTIA Overview | Comptia Security+ Book SY0-701
ComPTIA Overview | Comptia Security+ Book SY0-701ComPTIA Overview | Comptia Security+ Book SY0-701
ComPTIA Overview | Comptia Security+ Book SY0-701bronxfugly43
 
How to Create and Manage Wizard in Odoo 17
How to Create and Manage Wizard in Odoo 17How to Create and Manage Wizard in Odoo 17
How to Create and Manage Wizard in Odoo 17Celine George
 
Single or Multiple melodic lines structure
Single or Multiple melodic lines structureSingle or Multiple melodic lines structure
Single or Multiple melodic lines structuredhanjurrannsibayan2
 
Explore beautiful and ugly buildings. Mathematics helps us create beautiful d...
Explore beautiful and ugly buildings. Mathematics helps us create beautiful d...Explore beautiful and ugly buildings. Mathematics helps us create beautiful d...
Explore beautiful and ugly buildings. Mathematics helps us create beautiful d...christianmathematics
 
SKILL OF INTRODUCING THE LESSON MICRO SKILLS.pptx
SKILL OF INTRODUCING THE LESSON MICRO SKILLS.pptxSKILL OF INTRODUCING THE LESSON MICRO SKILLS.pptx
SKILL OF INTRODUCING THE LESSON MICRO SKILLS.pptxAmanpreet Kaur
 
Kodo Millet PPT made by Ghanshyam bairwa college of Agriculture kumher bhara...
Kodo Millet  PPT made by Ghanshyam bairwa college of Agriculture kumher bhara...Kodo Millet  PPT made by Ghanshyam bairwa college of Agriculture kumher bhara...
Kodo Millet PPT made by Ghanshyam bairwa college of Agriculture kumher bhara...pradhanghanshyam7136
 
Understanding Accommodations and Modifications
Understanding  Accommodations and ModificationsUnderstanding  Accommodations and Modifications
Understanding Accommodations and ModificationsMJDuyan
 
The basics of sentences session 3pptx.pptx
The basics of sentences session 3pptx.pptxThe basics of sentences session 3pptx.pptx
The basics of sentences session 3pptx.pptxheathfieldcps1
 
ICT role in 21st century education and it's challenges.
ICT role in 21st century education and it's challenges.ICT role in 21st century education and it's challenges.
ICT role in 21st century education and it's challenges.MaryamAhmad92
 
On National Teacher Day, meet the 2024-25 Kenan Fellows
On National Teacher Day, meet the 2024-25 Kenan FellowsOn National Teacher Day, meet the 2024-25 Kenan Fellows
On National Teacher Day, meet the 2024-25 Kenan FellowsMebane Rash
 
Salient Features of India constitution especially power and functions
Salient Features of India constitution especially power and functionsSalient Features of India constitution especially power and functions
Salient Features of India constitution especially power and functionsKarakKing
 
Activity 01 - Artificial Culture (1).pdf
Activity 01 - Artificial Culture (1).pdfActivity 01 - Artificial Culture (1).pdf
Activity 01 - Artificial Culture (1).pdfciinovamais
 
Food safety_Challenges food safety laboratories_.pdf
Food safety_Challenges food safety laboratories_.pdfFood safety_Challenges food safety laboratories_.pdf
Food safety_Challenges food safety laboratories_.pdfSherif Taha
 
2024-NATIONAL-LEARNING-CAMP-AND-OTHER.pptx
2024-NATIONAL-LEARNING-CAMP-AND-OTHER.pptx2024-NATIONAL-LEARNING-CAMP-AND-OTHER.pptx
2024-NATIONAL-LEARNING-CAMP-AND-OTHER.pptxMaritesTamaniVerdade
 
Spellings Wk 3 English CAPS CARES Please Practise
Spellings Wk 3 English CAPS CARES Please PractiseSpellings Wk 3 English CAPS CARES Please Practise
Spellings Wk 3 English CAPS CARES Please PractiseAnaAcapella
 
Application orientated numerical on hev.ppt
Application orientated numerical on hev.pptApplication orientated numerical on hev.ppt
Application orientated numerical on hev.pptRamjanShidvankar
 
Graduate Outcomes Presentation Slides - English
Graduate Outcomes Presentation Slides - EnglishGraduate Outcomes Presentation Slides - English
Graduate Outcomes Presentation Slides - Englishneillewis46
 

Kürzlich hochgeladen (20)

General Principles of Intellectual Property: Concepts of Intellectual Proper...
General Principles of Intellectual Property: Concepts of Intellectual  Proper...General Principles of Intellectual Property: Concepts of Intellectual  Proper...
General Principles of Intellectual Property: Concepts of Intellectual Proper...
 
Micro-Scholarship, What it is, How can it help me.pdf
Micro-Scholarship, What it is, How can it help me.pdfMicro-Scholarship, What it is, How can it help me.pdf
Micro-Scholarship, What it is, How can it help me.pdf
 
ComPTIA Overview | Comptia Security+ Book SY0-701
ComPTIA Overview | Comptia Security+ Book SY0-701ComPTIA Overview | Comptia Security+ Book SY0-701
ComPTIA Overview | Comptia Security+ Book SY0-701
 
How to Create and Manage Wizard in Odoo 17
How to Create and Manage Wizard in Odoo 17How to Create and Manage Wizard in Odoo 17
How to Create and Manage Wizard in Odoo 17
 
Single or Multiple melodic lines structure
Single or Multiple melodic lines structureSingle or Multiple melodic lines structure
Single or Multiple melodic lines structure
 
Explore beautiful and ugly buildings. Mathematics helps us create beautiful d...
Explore beautiful and ugly buildings. Mathematics helps us create beautiful d...Explore beautiful and ugly buildings. Mathematics helps us create beautiful d...
Explore beautiful and ugly buildings. Mathematics helps us create beautiful d...
 
SKILL OF INTRODUCING THE LESSON MICRO SKILLS.pptx
SKILL OF INTRODUCING THE LESSON MICRO SKILLS.pptxSKILL OF INTRODUCING THE LESSON MICRO SKILLS.pptx
SKILL OF INTRODUCING THE LESSON MICRO SKILLS.pptx
 
Kodo Millet PPT made by Ghanshyam bairwa college of Agriculture kumher bhara...
Kodo Millet  PPT made by Ghanshyam bairwa college of Agriculture kumher bhara...Kodo Millet  PPT made by Ghanshyam bairwa college of Agriculture kumher bhara...
Kodo Millet PPT made by Ghanshyam bairwa college of Agriculture kumher bhara...
 
Understanding Accommodations and Modifications
Understanding  Accommodations and ModificationsUnderstanding  Accommodations and Modifications
Understanding Accommodations and Modifications
 
The basics of sentences session 3pptx.pptx
The basics of sentences session 3pptx.pptxThe basics of sentences session 3pptx.pptx
The basics of sentences session 3pptx.pptx
 
ICT role in 21st century education and it's challenges.
ICT role in 21st century education and it's challenges.ICT role in 21st century education and it's challenges.
ICT role in 21st century education and it's challenges.
 
On National Teacher Day, meet the 2024-25 Kenan Fellows
On National Teacher Day, meet the 2024-25 Kenan FellowsOn National Teacher Day, meet the 2024-25 Kenan Fellows
On National Teacher Day, meet the 2024-25 Kenan Fellows
 
Salient Features of India constitution especially power and functions
Salient Features of India constitution especially power and functionsSalient Features of India constitution especially power and functions
Salient Features of India constitution especially power and functions
 
Activity 01 - Artificial Culture (1).pdf
Activity 01 - Artificial Culture (1).pdfActivity 01 - Artificial Culture (1).pdf
Activity 01 - Artificial Culture (1).pdf
 
Food safety_Challenges food safety laboratories_.pdf
Food safety_Challenges food safety laboratories_.pdfFood safety_Challenges food safety laboratories_.pdf
Food safety_Challenges food safety laboratories_.pdf
 
2024-NATIONAL-LEARNING-CAMP-AND-OTHER.pptx
2024-NATIONAL-LEARNING-CAMP-AND-OTHER.pptx2024-NATIONAL-LEARNING-CAMP-AND-OTHER.pptx
2024-NATIONAL-LEARNING-CAMP-AND-OTHER.pptx
 
Spellings Wk 3 English CAPS CARES Please Practise
Spellings Wk 3 English CAPS CARES Please PractiseSpellings Wk 3 English CAPS CARES Please Practise
Spellings Wk 3 English CAPS CARES Please Practise
 
Mehran University Newsletter Vol-X, Issue-I, 2024
Mehran University Newsletter Vol-X, Issue-I, 2024Mehran University Newsletter Vol-X, Issue-I, 2024
Mehran University Newsletter Vol-X, Issue-I, 2024
 
Application orientated numerical on hev.ppt
Application orientated numerical on hev.pptApplication orientated numerical on hev.ppt
Application orientated numerical on hev.ppt
 
Graduate Outcomes Presentation Slides - English
Graduate Outcomes Presentation Slides - EnglishGraduate Outcomes Presentation Slides - English
Graduate Outcomes Presentation Slides - English
 

Logarithmic Spirals

  • 2. What are they? Logarithmic spirals are spirals found in nature, unique because they are self-similar. Self-similarity means that a part of an object or image is the same as the whole. Self-similarity in a fern plant Fractals, which we learned about in class, are self-similar. The link here is to an animated Mandelbrot sequence zoom. You can see that as it zooms deeper and deeper into the fractal set, the image stays the same. Logarithmic spirals are also seen in the animation. Logarithmic spiral
  • 3. The Basics The basic spiral is the Archimedean spiral, in which the distance between the curves of the spiral is constant, as seen to the right. In logarithmic spirals, the distance between the curves increases in geometric size by a scale factor, but the angle at which each curve is formed is constant and the spiral retains its original shape. Archimedean spiral Logarithmic spiral in nature
  • 4. Spira Mirabilis This fact, that logarithmic spirals have the unique quality of increasing in size while retaining an unaltered shape, caused Jacob Bernoulli, in his studies, to call them spira mirabilis (“miraculous spiral”, in Latin). Interestingly, Jacob Bernoulli was so fascinated by logarithmic spirals that he wanted to have one put on his headstone, along with the Latin quote “Eadem mutata resurgo” (“Although changed, I shall arise the same”), which describes logarithmic spirals very well. Ironically, an Archimedean spiral was placed on his headstone by mistake. Spira mirabilis, as seen in a shell Spira mirabilis, as seen in a head of Romanesco broccoli
  • 5. Polar Coordinates Logarithmic spirals can be created on a polar coordinate graphing system, rather than the Cartesian coordinate system of graphing which we would use to graph normal functions. To graph polar functions, you would use a number that lies along the x-axis, just like with the Cartesian system, as your first point. But rather than using a number that lies along the y-axis as your second point, you would use an angle to determine where that point was.
  • 6. Logarithmic Formula In order to graph a logarithmic spiral (or any polar coordinates), you must find the values of r and theta (r,θ), just like how you would find the values for x and y (x,y) to graph a normal function. Logarithmic curves are expressed using the formula r=a . ebθ, where r is the radius, or distance from the center point (called the pole), e is the base for the logarithm, a and b are constants, and θ is the angle of the curve. You can use this formula, substituted with values on a graph for a and b, to create a logarithmic spiral. By increasing a, the distance of the curve from the pole on the graph, you are widening the spiral, but by leaving θ at a constant, you are keeping the angle the same; therefore, the spiral does not change shape.
  • 7. The Golden Spiral In class we learned about the golden ratio and how it can form a golden spiral, using the growth factor phi (ϕ). This sort of spiral increases in size by a rate that follows the Fibonacci sequence (1+0=1, 1+1=2, 2+1=3, 3+2=5, 5+3=8, 8+5=13, …). This spiral forms a golden rectangle, which is an example of the golden ratio at work, as well as the Fibonacci sequence; each square in the golden rectangle increases in size based on the next number in the Fibonacci sequence.
  • 8. Logarithmic Spirals in Nature The logarithmic spiral is a prime example of nature’s perfection in its fundamental structure. These spirals can be seen in many plants, animal shells, the path birds fly on to spiral in on prey, the formation of hurricanes and whirlpools, spiral galaxies (like the Milky Way), and many other things. Logarithmic spiral as seen in a whirlpool Logarithmic spiral as seen in the galaxy
  • 9. In Conclusion The prevalence of so many logarithmic and other similar spirals in nature can be taken as a philosophical statement on the similarity of all things, and teaches us that despite variations, there are some things that we all share. This, among other things, is one example of the link between mathematics and our tangible existence. Image designed by Alex Grey