Archimedes 1 2 1 – Elegant Mathematical Writing

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Archimedes 1 2 1 – Elegant Mathematical Writing

ArchimedesArchimedes 1 2 1 – Elegant Mathematical Writing Paper
Archimedes 1 2 1 – Elegant Mathematical Writing Prompts
Archimedes 1 2 1 – Elegant Mathematical Writing Worksheets
Archimedes 1 2 1 – Elegant Mathematical Writing Examples
The number π (/ p aɪ /) is a mathematical constant.It is defined as the ratio of a circle's circumference to its diameter, and it also has various equivalent definitions.It appears in many formulas in all areas of mathematics and physics.It is approximately equal to 3.14159. It has been represented by the Greek letter 'π' since the mid-18th century, and is spelled out as 'pi'.
There are six spirals, which you can describe with the functions f(x)=x^a a=2,1/2,-1/2,-1 and f(x)=exp(x), f(x)=ln(x). You distinguish two groups depending on how the parameter t grows from 0. If the absolute modulus of a function r(t) is increasing, the spirals run from inside to outside and go above all limits.
287–212 BC) was born in Syracuse, in the Greek colony of Sicily. He studied mathematics probably at the Museum in Alexandria and made important contributions to the field of mathematics. Archimedes discovered fundamental theorems concerning the centre of gravity of plane geometric shapes and solids. Archimedes: Mathematical Genius of the Ancient World. Enslow Publishing. ISBN 978-0-7660-2502-8. Archimedes: The Father of Mathematics. ISBN 978-1-4042-0774-5. Heath, Thomas L. Works of Archimedes. Dover Publications. ISBN 978-0-486-42084-4. Complete works of Archimedes in English.
Archimedes was one of the biggest mathematical geniuses this world has ever seen. Many of his inventions and ideas still play a role in the world we live in today. Archimedes was the son of an astronomer and mathematician named Phidias. He was born in the city of Syracuse on the island of Sicily in 287 BC. Syracuse was a focal point for art and science which were a strong influence on the maturing Archimedes. As he grew, he soon became obsessed in a curiosity for problem solving. After learning as much as he could in Syracuse, Archimedes traveled to Alexandria in Egypt to study for many years. He later returned with a strong education to push his limitations and study independently.
Archimedes foresaw modern calculus and analyzation by using concepts of infinitesimals and the method of exhaustion to derive and prove a range of geometrical theorems. This included the area of a circle, the surface area and volume of a sphere, and the area under a parabola. He also derived an accurate estimate of pi, creating a system using exponentiation for expressing very large numbers. He was also one of the first to apply mathematics to the physical world. He used his strong mathematical skills to become a pioneer in the world of physics.
During Archimedes' life he accomplished many feats and used his strong mind to assist his people. His king, Hiero II, was very fond of him and recognized his problem solving skills. Archimedes helped him on many occasions throughout his life. One such story tells of how King Hiero was unable to empty rainwater from the hull of one of his ships. The King called upon Archimedes for assistance. Archimedes' solution was to create a machine consisting of a hollow tube containing a spiral that could be turned by a handle at one end. When the lower end of the tube was placed into the hull and the handle turned, water was carried up the tube and out of the boat. The Archimedes Screw is still used as a method of irrigation in developing countries.
Another story tells of how King Hiero called on Archimedes to determine if a crown a blacksmith had made for him was completely gold or if silver had been used in it. This was no daunting task for Archimedes until the king requested that he figure it out without harming the crown. It took Archimedes a while to figure out a way to determine the material used until one day when he sat in his bath and saw the water rise he shouted 'Eureka!' He realized he could figure out the material by placing the crown in water and determining the displacement. He then continued to jump out of his bath and run naked through the streets to figure out what the crown had been made of.
Archimedes not only served the king but also his people. Syracuse had been under attack by the Romans for a long time in his old age. Archimedes developed many clever ways of defending his people through inventions used to batter and sink Roman ships. Eventually after many years the Romans finally invaded the city. A soldier had found Archimedes and demanded that he see his commander. Archimedes refused saying he was working on a difficult problem and the soldier angrily killed the 75 year old. Archimedes was one of the greatest mathematicians to ever live. He really proved the true meaning of math because it was his tool to solve many problems in his life. Archimedes once said 'Mathematics reveals its secrets only to those who approach it with pure love, for its own beauty.' I believe this was his message to us and that if we approach math with love and not hate we will see it as one of the greatest problem solving tools ever created by man instead of a bunch of useless numbers. 

Spirals
Contents of this PageWhat is a spiral?
Spirals by polar equations
..Archimedean Spiral
..Equiangular Spiral
..More Spirals
Clothoide
Spirals Made of Arcs
Spirals Made of Line SegmentsThree-dimensional Spirals
Loxodrome
Making of Spirals
Mandelbrot Set Spirals
Spirals Made of Metal
Spirals, Spirals, Spirals
Spirals on the Internet
References.
To the Main Page 'Mathematische Basteleien'
What is a spiral?
A spiral is a curve in the plane or in the space, which runs around a centre in a special way.
Different spirals follow. Most of them are produced by formulas. 
Spirals by Polar Equations topArchimedean Spiral top
You can make a spiral by two motions of a point: There is a uniform motion in a fixed direction and a motion in a circle with constant speed. Both motions start at the same point. ..............
(1) The uniform motion on the left moves a point to the right. - There are nine snapshots.
(2) The motion with a constant angular velocity moves the point on a spiral at the same time. - There is a point every 8th turn. 
(3) A spiral as a curve comes, if you draw the point at every turn. 
You get formulas analogic to the circle equations. 
Circle...Let P be a point of a circle with the radius R, which is given by an equation in the centre position.There are three essential descriptions of the circle:
(1) Central equation: x²+y² = R² or [y = sqr(R²-x²) und y = -sqr(R²-x²)],
(2) Parameter form: x(t) = R cos(t), y(t) = R sin(t),
(3) Polar equation: r(t) = R.
You give a point by a pair (radius OP, angle t) in the (simple) polar equation. The radius is the distance of the point from the origin (0|0). The angle lies between the radius and the positive x-axis, its vertex in the origin.
Spiral
The radius r(t) and the angle t are proportional for the simpliest spiral, the spiral of Archimedes. Therefore the equation is:
(3) Polar equation: r(t) = at [a is constant].
From this follows
(2) Parameter form: x(t) = at cos(t), y(t) = at sin(t),
(1) Central equation: x²+y² = a²[arc tan (y/x)]².
...The Archimedean spiral starts in the origin and makes a curve with three rounds.The distances between the spiral branches are the same. 
More exact: The distances of intersection points along a line through the origin are the same. 
...If you reflect an Archimedean spiral on a straight line, you get a new spiral with the opposite direction. 
Both spirals go outwards. If you look at the spirals, the left one forms a curve going to the left, the right one forms a curve going to the right. 
If you connect both spirals by a straight (red) or a bowed curve, a double spiral develops. Equiangular Spiral (Logarithmic Spiral, Bernoulli's Spiral) top...(1) Polar equation:  r(t) = exp(t). 
(2) Parameter form: x(t) = exp(t) cos(t), y(t) = exp(t) sin(t). 
(3) Central equation:  y = x tan[ln(sqr(x²+y²))]. The logarithmic spiral also goes outwards. 
The spiral has a characteristic feature: Each line starting in the origin (red) cuts the spiral with the same angle. 
More Spirals top
If you replace the term r(t)=at of the Archimedean spiral by other terms, you get a number of new spirals. There are six spirals, which you can describe with the functions f(x)=x^a [a=2,1/2,-1/2,-1] and f(x)=exp(x), f(x)=ln(x). You distinguish two groups depending on how the parameter t grows from 0. ......If the absolute modulus of a function r(t) is increasing, the spirals run from inside to outside and go above all limits. The spiral 1 is called parabolic spiral or Fermat's spiral.
....If the absolute modulus of a function r(t) is decreasing, the spirals run from outside to inside. They generally run to the centre, but they don't reach it. There is a pole. Spiral 2 is called the Lituus (crooked staff).
I chose equations for the different spiral formulas suitable for plotting.
Clothoide (Cornu Spiral)top....The clothoid or double spiral is a curve, whose curvature grows with the distance from the origin. The radius of curvature is opposite proportional to its arc measured from the origin.The parameter form consists of two equations with Fresnel's integrals, which can only be solved approximately.
You use the Cornu spiral to describe the energy distribution of Fresnel's diffraction at a single slit in the wave theory. 
Spirals Made of Arcs top
Half circle spirals ...You can add half circles growing step by step to get spirals. The radii have the ratios 1 : 1.5 : 2 : 2.5 : 3...
Fibonacci Spiral ...Draw two small squares on top of each other. Add a sequence of growing squares counter clockwise. Draw quarter circles inside the squares (black).
They form the Fibonacci Spiral.
The Fibonacci spiral is called after its numbers. If you take the length of the square sides in the order, you get the sequence 1,1,2,3,5,8,13,21, .. These are the Fibonacci numbers, which you can find by the recursive formula a(n)=a(n-1)+a(n-2) with [a(1)=1, a(2)=1, n>2]. 
Spirals Made of Line Segments top...The spiral is made by line segments with the lengths 1,1,2,2,3,3,4,4,.. Lines meet one another at right angles. 
...Draw a spiral in a crossing with four intersecting straight lines, which form 45° angles. Start with the horizontal line 1 and bend the next line perpendicularly to the straight line. The line segments form a geometric sequence with the common ratio sqr(2). If you draw a spiral into a straight line bundle, you approach the logarithmic spiral, if the angles become smaller and smaller.
...The next spiral is formed by a chain of right angled triangles, which have a common side. The hypotenuse of one triangle becomes the leg of the next. First link is a 1-1-sqr(2)-triangle. The free legs form the spiral.
It is special that the triangles touch in line segments. Wolf 1 34 5 download free. Their lengths are the roots of the natural numbers. You can proof this with the Pythagorean theorem. 
This figure is called root spiral or root snail or wheel of Theodorus.
...Squares are turned around their centre with 10° and compressed at the same time, so that their corners stay at the sides of their preceding square. 
Result: The corners form four spiral arms. The spiral is similar to the logarithm spiral, if the angles get smaller and smaller.
You can also turn other regular polygons e.g. an equilateral triangle. You get similar figures.
This picture reminds me of the programming language LOGO of the early days of computing (C64-nostalgia).
Three-dimensional Spirals top
Helix ...If you draw a circle with x=cos(t) and y=sin(t) and pull it evenly in z-direction, you get a spatial spiral called cylindrical spiral or helix.
The picture pair makes a 3D view possible....Reflect the 3D-spiral on a vertical plane. You get a new spiral (red) with the opposite direction. If you hold your right hand around the right spiral and if your thumb points in direction of the spiral axis, the spiral runs clockwise upward. It is right circular. 
You must use your left hand for the left spiral. It is left circular. The rotation is counter clockwise. 
Example: Nearly all screws have a clockwise rotation, because most of the people are right-handed. 
...In the 'technical' literature the right circular spiral is explained as follows: You wind a right- angled triangle around a cylinder. A clockwise rotating spiral develops, if the triangle increases to the right.
Conical Helix top
You can make the conical helix with the Archimedean spiral or equiangular spiral.The picture pairs make 3D views possible.Loxodrome, Spherical Helix ...The loxodrome is a curve on the sphere, which cuts the meridians at a constant angle. They appear on the Mercator projection as straight lines.
The parametric representation is 
x=cos(t) cos [tan-1(at)]
y=sin(t) cos[tan-1(at)]
z= -sin [tan-1(at)] (a is constant)
You can find out x²+y²+z²=1. This equation means that the loxodrome is lying on the sphere. 
Generally there is a loxodrome at every solid made by rotation about an axis. 
Making of Spirals top...A strip of paper becomes a spiral, if you pull the strip between the thumb and the edge of a knife, pressing hard. The spiral becomes a curl where gravity is present.
You use this effect to decorate the ends of synthetic materials, such as the narrow colourful strips or ribbons used in gift-wrapping. 
I suppose that you have to explain this effect in the same way as a bimetallic bar. You create a bimetallic bar by glueing together two strips, each made of a different metal. Once this bimetallic bar is heated, one metal strip expands more than the other causing the bar to bend.
The reason that the strip of paper bends is not so much to do with the difference in temperature between the top and bottom side. The knife changes the structure of the surface of the paper. This side becomes 'shorter'. 
Incidentally, a strip of paper will bend slightly if you hold it in the heat of a candle flame.
...Forming curls reminds me of an old children's game: Take a dandelion flower and cut the stem into two or four strips, keeping the head intact. If you place the flower into some water, so that the head floats on the surface, the strips of the stem will curl up. (Mind the spots.)A possible explanation: Perhaps the different absorption of water on each side of the strips causes them to curl up.
Mandelbrot Set Spirals top
The coordinates belong to the centre of the pictures. You also find nice spirals as Julia Sets. Here is an example:You find more about these graphics on my page Mandelbrot Set.
Spirals Made of Metal top
You find nice spirals as a decoration of barred windows, fences, gates or doors. You can see them everywhere, if you are look around....I found spirals worth to show at New Ulm, Minnesota, USA. Americans with German ancestry built a copy of the Herman monument near Detmold/Germany in about 1900. 
Iron railings with many spirals decorate the stairs (photo). 
More about the American and German Herman on Wikipedia-pages (URL below)
Costume jewelleries also take spirals as motive. ...Annette's spiral
Spirals, Spirals, Spirals top
Ammonites, antlers of wild sheep, Archimedes' water spiral, area of high or low pressure, arrangement of the sunflower cores, @, bimetal thermometer, bishop staff, Brittany sign, circles of a sea-eagle, climbs, clockwise rotating lactic acid, clouds of smoke, coil, coil spring, corkscrew, creepers (plants), curl, depression in meteorology, disc of Festós, double filament of the bulb, double helix of the DNA, double spiral, electron rays in the magnetic longitudinal field, electrons in cyclotron, Exner spiral, finger mark, fir cone, glider ascending, groove of a record, head of the music instrument violin, heating wire inside a hotplate, heat spiral, herb spiral, inflation spiral, intestine of a tadpole, knowledge spiral, licorice snail, life spiral, Lorenz attractor, minaret at Samarra (Iraq), music instrument horn, pendulum body of the Galilei pendulum, relief strip of the Trajan's column at Rome or the Bernward column at Hildesheim, poppy snail, road of a cone mountain, role (wire, thread, cable, hose, tape measure, paper, bandage), screw threads, simple pendulum with friction, snake in resting position, snake of Aesculapius, snail of the interior ear, scrolls, screw alga, snail-shell, spider net, spiral exercise book, spiral nebula, spiral staircase (e.g. the two spiral stairs in the glass dome of the Reichstag in Berlin), Spirallala ;-), Spirelli noodles, Spirills (e.g. Cholera bacillus), springs of a mattress, suction trunk (lower jaw) of the cabbage white butterfly, tail of the sea-horse, taps of conifers, tongue and tail of the chamaeleon, traces on CD or DVD, treble clef, tusks of giants, viruses, volute, watch spring and balance spring of mechanical clocks, whirlpool, whirlwind.
Spirals on the Internet top
German
Asti
BEWEGUNGSFUNKTIONEN Spiralen
D.H.O. Braasch
Spiralen als Symbol der Sonnenbahn
Jürgen Berkemeier
Fibonacci-Spiralen
Matheprisma
Bewegungsfunktionen (Spiralen 1 ) - (Spiralen online zeichnen)
Michael Komma
Fresnel-Beugung am Einzelspalt (Cornu-Spirale)
Susanne Helbig, Kareen Henkel und Jan Kriener
Spiralen in Naturwissenschaft, Technik und Kunst
Stephan Jaeckel und Sergej Amboni
Spiralen in Natur, Technik und Kunst 
(Referenz: Heitzer J, Spiralen, ein Kapitel phänomenaler Mathematik, Leipzig 1998)
Wikipedia
Spirale, Klothoide, Logarithmische Spirale, Fibonacci Folge, Loxodrome, Ulam-Spirale
Hermannsdenkmal, Hermann Heights Monument
Archimedes 1 2 1 – Elegant Mathematical Writing Paper
EnglishArchimedes 1 2 1 – Elegant Mathematical Writing PromptsAyhan Kursat ERBAS
Equiangular Spiral
Bob Allanson
This is a logarithmic spiral
David Eppstein (Geometry Junkyard)
Spirals, (Links)
Eric W. Weisstein (MathWorld)
Spirals: 
Archimedean Spiral, Circle Involute, Conical Spiral, Cornu Spiral, Curlicue Fractal, Fermat's Spiral, Helix, Hyperbolic Spiral, Logarithmic Spiral, Mice Problem, Nielsen's Spiral, Polygonal Spiral, Prime Spiral, Rational Spiral, Seashell, Spherical Spiral
Hop David (Hop's Gallery)
Riemann sphere, Ram's Horn, Spiral Tile
Jan Wassenaar
spiral
John Macnab
Sculptures
Keith Devlin 
The Double Helix
Mark Newbold 
Counter-Rotating Spirals Illusion
Richard Parris (Freeware-Program WINPLOT) 
The officiell Website is closed. Download of the German Program at heise for example
Xah Lee
Equiangular Spiral, Archimedean Spiral, Lituus, Cornu Spiral
Archimedes 1 2 1 – Elegant Mathematical Writing WorksheetsWikipedia
Spiral, Archimedean spiral, Cornu spiral, Fermat's spiral, Hyperbolic spiral, Lituus, Logarithmic spiral, 
Fibonacci spiral, Golden spiral, Rhumb line, Ulam spiral, 
Hermann Heights Monument, Hermannsdenkmal
FrenchRobert FERRÉOL (COURBES 2D )
SPIRALE
COURBES 3D (SPHÉRO-CYLINDRIQUE, SPIRALE CONIQUE DE PAPPUS, SPIRALE CONIQUE DE PIRONDINI, SPIRALE SPHÉRIQUE)

Different languages
Readers of this page sent me the following links. I am not responsible for the contents.
You can read this page in Haitian Creole translation. 

Archimedes 1 2 1 – elegant mathematical writing worksheets

Spirals
Contents of this PageWhat is a spiral?
Spirals by polar equations
..Archimedean Spiral
..Equiangular Spiral
..More Spirals
Clothoide
Spirals Made of Arcs
Spirals Made of Line SegmentsThree-dimensional Spirals
Loxodrome
Making of Spirals
Mandelbrot Set Spirals
Spirals Made of Metal
Spirals, Spirals, Spirals
Spirals on the Internet
References.
To the Main Page 'Mathematische Basteleien'
What is a spiral?
A spiral is a curve in the plane or in the space, which runs around a centre in a special way.
Different spirals follow. Most of them are produced by formulas. 
Spirals by Polar Equations topArchimedean Spiral top
You can make a spiral by two motions of a point: There is a uniform motion in a fixed direction and a motion in a circle with constant speed. Both motions start at the same point. ..............
(1) The uniform motion on the left moves a point to the right. - There are nine snapshots.
(2) The motion with a constant angular velocity moves the point on a spiral at the same time. - There is a point every 8th turn. 
(3) A spiral as a curve comes, if you draw the point at every turn. 
You get formulas analogic to the circle equations. 
Circle...Let P be a point of a circle with the radius R, which is given by an equation in the centre position.There are three essential descriptions of the circle:
(1) Central equation: x²+y² = R² or [y = sqr(R²-x²) und y = -sqr(R²-x²)],
(2) Parameter form: x(t) = R cos(t), y(t) = R sin(t),
(3) Polar equation: r(t) = R.
You give a point by a pair (radius OP, angle t) in the (simple) polar equation. The radius is the distance of the point from the origin (0|0). The angle lies between the radius and the positive x-axis, its vertex in the origin.
Spiral
The radius r(t) and the angle t are proportional for the simpliest spiral, the spiral of Archimedes. Therefore the equation is:
(3) Polar equation: r(t) = at [a is constant].
From this follows
(2) Parameter form: x(t) = at cos(t), y(t) = at sin(t),
(1) Central equation: x²+y² = a²[arc tan (y/x)]².
...The Archimedean spiral starts in the origin and makes a curve with three rounds.The distances between the spiral branches are the same. 
More exact: The distances of intersection points along a line through the origin are the same. 
...If you reflect an Archimedean spiral on a straight line, you get a new spiral with the opposite direction. 
Both spirals go outwards. If you look at the spirals, the left one forms a curve going to the left, the right one forms a curve going to the right. 
If you connect both spirals by a straight (red) or a bowed curve, a double spiral develops. Equiangular Spiral (Logarithmic Spiral, Bernoulli's Spiral) top...(1) Polar equation:  r(t) = exp(t). 
(2) Parameter form: x(t) = exp(t) cos(t), y(t) = exp(t) sin(t). 
(3) Central equation:  y = x tan[ln(sqr(x²+y²))]. The logarithmic spiral also goes outwards. 
The spiral has a characteristic feature: Each line starting in the origin (red) cuts the spiral with the same angle. 
More Spirals top
If you replace the term r(t)=at of the Archimedean spiral by other terms, you get a number of new spirals. There are six spirals, which you can describe with the functions f(x)=x^a [a=2,1/2,-1/2,-1] and f(x)=exp(x), f(x)=ln(x). You distinguish two groups depending on how the parameter t grows from 0. ......If the absolute modulus of a function r(t) is increasing, the spirals run from inside to outside and go above all limits. The spiral 1 is called parabolic spiral or Fermat's spiral.
....If the absolute modulus of a function r(t) is decreasing, the spirals run from outside to inside. They generally run to the centre, but they don't reach it. There is a pole. Spiral 2 is called the Lituus (crooked staff).
I chose equations for the different spiral formulas suitable for plotting.
Clothoide (Cornu Spiral)top....The clothoid or double spiral is a curve, whose curvature grows with the distance from the origin. The radius of curvature is opposite proportional to its arc measured from the origin.The parameter form consists of two equations with Fresnel's integrals, which can only be solved approximately.
You use the Cornu spiral to describe the energy distribution of Fresnel's diffraction at a single slit in the wave theory. 
Spirals Made of Arcs top
Half circle spirals ...You can add half circles growing step by step to get spirals. The radii have the ratios 1 : 1.5 : 2 : 2.5 : 3...
Fibonacci Spiral ...Draw two small squares on top of each other. Add a sequence of growing squares counter clockwise. Draw quarter circles inside the squares (black).
They form the Fibonacci Spiral.
The Fibonacci spiral is called after its numbers. If you take the length of the square sides in the order, you get the sequence 1,1,2,3,5,8,13,21, .. These are the Fibonacci numbers, which you can find by the recursive formula a(n)=a(n-1)+a(n-2) with [a(1)=1, a(2)=1, n>2]. 
Spirals Made of Line Segments top...The spiral is made by line segments with the lengths 1,1,2,2,3,3,4,4,.. Lines meet one another at right angles. 
...Draw a spiral in a crossing with four intersecting straight lines, which form 45° angles. Start with the horizontal line 1 and bend the next line perpendicularly to the straight line. The line segments form a geometric sequence with the common ratio sqr(2). If you draw a spiral into a straight line bundle, you approach the logarithmic spiral, if the angles become smaller and smaller.
...The next spiral is formed by a chain of right angled triangles, which have a common side. The hypotenuse of one triangle becomes the leg of the next. First link is a 1-1-sqr(2)-triangle. The free legs form the spiral.
It is special that the triangles touch in line segments. Wolf 1 34 5 download free. Their lengths are the roots of the natural numbers. You can proof this with the Pythagorean theorem. 
This figure is called root spiral or root snail or wheel of Theodorus.
...Squares are turned around their centre with 10° and compressed at the same time, so that their corners stay at the sides of their preceding square. 
Result: The corners form four spiral arms. The spiral is similar to the logarithm spiral, if the angles get smaller and smaller.
You can also turn other regular polygons e.g. an equilateral triangle. You get similar figures.
This picture reminds me of the programming language LOGO of the early days of computing (C64-nostalgia).
Three-dimensional Spirals top
Helix ...If you draw a circle with x=cos(t) and y=sin(t) and pull it evenly in z-direction, you get a spatial spiral called cylindrical spiral or helix.
The picture pair makes a 3D view possible....Reflect the 3D-spiral on a vertical plane. You get a new spiral (red) with the opposite direction. If you hold your right hand around the right spiral and if your thumb points in direction of the spiral axis, the spiral runs clockwise upward. It is right circular. 
You must use your left hand for the left spiral. It is left circular. The rotation is counter clockwise. 
Example: Nearly all screws have a clockwise rotation, because most of the people are right-handed. 
...In the 'technical' literature the right circular spiral is explained as follows: You wind a right- angled triangle around a cylinder. A clockwise rotating spiral develops, if the triangle increases to the right.
Conical Helix top
You can make the conical helix with the Archimedean spiral or equiangular spiral.The picture pairs make 3D views possible.Loxodrome, Spherical Helix ...The loxodrome is a curve on the sphere, which cuts the meridians at a constant angle. They appear on the Mercator projection as straight lines.
The parametric representation is 
x=cos(t) cos [tan-1(at)]
y=sin(t) cos[tan-1(at)]
z= -sin [tan-1(at)] (a is constant)
You can find out x²+y²+z²=1. This equation means that the loxodrome is lying on the sphere. 
Generally there is a loxodrome at every solid made by rotation about an axis. 
Making of Spirals top...A strip of paper becomes a spiral, if you pull the strip between the thumb and the edge of a knife, pressing hard. The spiral becomes a curl where gravity is present.
You use this effect to decorate the ends of synthetic materials, such as the narrow colourful strips or ribbons used in gift-wrapping. 
I suppose that you have to explain this effect in the same way as a bimetallic bar. You create a bimetallic bar by glueing together two strips, each made of a different metal. Once this bimetallic bar is heated, one metal strip expands more than the other causing the bar to bend.
The reason that the strip of paper bends is not so much to do with the difference in temperature between the top and bottom side. The knife changes the structure of the surface of the paper. This side becomes 'shorter'. 
Incidentally, a strip of paper will bend slightly if you hold it in the heat of a candle flame.
...Forming curls reminds me of an old children's game: Take a dandelion flower and cut the stem into two or four strips, keeping the head intact. If you place the flower into some water, so that the head floats on the surface, the strips of the stem will curl up. (Mind the spots.)A possible explanation: Perhaps the different absorption of water on each side of the strips causes them to curl up.
Mandelbrot Set Spirals top
The coordinates belong to the centre of the pictures. You also find nice spirals as Julia Sets. Here is an example:You find more about these graphics on my page Mandelbrot Set.
Spirals Made of Metal top
You find nice spirals as a decoration of barred windows, fences, gates or doors. You can see them everywhere, if you are look around....I found spirals worth to show at New Ulm, Minnesota, USA. Americans with German ancestry built a copy of the Herman monument near Detmold/Germany in about 1900. 
Iron railings with many spirals decorate the stairs (photo). 
More about the American and German Herman on Wikipedia-pages (URL below)
Costume jewelleries also take spirals as motive. ...Annette's spiral
Spirals, Spirals, Spirals top
Ammonites, antlers of wild sheep, Archimedes' water spiral, area of high or low pressure, arrangement of the sunflower cores, @, bimetal thermometer, bishop staff, Brittany sign, circles of a sea-eagle, climbs, clockwise rotating lactic acid, clouds of smoke, coil, coil spring, corkscrew, creepers (plants), curl, depression in meteorology, disc of Festós, double filament of the bulb, double helix of the DNA, double spiral, electron rays in the magnetic longitudinal field, electrons in cyclotron, Exner spiral, finger mark, fir cone, glider ascending, groove of a record, head of the music instrument violin, heating wire inside a hotplate, heat spiral, herb spiral, inflation spiral, intestine of a tadpole, knowledge spiral, licorice snail, life spiral, Lorenz attractor, minaret at Samarra (Iraq), music instrument horn, pendulum body of the Galilei pendulum, relief strip of the Trajan's column at Rome or the Bernward column at Hildesheim, poppy snail, road of a cone mountain, role (wire, thread, cable, hose, tape measure, paper, bandage), screw threads, simple pendulum with friction, snake in resting position, snake of Aesculapius, snail of the interior ear, scrolls, screw alga, snail-shell, spider net, spiral exercise book, spiral nebula, spiral staircase (e.g. the two spiral stairs in the glass dome of the Reichstag in Berlin), Spirallala ;-), Spirelli noodles, Spirills (e.g. Cholera bacillus), springs of a mattress, suction trunk (lower jaw) of the cabbage white butterfly, tail of the sea-horse, taps of conifers, tongue and tail of the chamaeleon, traces on CD or DVD, treble clef, tusks of giants, viruses, volute, watch spring and balance spring of mechanical clocks, whirlpool, whirlwind.
Spirals on the Internet top
German
Asti
BEWEGUNGSFUNKTIONEN Spiralen
D.H.O. Braasch
Spiralen als Symbol der Sonnenbahn
Jürgen Berkemeier
Fibonacci-Spiralen
Matheprisma
Bewegungsfunktionen (Spiralen 1 ) - (Spiralen online zeichnen)
Michael Komma
Fresnel-Beugung am Einzelspalt (Cornu-Spirale)
Susanne Helbig, Kareen Henkel und Jan Kriener
Spiralen in Naturwissenschaft, Technik und Kunst
Stephan Jaeckel und Sergej Amboni
Spiralen in Natur, Technik und Kunst 
(Referenz: Heitzer J, Spiralen, ein Kapitel phänomenaler Mathematik, Leipzig 1998)
Wikipedia
Spirale, Klothoide, Logarithmische Spirale, Fibonacci Folge, Loxodrome, Ulam-Spirale
Hermannsdenkmal, Hermann Heights Monument
Archimedes 1 2 1 – Elegant Mathematical Writing Paper
EnglishArchimedes 1 2 1 – Elegant Mathematical Writing PromptsAyhan Kursat ERBAS
Equiangular Spiral
Bob Allanson
This is a logarithmic spiral
David Eppstein (Geometry Junkyard)
Spirals, (Links)
Eric W. Weisstein (MathWorld)
Spirals: 
Archimedean Spiral, Circle Involute, Conical Spiral, Cornu Spiral, Curlicue Fractal, Fermat's Spiral, Helix, Hyperbolic Spiral, Logarithmic Spiral, Mice Problem, Nielsen's Spiral, Polygonal Spiral, Prime Spiral, Rational Spiral, Seashell, Spherical Spiral
Hop David (Hop's Gallery)
Riemann sphere, Ram's Horn, Spiral Tile
Jan Wassenaar
spiral
John Macnab
Sculptures
Keith Devlin 
The Double Helix
Mark Newbold 
Counter-Rotating Spirals Illusion
Richard Parris (Freeware-Program WINPLOT) 
The officiell Website is closed. Download of the German Program at heise for example
Xah Lee
Equiangular Spiral, Archimedean Spiral, Lituus, Cornu Spiral
Archimedes 1 2 1 – Elegant Mathematical Writing WorksheetsWikipedia
Spiral, Archimedean spiral, Cornu spiral, Fermat's spiral, Hyperbolic spiral, Lituus, Logarithmic spiral, 
Fibonacci spiral, Golden spiral, Rhumb line, Ulam spiral, 
Hermann Heights Monument, Hermannsdenkmal
FrenchRobert FERRÉOL (COURBES 2D )
SPIRALE
COURBES 3D (SPHÉRO-CYLINDRIQUE, SPIRALE CONIQUE DE PAPPUS, SPIRALE CONIQUE DE PIRONDINI, SPIRALE SPHÉRIQUE)

Different languages
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You can read this page in Haitian Creole translation. 
You can also read this page in Hindi.
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You can also read this page in Filipino.Archimedes 1 2 1 – Elegant Mathematical Writing Examples
References top
(1) Martin Gardener: Unsere gespiegelte Welt, Ullstein, Berlin, 1982 [ISBN 3-550-07709-2]
(2) Rainer und Patrick Gaitzsch: Computer-Lösungen für Schule und Studium, Band 2, Landsberg am Lech, 1985
(3) Jan Gullberg: Mathematics - From the Birth of Numbers, New York / London (1997) [ISBN 0-393-04002-X]
(4) Khristo N. Boyadzhiev: Spirals and Conchospirals in the Flight of Insects, The College Mathematics Journal, 
 Vol.30, No.1 (Jan.,1999) pp.23-31
(5) Jill Purce: the mystic spiral - Journey of the Soul, Thames and Hudson, 1972, reprinted 1992
Feedback: Email address on my main pageThis page is also available in German
URL of my Homepage:
http://www.mathematische-basteleien.de/
© Jürgen Köller 2002
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