The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line that rotates with constant angular velocity. Equivalently, in polar coordinates $(r, θ)$ it can be described by the equation

r=b\cdot \theta

with real number

b

. Changing the parameter

b

controls the distance between loops.

From the above equation, it can thus be stated: position of particle from point of start is proportional to angle $θ$ as time elapses.

Archimedes described such a spiral in his book On Spirals. Conon of Samos was a friend of his and Pappus states that this spiral was discovered by Conon.^[1]

Derivation of general equation of spiral[edit]

A physical approach is used below to understand the notion of Archimedean spirals.

Suppose a point object moves in the Cartesian system with a constant velocity $v$ directed parallel to the $x$ -axis, with respect to the $xy$ -plane. Let at time $t = 0$ , the object was at an arbitrary point $(c, 0, 0)$ . If the $xy$ plane rotates with a constant angular velocity $ω$ about the $z$ -axis, then the velocity of the point with respect to $z$ -axis may be written as:

{\begin{aligned}|v_{0}|&={\sqrt {v^{2}+\omega ^{2}(vt+c)^{2}}}\\v_{x}&=v\cos \omega t-\omega (vt+c)\sin \omega t\\v_{y}&=v\sin \omega t+\omega (vt+c)\cos \omega t\end{aligned}}

Here $vt + c$ is the modulus of the position vector of the particle at any time $t$ , $v x$ is the velocity component along the $x$ -axis and $v y$ is the component along the $y$ -axis. The figure shown alongside explains this.

{\begin{aligned}\int v_{x}\,dt&=x\\\int v_{y}\,dt&=y\end{aligned}}

The above equations can be integrated by applying integration by parts, leading to the following parametric equations:

{\begin{aligned}x&=(vt+c)\cos \omega t\\y&=(vt+c)\sin \omega t\end{aligned}}

Squaring the two equations and then adding (and some small alterations) results in the Cartesian equation

{\sqrt {x^{2}+y^{2}}}={\frac {v}{\omega }}\cdot \arctan {\frac {y}{x}}+c

(using the fact that

ωt = θ

and

θ = arctan y / x

) or

\tan \left(\left({\sqrt {x^{2}+y^{2}}}-c\right)\cdot {\frac {\omega }{v}}\right)={\frac {y}{x}}

Its polar form is

r={\frac {v}{\omega }}\cdot \theta +c.

Arc length and curvature [edit]

Given the parametrization in cartesian coordinates

f\colon \theta \mapsto (r\,\cos \theta ,r\,\sin \theta )=(b\,\theta \,\cos \theta ,b\,\theta \,\sin \theta )

the arc length from

θ 1

to

θ 2

is

{\frac {b}{2}}\left[\theta \,{\sqrt {1+\theta ^{2}}}+\ln \left(\theta +{\sqrt {1+\theta ^{2}}}\right)\right]_{\theta _{1}}^{\theta _{2}}

or, equivalently:

{\frac {b}{2}}\left[\theta \,{\sqrt {1+\theta ^{2}}}+\operatorname {arsinh} \theta \right]_{\theta _{1}}^{\theta _{2}}.

The total length from

θ 1 = 0

to

θ 2 = θ

is therefore

{\frac {b}{2}}\left[\theta \,{\sqrt {1+\theta ^{2}}}+\ln \left(\theta +{\sqrt {1+\theta ^{2}}}\right)\right].

The curvature is given by

\kappa ={\frac {\theta ^{2}+2}{b\left(\theta ^{2}+1\right)^{\frac {3}{2}}}}

Characteristics[edit]

The Archimedean spiral has the property that any ray from the origin intersects successive turnings of the spiral in points with a constant separation distance (equal to $2 πb$ if $θ$ is measured in radians), hence the name "arithmetic spiral". In contrast to this, in a logarithmic spiral these distances, as well as the distances of the intersection points measured from the origin, form a geometric progression.

The Archimedean spiral has two arms, one for $θ > 0$ and one for $θ < 0$ . The two arms are smoothly connected at the origin. Only one arm is shown on the accompanying graph. Taking the mirror image of this arm across the $y$ -axis will yield the other arm.

For large $θ$ a point moves with well-approximated uniform acceleration along the Archimedean spiral while the spiral corresponds to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity^[2] (see contribution from Mikhail Gaichenkov).

As the Archimedean spiral grows, its evolute asymptotically approaches a circle with radius $| v | / ω$ .

General Archimedean spiral[edit]

Sometimes the term Archimedean spiral is used for the more general group of spirals

r=a+b\cdot \theta ^{\frac {1}{c}}.

The normal Archimedean spiral occurs when $c = 1$ . Other spirals falling into this group include the hyperbolic spiral ( $c = -1$ ), Fermat's spiral ( $c = 2$ ), and the lituus ( $c = -2$ ).

Applications[edit]

One method of squaring the circle, due to Archimedes, makes use of an Archimedean spiral. Archimedes also showed how the spiral can be used to trisect an angle. Both approaches relax the traditional limitations on the use of straightedge and compass in ancient Greek geometric proofs.^[3]

The Archimedean spiral has a variety of real-world applications. Scroll compressors, used for compressing gases, have rotors that can be made from two interleaved Archimedean spirals, involutes of a circle of the same size that almost resemble Archimedean spirals,^[4] or hybrid curves.

Archimedean spirals can be found in spiral antenna, which can be operated over a wide range of frequencies.

The coils of watch balance springs and the grooves of very early gramophone records form Archimedean spirals, making the grooves evenly spaced (although variable track spacing was later introduced to maximize the amount of music that could be cut onto a record).^[5]

Asking for a patient to draw an Archimedean spiral is a way of quantifying human tremor; this information helps in diagnosing neurological diseases.

Archimedean spirals are also used in digital light processing (DLP) projection systems to minimize the "rainbow effect", making it look as if multiple colors are displayed at the same time, when in reality red, green, and blue are being cycled extremely quickly.^[6] Additionally, Archimedean spirals are used in food microbiology to quantify bacterial concentration through a spiral platter.^[7]

You searched for

"WIND" in the KJV Bible

Related words:

Winding Winds Windy

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Psalms 78:26chapter context similar meaning copy save: He caused an east wind to blow in the heaven: and by his power he brought in the south wind.
Isaiah 27:8chapter context similar meaning copy save: In measure, when it shooteth forth, thou wilt debate with it: he stayeth his rough wind in the day of the east wind.
Hosea 12:1chapter context similar meaning copy save: Ephraim feedeth on wind, and followeth after the east wind: he daily increaseth lies and desolation; and they do make a covenant with the Assyrians, and oil is carried into Egypt.
Mark 4:39chapter context similar meaning copy save: And he arose, and rebuked the wind, and said unto the sea, Peace, be still. And the wind ceased, and there was a great calm.
Ecclesiastes 1:6chapter context similar meaning copy save: The wind goeth toward the south, and turneth about unto the north; it whirleth about continually, and the wind returneth again according to his circuits.
1 Kings 19:11chapter context similar meaning copy save: And he said, Go forth, and stand upon the mount before the LORD. And, behold, the LORD passed by, and a great and strong wind rent the mountains, and brake in pieces the rocks before the LORD; but the LORD was not in the wind: and after the wind an earthquake; but the LORD was not in the earthquake:
Ezekiel 37:9chapter context similar meaning copy save: Then said he unto me, Prophesy unto the wind, prophesy, son of man, and say to the wind, Thus saith the Lord GOD; Come from the four winds, O breath, and breathe upon these slain, that they may live.
Hosea 13:15chapter context similar meaning copy save: Though he be fruitful among his brethren, an east wind shall come, the wind of the LORD shall come up from the wilderness, and his spring shall become dry, and his fountain shall be dried up: he shall spoil the treasure of all pleasant vessels.
Exodus 10:13chapter context similar meaning copy save: And Moses stretched forth his rod over the land of Egypt, and the LORD brought an east wind upon the land all that day, and all that night; and when it was morning, the east wind brought the locusts.
Psalms 48:7chapter context similar meaning copy save: Thou breakest the ships of Tarshish with an east wind.
John 6:18chapter context similar meaning copy save: And the sea arose by reason of a great wind that blew.
Job 7:7chapter context similar meaning copy save: O remember that my life is wind: mine eye shall no more see good.
Proverbs 25:23chapter context similar meaning copy save: The north wind driveth away rain: so doth an angry countenance a backbiting tongue.
Proverbs 25:14chapter context similar meaning copy save: Whoso boasteth himself of a false gift is like clouds and wind without rain.
Acts 27:14chapter context similar meaning copy save: But not long after there arose against it a tempestuous wind, called Euroclydon.
Job 6:26chapter context similar meaning copy save: Do ye imagine to reprove words, and the speeches of one that is desperate, which are as wind?
Job 30:15chapter context similar meaning copy save: Terrors are turned upon me: they pursue my soul as the wind: and my welfare passeth away as a cloud.
Job 15:2chapter context similar meaning copy save: Should a wise man utter vain knowledge, and fill his belly with the east wind?
Matthew 14:30chapter context similar meaning copy save: But when he saw the wind boisterous, he was afraid; and beginning to sink, he cried, saying, Lord, save me.
Jeremiah 4:12chapter context similar meaning copy save: Even a full wind from those places shall come unto me: now also will I give sentence against them.
Psalms 83:13chapter context similar meaning copy save: O my God, make them like a wheel; as the stubble before the wind.
Job 30:22chapter context similar meaning copy save: Thou liftest me up to the wind; thou causest me to ride upon it, and dissolvest my substance.
Matthew 14:32chapter context similar meaning copy save: And when they were come into the ship, the wind ceased.
Psalms 148:8chapter context similar meaning copy save: Fire, and hail; snow, and vapour; stormy wind fulfilling his word:
Job 37:17chapter context similar meaning copy save: How thy garments are warm, when he quieteth the earth by the south wind?
Psalms 107:25chapter context similar meaning copy save: For he commandeth, and raiseth the stormy wind, which lifteth up the waves thereof.
Job 21:18chapter context similar meaning copy save: They are as stubble before the wind, and as chaff that the storm carrieth away.
Hosea 4:19chapter context similar meaning copy save: The wind hath bound her up in her wings, and they shall be ashamed because of their sacrifices.
Genesis 41:6chapter context similar meaning copy save: And, behold, seven thin ears and blasted with the east wind sprung up after them.
Luke 12:55chapter context similar meaning copy save: And when ye see the south wind blow, ye say, There will be heat; and it cometh to pass.

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You searched for

"NOISE" in the KJV Bible

Related words:

Noised Noisome

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Ezekiel 3:13chapter context similar meaning copy save: I heard also the noise of the wings of the living creatures that touched one another, and the noise of the wheels over against them, and a noise of a great rushing.
Ezra 3:13chapter context similar meaning copy save: So that the people could not discern the noise of the shout of joy from the noise of the weeping of the people: for the people shouted with a loud shout, and the noisewas heard afar off.
2 Kings 7:6chapter context similar meaning copy save: For the Lord had made the host of the Syrians to hear a noise of chariots, and a noise of horses, even the noise of a great host: and they said one to another, Lo, the king of Israel hath hired against us the kings of the Hittites, and the kings of the Egyptians, to come upon us.
Exodus 32:17chapter context similar meaning copy save: And when Joshua heard the noise of the people as they shouted, he said unto Moses, There is a noise of war in the camp.
Ezekiel 1:24chapter context similar meaning copy save: And when they went, I heard the noise of their wings, like the noise of great waters, as the voice of the Almighty, the voice of speech, as the noise of an host: when they stood, they let down their wings.
Jeremiah 49:21chapter context similar meaning copy save: The earth is moved at the noise of their fall, at the cry the noise thereof was heard in the Red sea.
Isaiah 13:4chapter context similar meaning copy save: The noise of a multitude in the mountains, like as of a great people; a tumultuous noise of the kingdoms of nations gathered together: the LORD of hosts mustereth the host of the battle.
Psalms 65:7chapter context similar meaning copy save: Which stilleth the noise of the seas, the noise of their waves, and the tumult of the people.
Psalms 98:4chapter context similar meaning copy save: Make a joyful noise unto the LORD, all the earth: make a loud noise, and rejoice, and sing praise.
Joel 2:5chapter context similar meaning copy save: Like the noise of chariots on the tops of mountains shall they leap, like the noise of a flame of fire that devoureth the stubble, as a strong people set in battle array.
1 Samuel 4:14chapter context similar meaning copy save: And when Eli heard the noise of the crying, he said, What meaneth the noise of this tumult? And the man came in hastily, and told Eli.
Nahum 3:2chapter context similar meaning copy save: The noise of a whip, and the noise of the rattling of the wheels, and of the pransing horses, and of the jumping chariots.
Isaiah 17:12chapter context similar meaning copy save: Woe to the multitude of many people, which make a noise like the noise of the seas; and to the rushing of nations, that make a rushing like the rushing of mighty waters!
1 Samuel 4:6chapter context similar meaning copy save: And when the Philistines heard the noise of the shout, they said, What meaneth the noise of this great shout in the camp of the Hebrews? And they understood that the ark of the LORD was come into the camp.
Psalms 33:3chapter context similar meaning copy save: Sing unto him a new song; play skilfully with a loud noise.
Psalms 100:1chapter context similar meaning copy save: (A Psalm of praise.) Make a joyful noise unto the LORD, all ye lands.
Psalms 66:1chapter context similar meaning copy save: (To the chief Musician, A Song or Psalm.) Make a joyful noise unto God, all ye lands:
Jeremiah 46:17chapter context similar meaning copy save: They did cry there, Pharaoh king of Egypt is but a noise; he hath passed the time appointed.
Psalms 55:2chapter context similar meaning copy save: Attend unto me, and hear me: I mourn in my complaint, and make a noise;
Psalms 98:6chapter context similar meaning copy save: With trumpets and sound of cornet make a joyful noise before the LORD, the King.
Psalms 59:6chapter context similar meaning copy save: They return at evening: they make a noise like a dog, and go round about the city.
Psalms 95:2chapter context similar meaning copy save: Let us come before his presence with thanksgiving, and make a joyful noise unto him with psalms.
Job 36:29chapter context similar meaning copy save: Also can any understand the spreadings of the clouds, or the noise of his tabernacle?
Job 37:2chapter context similar meaning copy save: Hear attentively the noise of his voice, and the sound that goeth out of his mouth.
Psalms 95:1chapter context similar meaning copy save: O come, let us sing unto the LORD: let us make a joyful noise to the rock of our salvation.
Amos 5:23chapter context similar meaning copy save: Take thou away from me the noise of thy songs; for I will not hear the melody of thy viols.
Jeremiah 50:46chapter context similar meaning copy save: At the noise of the taking of Babylon the earth is moved, and the cry is heard among the nations.
Isaiah 33:3chapter context similar meaning copy save: At the noise of the tumult the people fled; at the lifting up of thyself the nations were scattered.
Isaiah 9:5chapter context similar meaning copy save: For every battle of the warrior is with confused noise, and garments rolled in blood; but this shall be with burning and fuel of fire.
Psalms 42:7chapter context similar meaning copy save: Deep calleth unto deep at the noise of thy waterspouts: all thy waves and thy billows are gone over me.

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Bell

Hiss

Tophet

Samaria

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[1]

[2]

[3]

[4]

[5]

[6]

[7]

Sequoioideae Temporal range: Jurassic–Present PreꞒ Ꞓ O S D C P T J K Pg N

The trunk of a Metasequoia glyptostroboides
Scientific classification
Kingdom:	Plantae
Clade:	Tracheophytes
Clade:	Gymnospermae
Division:	Pinophyta
Class:	Pinopsida
Order:	Cupressales
Family:	Cupressaceae
Subfamily:	Sequoioideae
Genera
Sequoia Sequoiadendron Metasequoia †Austrosequoia'

UNESCO World Heritage Site
Depiction of aurochs, horses and deer
Location	Montignac, France
Part of	Prehistoric Sites and Decorated Caves of the Vézère Valley
Criteria	Cultural: i, iii
Reference	85-011
Inscription	1979 (3rd Session)
Area	34.34 ha (84+7⁄8 acres)
Coordinates	45°03′13″N 1°10′12″E
Location in the Dordogne Show map of DordogneShow map of FranceShow all

Lascaux
Wikimedia \| © OpenStreetMap
Coordinates	45°03′13″N 1°10′12″E Lascaux (English: /læˈskoʊ/ la-SKOH,^[1] US also /lɑːˈskoʊ/ lah-SKOH;^[2] French: Grotte de Lascaux [ɡʁɔt dəlasko],^[3] "Lascaux Cave") is a network of caves near the village of Montignac, in the department of Dordogne in southwestern France. Over 600 parietal wall paintings cover the interior walls and ceilings of the cave. The paintings represent primarily large animals, typical local contemporary fauna that correspond with the fossil record of the Upper Paleolithic in the area. They are the combined effort of many generations and, with continued debate, the age of the paintings is now usually estimated at around 17,000 years (early Magdalenian).^[4]^[5]^[6] Because of the outstanding prehistoric art in the cave, Lascaux was inducted into the UNESCO World Heritage List in 1979, as an element of the Prehistoric Sites and Decorated Caves of the Vézère Valley.^[7] The original caves have been closed to the public since 1963, as their condition was deteriorating, but there are now a number of replicas. History since rediscovery[edit] Modern entrance to the Lascaux cave On 12 September 1940, the entrance to the Lascaux Cave was discovered on the La Rochefoucauld-Montbel lands by 18-year-old Marcel Ravidat when his dog, Robot, investigated a hole left by an uprooted tree (Ravidat would embellish the story in later retellings, saying Robot had fallen into the cave.)^[8]^[9]Ravidat returned to the scene with three friends, Jacques Marsal, Georges Agnel, and Simon Coencas. They entered the cave through a 15-metre-deep (50-foot) shaft that they believed might be a legendary secret passage to the nearby Lascaux Manor.^[9]^[10]^[11] The teenagers discovered that the cave walls were covered with depictions of animals.^[12]^[13] Galleries that suggest continuity, context or simply represent a cavern were given names. Those include the Hall of the Bulls, the Passageway, the Shaft, the Nave, the Apse, and the Chamber of Felines. They returned along with the Abbé Henri Breuil on 21 September 1940; Breuil would make many sketches of the cave, some of which are used as study material today due to the extreme degradation of many of the paintings. Breuil was accompanied by Denis Peyrony, curator of Les eyzies (Prehistory Museum) at Les Eyzies, Jean Bouyssonie and Dr Cheynier. The cave complex was opened to the public on 14 July 1948, and initial archaeological investigations began a year later, focusing on the Shaft. By 1955, carbon dioxide, heat, humidity, and other contaminants produced by 1,200 visitors per day had visibly damaged the paintings. As air condition deteriorated, fungi and lichen increasingly infested the walls. Consequently, the cave was closed to the public in 1963, the paintings were restored to their original state, and a monitoring system on a daily basis was introduced. Replicas[edit] Part of Lascaux IV Conservation problems in the original cave have made the creation of replicas more important. Lascaux II[edit] Lascaux II, an exact copy of the Great Hall of the Bulls and the Painted Gallery was displayed at the Grand Palais in Paris, before being displayed from 1983 in the cave's vicinity (about 200 m or 660 ft away from the original cave), a compromise and attempt to present an impression of the paintings' scale and composition for the public without harming the originals.^[10]^[13] A full range of Lascaux's parietal art is presented a few kilometres from the site at the Centre of Prehistoric Art, Le Parc du Thot, where there are also live animals representing ice-age fauna.^[14] The paintings for this site were duplicated with the same type of materials (such as iron oxide, charcoal, and ochre) which were believed to be used 19,000 years ago.^[9]^[15]^[16]^[17] Other facsimiles of Lascaux have also been produced over the years. Lascaux III[edit] Lascaux III is a series of five exact reproductions of the cave art (the Nave and Shaft) that, since 2012, have traveled around the world allowing knowledge of Lascaux to be shared far from the original. Lascaux IV[edit] Lascaux IV is a new copy of all the painted areas of the cave that forms part of the International Centre for Parietal Art (Centre International de l'Art Pariétal). Since December 2016 this larger and more accurate replica which integrates digital technology into the display is presented in a new museum built by Snøhetta inside the hill overlooking Montignac.^[18]^[19]

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Archimedean spiral

Derivation of general equation of spiral[edit]

Arc length and curvature [edit]

Characteristics[edit]

General Archimedean spiral[edit]

Applications[edit]

You searched for

"WIND" in the KJV Bible

You searched for

"NOISE" in the KJV Bible

Related Bible Terms

Sequoioideae

Description[edit]

Zodiac

Name

Lascaux

History since rediscovery[edit]

Replicas[edit]

Lascaux II[edit]

Lascaux III[edit]

Lascaux IV[edit]

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