A re-arrangement of a major, minor, dimished, augmented, or any other chord. When dealing with triads, there are 3 ways to arrange the notes ( 1 root position, and 2 inversions ). For our example, we shall use a C major chord.

```            1st 2nd inversions
Root  \/  \/
-----------------------|
E |
-----------------------|
C   C |
-----------------------|
|
-----------G-----G---G-|  - V (Fifth degree)
|
-----------E-----E-----|  - III (Third degree)

-----------C---------     - I (Root Note)```
This is the first, or root, position of the chord. A first inversion would move the root to the top of the chord, like so:
C
G
E

A second inversion would move the third and the root up an octave:
E
C
G

Now, replace the root, third, and fifth with their equivalent members of other scales (dim, aug, etc). 'Hoo ray' for inversions!
An inversion is any element of a roller coaster that changes the train's orientation so the riders are upside down, even if only briefly. Common inversions include the vertical loop, corkscrew, and cobra roll; less-common inversions include the cutback, heartline flip, and Immelman.

The naming of inversions is actually a complex matter because various companies will assign different names to the same element. What Arrow Dynamics calls a boomerang is called a sidewinder by Vekoma; what Vekoma calls a boomerang is called a cobra roll by Intamin and Bolliger & Mabillard and a batwing by Arrow - B&M also has an element called the batwing, but it is not the same as Arrow's nor is it a cobra roll. Vekoma does have an element called the cobra roll, but it doesn't resemble a B&M cobra roll or an Arrow batwing! And so on.

An inversion, as defined by Scott Kim, "is a word or name written so it reads in more than one way." A strict example would be a word that is calligraphed in such a way that it reads the same whether right-side-up or upside-down. But it is also possible to design inversions that read as one word right-side-up and a different word upside-down; or as the same word whether mirrored or not; or the same word readable in multiple directions (the Sun logo could be considered an example of this). There are a limitless number of variations on the theme. Visit Scott Kim's "Inversions" page on the Web at http://www.scottkim.com/inversions/ for more examples.

Inversion is the geometrical process by which points P are transformed to their corresponding inverses P'. Inversion is performed around a circle of inversion, in which all points outside the circle go inside, and all inside go outside. Points on the circle stay put.

To invert a point P outside the circle centered at O with radius r, (you might want to get out a piece of paper for this) draw the line segment OP. Now draw a tangent line to the circle that includes P (call the point it touches the circle T). Finally, draw the line segment OT. The foot of the altitude of OTP from T is the inverted point P' (That is, where the perpendicular to TP intersects OP).

Similarly, to invert a point P inside the circle, first draw the line containing OP. Draw a perpendicular line to OP at P. The point at which this intersects the circle is T. Finally, draw a tangent from the circle at T, and the point where T intersects the line containing OP is P', the inverted point.

Practice it a few times, just to get a feel for it. I'd include a diagram if I could.

Inversion does a lot of cool stuff. For instance, inverting an (infinitely long) line outside the circle of inversion turns it into a circle inside the inversion circle. Also, when inverting anything, intersecting points still intersect. For instance, inverting two intersecting lines gives two tangential circles.

Try inverting a square.

Analytically, inversion about the origin with radius r can be described with the vector equation:

x' = r2x / |x|2    (Eric Weisstein's World of Mathematics, http://mathworld.wolfram.com/Inversion.html)

Naturally, this can be extended to any circle of inversion through simple translation.

If lines are treated as circles of infinite radius, all circles invert to circles.

Many geometrical proofs can be done by inverting the figure in question about a suitable circle of inversion. It can turn massively complex concepts into relatively simple ones.

Inverting a parabola about its focus gives a cardioid. Inverting a logarithmic spiral about its center gives yet another logarithmic spiral.

What happens when you invert a hyperbola?

In addition to Soujirou's writeup about inversions in music, there is a shorthand for writing inversions, and an additional inversion for every extension note added to the chord.
First, this shorthand involves the chord name followed by a subscript number. If the chord is G Major and it is in first inversion, then we can call it a G6, since the third on bottom and root on top makes a sixth, and the third to fifth makes a third, but that is implied, so we simply call it a G6 chord.
The Second inversion can be called a G64, because there is a sixth between the fifth on bottom and the third on top, and a fourth between the fifth on bottom and the root in the middle.
( As a side note, people often make the two numbers vertical instead of side by side.)
In Seventh chords, we get another inversion, because there is another note. Now, first inversions are shortened to 65, to imply the seventh now included... A seventh chord Second inversion is a 653, but shortened to 63. And the new Third inversion is a 643 chord, but shortened to 43.
There is also One new inversion for the Ninth chord, another for the Eleventh, and another for the Thirteenth chord. These may be talked about later.....

An inversion is a section of roller coaster track with a goal of flipping over the rider. Inversion comes from the Latin word invertere meaning "to reverse" or "flip over". How much flipping must take place for an element to be truly considered an inversion is debatable. While some parks consider exceeding 90-degrees to be an inversions, Millennium Force (MF) at Cedar Point (CP) on the other hand, has three sections of track that exceed the 90-degree mark, but the park lists it as having no inversions.

Early Inversion History

The first inversion was on the Centrifugal Railway built in Paris, France in 1848. The ride consisted of a nearly perfectly circular vertical loop. Claims were that it reached speeds in excess of 150 mph, which is clearly an exaggeration.

The vertical loop inversion was greatly refined in 1968 when Karl Bacon of Arrow conceived of the corkscrew. Arrow dared to build a prototype of a corkscrew and the concept of inversion took on new meaning. Besides reinventing the inversion, this prototype introduced the idea of steel tubular track to ease the inversion elements. In 1975 Arrow installed the first three corkscrew coasters located at Knott's Berry Farm, Opryland, and Old Chicago. In 1976 this was followed by Arrow constructing Corkscrew at Cedar Point being the first coaster with three inversions. Intamin, in the same year, working with Anton Schwarzkopf, designed the first modern vertical looping coaster: Great American Revolution, and that it was.

In 1978 Arrow one-upped itself by building Loch Ness Monster at Busch Gardens Williamsburg. Loch Ness Monster debuted with two interlocking vertical loops. Schwarzkopf answered this in the same year by building two intense (at the time) double-looping coasters: Shockwave at Six Flags Over Texas and Mind Bender at Six Flags Over Georgia. What no one realized is the eighties were going to be an entire decade of "one-upping".

Highlights of early advances in inversions
1848 - Centrifugal Railway was the first inversion
1968 - Prototype corkscrew built
1975 - First three coskscrews installed by Arrow
1976 - Corkscrew built at CP setting new inversion record (3) and the first modern vertical looper: Great American Revolution by Schwarzkopf.
1978 - Loch Ness Monster by Arrow with interlocking vertical loops
1978 - Schwarzkopf opened two double loopers: Shockwave and Mind Bender

Inversion History of the 1980s

In 1980 the first four-inversion coasters appeared. Two vertical loops were added to the popular Turn of the Century corkscrew coasters at both Great America parks bringing their totals to four inversions. They were also both renamed Demon at this time. The Carolina Cyclone, built in 1980, featured two vertical loops and two corkscrews, similar to the newly enhanced Demons. The other four-inversion coaster to open that year was the Orient Express at Worlds of Fun. It featured the world's first two-inversion element: The Kamikaze Curve (now known as an Arrow Boomerang). It was Orient Express's third and fourth inversions.

In 1981, Vekoma introduced the Boomerang coaster and element, which differs a great deal from the Arrow element of the same name. The first Vekoma Boomerang was Escorpion at Reino Aventura. Since then, this type of coaster has become the most cloned coaster ever, also called an off the shelf coaster.

In 1982 Arrow one-upped to the five-inversion coaster Viper, at Darien Lake, with a vertical loop, a 2-inversion boomerang, and a double corkscrew.

1983 was the first year that coaster enthusiasts could invert standing up. On May 31, 1983, Worlds of Fun put stand-up trains on Screamroller, their Arrow Corkscrew coaster, and renamed it Extremeroller.

The first barrel roll inversion, and pipeline-like coaster, came to be in 1985 when TOGO built the Ultra Twister at Nagashima Spaland.

In 1986 the Schwarzkopf looper Thriller was built. It featured circular loops, and G forces of up to 6.7 G. Most notably aside from the g-forces, Thriller is the world's largest portable coaster, and, sadly, the last independent Schwarzkopf design. Thriller was toured around Germany until 1998 and now is a permanent fixture at Six Flags Astroworld as Taz's Texas Tornado. 1986 also saw the advent of the new element: the Arrow sidewinder. It was not technically new, but was new on its own merit using one half of a boomerang. This element debuted on Scream Machine at Expo '86 but has found a home as Ninja at Six Flags St. Louis.

In 1987 we skipped to "two-upping" when the revolutionary Vortex was opened at Paramount's Kings Island (PKI). Vortex shocked riders by being the first coaster to feature six inversions. Arrow was happy to up themselves the following year in 1988 when they built not only the first seven inversion coaster, but at the time, the world's tallest roller coaster: Shockwave at Six Flags Great America (SFGAm).

Vekoma joined the inversion game in 1989 when they built the seven time inverting Goudurix at Parc Asterix. It not only tied the world record for inversions, but also had a new inversion element: the butterfly. This element begins like a normal loop, but on the way up it twists slightly to one side. Then, on the way down, it twists again and crosses over its entry. Then the element is repeated in reverse. In the same year Schwarzkopf returned and designed Olympia Looping featuring five vertical loops, more than any other roller coaster. With the fury of the eighties numbers game closing the nineties would see inversions taking twists (pun intended) beyond imagination.

Highlights of inversion advancements in the 1980s
1980 - First four inversion coasters: Demon
1981 - Vekoma Boomerang premiered
1982 - Arrow opened first five inversion coaster - Viper
1983 - First stand-up loop: Extremeroller1985 - First barrel roll: Ultra Twister
1986 - Debut of Arrow Sidewinder element: Ninja
1987 - First six-inversion coaster: Vortex
1988 - Tallest and first seven-inversion coaster: Shockwave
1989 - Vekoma Butterfly element introduced: Goudurix

Inversion History of the 1990s

In 1990, Walter Bolliger and Claude Mabillard (B&M) broke off from their parent company, Intamin. This was the start of something wonderful for coaster riders everywhere. In the years to come, B&M would become famous for their very large and smooth inversion elements.

In 1992 the concept of inversion was revolutionized, as were coasters in general with the advent of the inverted coaster. B&M built the first inverted roller coaster, Batman The Ride, at Six Flags Great America (SFGAm). This introduced the first outside loops, the first in-line twist, and the first outside corkscrews called Wingovers. Also in 1992, Arrow designed Drachen Fire at Busch Gardens Williamsburg featuring three unique elements: a wraparound corkscrew on the first drop, a batwing, and a cutback, which is like a corkscrew with its second half reversed creating a 180ยบ turn.

Busch Gardens, wanting not to lose out, began designing an inverting marvel for their park in Tampa Bay, Florida. In 1993 they built one of the most revolutionary coasters to ever be designed: Kumba. Off the top of the lift, Kumba has a curving first drop straight into a loop around the lift hill. Next it roars into a diving loop, an element in which the train dives sideways into a huge loop. After that, it has a camelback, the element Busch wanted to include on Drachen Fire. Following that, riders fly into a cobra roll, (a B&M version of the Arrow batwing) and a pair of interlocking flat spins (the B&M name for a corkscrew). Kumba introduced the world to the largest vertical loop at the time as well as the first diving loop, camelback, cobra roll, and interlocking flat spins.

In 1994 B&M tackled inverted coasters and inversion with the beast of a coaster at Cedar Point known as Raptor. Raptor features a vertical loop, in-line twist, two flat spins, and the first Cobra Roll on an inverted coaster. Also in 1994, Intamin debuted 7up Shockwave at Drayton Manor. This took the record as the first stand-up coaster with four inversions, and the first stand-up coaster to feature a double corkscrew. Furthermore, this Shockwave coaster was and still is the only stand-up coaster to feature a revolution (Intamin's name for a camelback).

In 1995 the last inversion record breaker of the 20th century came out thanks to B&M with Dragon Khan. This massive flip-a-thon clocks in at eight inversions. Dragon Khan has the same inversions as Kumba, save for an extra vertical loop inserted between the cobra roll and interlocking flat spins. Intamin also built their first pipeline coaster in 1995: Spiral Coaster at Lotte World Sky Plaza featuring two diving loops and two barrel rolls.

Intamin tried these new elements on steel loopers and produced Lethal Weapon Pursuit at Warner Brothers Movie World in 1996. This dual track coaster begins on both tracks with vertical loops. Their second inversions, however, are both Zero-G-Heart Rolls, similar to heartline rolls. The two Heart Rolls are side by side, and the trains roll towards each other. Also in 1996 B&M set the new largest vertical loop record on their stand-up mammoth Mantis at Cedar Point. B&M introduced new inversions this same year on Montu an Immelman Turn and a B&M Batwing. Montu also clinched the inversion record for inverted coasters with seven.

In 1997 B&M built a two more record breaking coasters with Chang with the stand-up inversion record of five and Alpengeist with the world's tallest inverted coaster, tallest outside loop, and largest Cobra Roll.

Premier refreshed the inversion market in 1998 with the Top Hat element. This element was included on the two Mr. Freeze coasters rides built in 1997 and opened in 1998 due to delays. Dragon Kahn went for three years without a match to its inversion record until Intamin built Monte Makaya. Its eight inversions are, respectively: an 82-foot vertical loop, a double inversion cobra roll, a double corkscrew, and three zero-g-heart rolls.

The year 1999 can make inversion lovers weep. It was a weak year compared to the preceding score. One coaster of note is Dueling Dragons at Universal Studios Islands of Adventure (IOA). The vertical loops on the "dueling" tracks race toward each other before looping missing each other by less than 20 inches . Also in 1999, Vekoma and Disney's Rock 'n Roller Coaster featured the first Roll-over. This element resembles the Vekoma boomerang except that the second half is mirrored.

Highlights of inversion advancements in the 1990s
1990 - B&M broke off from Intamin
1992 - First inverted coaster and inverted looped: Batman the Ride
1992 - Debut of the Arrow batwing and cutback elements: Drachen Fire
1993 - First camelback (and other element introductions): Kumba
1994 - First inverted cobra roll: Raptor
1996 - First immelmann turn and B&M batwing: Montu
1997 - B&M's record holding tallest inverted coaster, tallest outside loop, and largest cobra roll: Alpengeist
1998 - First Top Hat element: Mr. Freeze
1998 - Eight inversion record holder: Monte Makaya

As the new century began the wooden loop reappeared for the first time in almost 70 years. Son of Beast at Paramount's Kings Island (PKI) premiered as the world's tallest, fastest, and only looping wooden roller coaster with the vertical loop measuring 118 feet. B&M's Medusa at Six Flags Marine World opened in 2000 with the first Sea Serpent element basically identical to a Vekoma Roll-over.

Record Holders for Most Inversions
(Coaster - designer - park - year opened - # of inversions)
Colossus - Intamin - Thorpe Park - 2002 - 10
Avalancha - Intamin - Xetulul - 2002 - 8
Dragon Kahn - B&M - Prot Aventura - 1995 - 8
Monte Makaya - Intamin - Terra Encantada - 1998 - 8
Goudurix - Vekoma - Parc Asterix - 1989 - 7
[Great American Scream Machine - Arrow - Six Flags Great Adventure - 1999 - 7
Incredible Hulk - B&M - Universal Islands of Adventure - 1999 - 7
Kraken - B&M - Sea World of Florida - 1999 - 7
Kumba - B&M - Busch Gardens Tampa - 1993 - 7
Medusa - B&M - Six Flags Great America - 1999 - 7
Medusa - B&M - Six Flags Marine World - 2000 - 7
Shockwave - Arrow - Six Flags Great America - 1988 - 7
Superman - B&M - Warner Brothers Movie World - 2002 - 7
Viper - Arrow - Six Flags Magic Mountain - 1990 - 7

Everyone from riders to designers knows the surface is only scratched. Innovations in the coaster world are endless. Inversions will continue to be refined not only in steel and wooden basic structures but with the tons of new coaster types constantly being developed like the vertical coaster and 4-D coaster. Next time you find yourself upside down, in midst of scream, remember the wonders of Intamin, B&M, Vekoma, Arrow, and especially Anton Schwarzkopf that got you there. Well, you may be too preoccupied to remember the history behind your head-over-heels experience. Either way, enjoy!

Resources include: Coaster Globe, roller coaster database, rec.roller-coasters, and many of tunnel vision runs through nearly every inversion element available.

In*ver"sion (?), n. [L. inversio: cf. F. inversion. See Invert.]

1.

The act of inverting, or turning over or backward, or the state of being inverted.

2.

A change by inverted order; a reversed position or arrangement of things; transposition.

It is just the inversion of an act of Parliament; your lordship first signed it, and then it was passed among the Lords and Commons. Dryden.

3. Mil.

A movement in tactics by which the order of companies in line is inverted, the right being on the left, the left on the right, and so on.

4. Math.

A change in the order of the terms of a proportion, so that the second takes the place of the first, and the fourth of the third.

5. Geom.

A peculiar method of transformation, in which a figure is replaced by its inverse figure. Propositions that are true for the original figure thus furnish new propositions that are true in the inverse figure. See Inverse figures, under Inverse.

6. Gram.

A change of the usual order of words or phrases; as, "of all vices, impurity is one of the most detestable," instead of, "impurity is one of the most detestable of all vices."

7. Rhet.

A method of reasoning in which the orator shows that arguments advanced by his adversary in opposition to him are really favorable to his cause.

8. Mus. (a)

Said of intervals, when the lower tone is placed an octave higher, so that fifths become fourths, thirds sixths, etc.

(b)

Said of a chord, when one of its notes, other than its root, is made the bass.

(c)

Said of a subject, or phrase, when the intervals of which it consists are repeated in the contrary direction, rising instead of falling, or vice versa.

(d)

Said of double counterpoint, when an upper and a lower part change places.

9. Geol.

The folding back of strata upon themselves, as by upheaval, in such a manner that the order of succession appears to be reversed.

10. Chem.

The act or process by which cane sugar (sucrose), under the action of heat and acids or ferments (as diastase), is broken or split up into grape sugar (dextrose), and fruit sugar (levulose); also, less properly, the process by which starch is converted into grape sugar (dextrose).

⇒ The terms invert and inversion, in this sense, owe their meaning to the fact that the plane of polarization of light, which is rotated to the right by cane sugar, is turned toward the left by levulose.

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