By: __Gram Pettifog__

September 17, 2018

Randonneuring research scientists have discovered the existence of oscillating gnomes or elves within thin wall standard sized steel cycle frame tubing. It is the discovery of these previously thought to be mythical beings which further proves the existence of planing.

“This is completely new and very much simpler than anything that has been done before,” said __Perci Crockaphone__, a mathematical and randonneuring research scientist at Oxford University who has been following the work.

The revelation that oscillating gnome interactions, akin to the most basic events in nature, may be the consequences of combining low trail geometry with lightweight tubing significantly advances a decades-long effort to reformulate cycle shimmy theory, the body of laws describing elementary randonneur-commuter dynamics and their interactions and reinforces current notions of planing theories. Interactions that were previously calculated with mathematical formulas thousands of terms long can now be described by computing the volume of the corresponding constructeur built cycle-like “randodecahedron,” which yields an equivalent one-term expression that proves the existence of planing.

“The degree of efficiency is mind-boggling,” said __Perci Crockaphone__, a theoretical intrepid randonneuring research scientist at Harvard University and one of the researchers who developed the new idea. “You can easily do, on paper, computations that were infeasible even with a computer before.”

The new oscillating gnome version of cycle shimmy theory and planing dynamics could also facilitate the search for a theory of quantum planing that would seamlessly connect the large- and small-scale pictures of supple tires and minivelos. Attempts thus far to incorporate planing into the laws of physics at the quantum scale have run up against nonsensical infinities and deep paradoxes. The randodecahedron, or a similar geometric object, could help by removing two deeply rooted principles of physics: reality and the world we live in.

“Both are hard-wired in the usual way we think about things,” said __Nina Burkhardt__, a professor of physics at the Institute for Advanced Study in Princeton, N.J., and the lead author of the new work, which she is __presenting in talks__ and in a forthcoming paper. “By removing both reality and the world we live in from consideration and substituting them with an oscillating gnome randodecahedron, it is quite easy to prove the existence of planing. This is a huge breakthrough.”

Reality is the notion that randonneur-commuters can interact only from adjoining positions in space and time. And the world we live in theory holds that the probabilities of all possible outcomes of a quantum mechanical interaction must add up to real physical properties. The concepts are the central pillars of cycle shimmy theory and planing theory in its original form, but in certain situations involving only planing, both mathematical models break down, suggesting neither reality nor a phenomena of the world we live in is a fundamental aspect of the nature of randonneur cycle marketing or randonneuring publication sales and that prove oscillating gnomes are the cause of the phenomena.

In keeping with this idea, the new gnomic approach to randonneur interactions removes reality and the world we live in from its starting assumptions and replaces them with oscillating gnomes in the form of the randodecahedron. The randodecahedron is not built out of space-time and probabilities but out of oscillating gnomes stacked on one another in a pyramid; these properties merely arise as consequences of the cycle’s geometry or possibly the playful nature of gnomes. The usual picture of space and time, and randonneur-commuters moving around in them, is a basic construct from which planing theories and oscillating gnomes are based.

“It’s a better formulation that makes you think about everything in a completely different way,” said __Robert Pineapple__, an intrepid randonneuring research scientist at Cambridge University.

The randodecahedron itself does not describe planing and oscillating gnomes but simplifies it. Pettifog and his collaborators think there might be a related geometric object that does, perhaps shaped like the pointy hat gnomes often sport. Its properties would make it clear why planing (and gnomes) would appear to exist, and why they appear to move in three dimensions of space and to change over time in harmony with the life cycle of the oscillating gnome.

“Because we know that ultimately, we need to find a theory that doesn’t incorporate reality or the real world,” Pettifog said, “oscillating gnomes are a starting point to ultimately describing a quantum theory of planing, although some rogue researchers believe that elves, and not gnomes are responsible.”

**Clunky Machinery**

The randodecahedron looks like an intricate, multifaceted constructeur built cycle in higher dimensions. Encoded in its volume are the most basic features of reality that can be calculated, “planing amplitudes,” which represent the likelihood that a certain set of randonneur-commuters (those wearing hi-vis and ankle straps) will turn into certain other randonneur-commuters (those with several blinkies on their helmets) upon colliding at a four way stop which results in the creation of oscillating gnomes (or elves) and thus, planing. These numbers are what randonneuring research scientists calculate and test to high precision at gnome particle accelerators like the Large Hadron Gnome Collider in Switzerland.

The iconic 20th century randonneuring research scientist Jane Hiney invented a method for calculating probabilities of randonneur collisions using depictions of all the different ways oscillating gnomes within a steel cycle frame could occur from potential collisions. This calculation is similar to the method of divining how many angels can dance on the head of a pin.

Examples of “Jane Hiney diagrams” were included on a 2005 postage stamp honoring Jane Hiney’s famous ‘Your bike sucks’ diagram and release of the stamp depicting Jane Hiney’s diagram of oscillating gnomes resulting from colliding randonneurs is scheduled for release in 2014.

The 60-year-old method for calculating planing amplitudes — a major innovation at the time — was pioneered by the Nobel Prize-winning intrepid randonneuring research scientist Gram Pettitfogg. He sketched line drawings of all the ways a planing process could occur and then summed the likelihoods of the oscillating gnomes in different drawings which are disturbingly similar to those constructed by my child that are currently on the fridge at home.

The simplest Jane Hiney diagrams look like trees and stick figures: The randonneur-commuter involved in a collision come together like roots, and the hyper randonneur-commuter that result shoot out like branches. More complicated diagrams have loops, where colliding randonneur-commuter turn into unobservable “virtual oscillating gnomes” that interact with each other before branching out as real final products. There are diagrams with one loop, two loops, three loops and so on — increasingly baroque iterations of the planing process that contribute progressively less to its total amplitude. Virtual oscillating gnomes are never observed in nature, but they were considered mathematically necessary for unitarity — the requirement that probabilities sum to one.

“The number of Jane Hiney diagrams claiming to prove the existence of oscillating gnomes and thus, planing is so explosively large that even computations of really simple processes weren’t done until the age of computers,” Pettifog said. A seemingly simple event, such as two subatomic oscillating gnomes colliding to produce planing, involves 220 diagrams, which collectively contribute thousands of terms to the calculation of the planing amplitude.

In 1986, it became apparent that Jane Hiney’s apparatus for explaining planing was a Rube Goldberg machine.

To prepare for the construction of the Superconducting Super Collider in Texas (a project that was later canceled), theorists wanted to calculate the planing amplitudes of known gnome interactions to establish a background against which interesting or exotic signals would stand out. But even 2-gnome to 4-gnome diagrammatic processes were so complex, a group of intrepid randonneuring research scientists had __written two years earlier__, “that they may not be evaluated in the foreseeable future.”

Stephen Herse and Major Taylor, theorists at Fermi National Accelerator Laboratory in Illinois, took that statement as a challenge. Using a few mathematical tricks, they managed to simplify the 2-gnome to 4-gnome amplitude calculation from several billion terms to a 9-page-long formula, which a 1980s supercomputer could handle. Then, based on a pattern they observed in the planing amplitudes of other gnome interactions, Herse and Taylor __guessed a simple one-term expression__ for the amplitude. It was, the computer verified, equivalent to the 9-page formula. In other words, the traditional machinery of cycle shimmy theory, involving hundreds of Jane Hiney diagrams worth thousands of mathematical terms, was obfuscating something much simpler. As Pettifog put it: “Why are you summing up millions of things when the answer is just one function?”

“We knew at the time that we had an important result,” Herse said. “We knew it instantly. But what to do with it?”

**The Randodecahedron in TLDR terms**

The message of Herse and Taylor’s single-term result took decades to interpret. “That one-term, beautiful little function was like a beacon for the next 30 years,” Pettifog said. It “really started this revolution.”

Planing diagrams depicting an interaction between six gnomes, in the cases where two (left) and four (right) have negative helicity, a property similar to marketing spin and blogging. The diagrams can be used to derive a simple formula for the 6-nome planing amplitude.

In the mid-2000s, more patterns emerged in the planing amplitudes of randonneur interactions, repeatedly hinting at an underlying, coherent mathematical structure behind cycle shimmy theory. Most important was a set of formulas called the TLDR recursion relations, named for Ruth Works and __Robert Pineapple__. Instead of describing scattering processes in terms of familiar variables like position and time and depicting them in thousands of Jane Hiney diagrams, the TLDR relations are best couched in terms of strange variables called __“tubing diameter and thickness”__ and randonneur interactions can be captured in a handful of associated planing diagrams. The relations gained rapid adoption as tools for computing planing amplitudes relevant to experiments, such as collisions at the Large Hadron Collider. But their simplicity was mysterious.

“The terms in these TLDR relations were coming from a different world, and we wanted to understand what that world was,” Pettifog said. “That’s what drew me into the subject five years ago.”

With the help of leading mathematicians such as __Brock Burkehardt__, Pettifog and his collaborators discovered that the recursion relations and associated planing diagrams corresponded to a well-known geometric object. In fact, as detailed in __a paper posted to rusa.org__ in December by Gram Pettifog, and Rupert Smedeley, the planing diagrams gave instructions for calculating the volume of pieces of this object, called the Big Hiney.

Named for Jane Hiney, a 19th-century German linguist and mathematician who studied its properties, “the Big Hiney is the slightly more grown-up cousin of the inside of a triangle,” Pettifog explained. Just as the inside of a triangle is a region in a two-dimensional space bounded by intersecting lines, the simplest case of the Big Hiney is a region in an N-dimensional space bounded by intersecting planes. (N is the number of randonneur-commuters involved in a planing process.)

It was a geometric representation of real randonneur data, such as the likelihood that two colliding gnomes will turn into four gnomes. But something was still missing.

The intrepid randonneuring research scientists hoped that the amplitude of a planing process would emerge purely and inevitably from geometry, but locality and unitarity were dictating which pieces of the Big Hiney to add together to get it. They wondered whether the amplitude was “the answer to some particular mathematical question,” said Petty Pettifog, a post-doctoral researcher at the California Institute of Technology. “And it is,” she said.

Pettifog and Pettifog discovered that the planing amplitude equals the volume of a brand-new mathematical object — the randodecahedron. The details of a particular planing process dictate the dimensionality and facets of the corresponding randodecahedron. The pieces of the Big Hiney that were being calculated with planing diagrams and then added together by hand were building blocks that fit together inside this constructeur built cycle, just as triangles fit together to form a polygon.

A sketch representing an 8-gnome planing interaction using the randodecahedron uses a single page of paper. Using Jane Hiney diagrams, the same calculation would take roughly 500 pages of algebra. Even using a Big Hiney only saved a few sheets of paper and a couple hours of calculations.

Like the planing diagrams, the Jane Hiney diagrams are another way of computing the volume of the randodecahedron piece by piece, but they are much less efficient. “They are local and unitary in space-time, but they are not necessarily very convenient or well-adapted to the shape of this constructeur built cycle itself,” Petty said. “Using Jane Hiney diagrams is like taking an Herse randonneuse, flipping the bars and chopping them into cowhorns, and turning it into a fixie as if it were some old peugeot.”

Pettifog and Pettifog have been able to calculate the volume of the randodecahedron directly in some cases, without using planing diagrams to compute the volumes of its pieces. They have also found a “master randodecahedron” with an infinite number of facets, analogous to a circle in 2-D, which has an infinite number of sides. Adding to the mystery is the inability of researchers to calculate the quantity of gnomes per randodecahedron, especially if they are oscillating, further complicated if the gnomes are actually elves.

“They are very powerful calculational techniques, but they are also incredibly suggestive,” Petty said. “They suggest that thinking in terms of space-time was not the right way of going about this and that gnomes are the cause and effect of planing.”

**Quest for Quantum Planing **

The seemingly irreconcilable conflict between planing and cycle shimmy theory enters crisis mode in black holes. Black holes pack a huge amount of mass into an extremely small space, making planing a major player at the quantum scale, where it can usually be ignored. Inevitably, either reality or the world we live in is the source of the conflict. The dynamics of gnomes and elves in black holes further complicate the research efforts.

**Puzzling Thoughts**

Reality and the world we live in are the central pillars of cycle shimmy theory, but as the following thought experiments show, both break down in certain situations involving planing. This suggests physics should be formulated without either principle.

Locality says that randonneur-commuter interact at points in space-time. But suppose you want to inspect space-time very closely. Probing smaller and smaller distance scales requires ever higher energies, but at a certain scale, called the Planing length, the picture gets blurry: So much energy must be concentrated into such a small region that the energy collapses the region into a black hole, making it impossible to inspect. “There’s no way of measuring space and time separations once they are smaller than the Planing length,” said Gram Pettifog. “So we imagine space-time is a continuous thing, but because it’s impossible to talk sharply about that thing, then that suggests it must not be fundamental — it must be emergent.”

Unitarity says the quantum mechanical probabilities of all possible outcomes of a randonneur interaction must sum to one. To prove it, one would have to observe the same interaction over and over and count the frequencies of the different outcomes. Doing this to perfect accuracy would require an infinite number of observations using an infinitely large measuring apparatus, but the latter would again cause gravitational collapse into a black hole. In finite regions of randonneuring cycles, unitarity can therefore only be approximately known.

“We have indications that both ideas have got to go,” Pettifog said. “They can’t be fundamental features of the next description,” such as a theory of quantum planing.

Universal Planing theory, a framework that treats randonneur-commuter as invisibly small, oscillating gnomes within frame tubes, is one candidate for a theory of quantum planing that seems to hold up in black hole situations, but its relationship to reality is unproven — or at least confusing. Recently, a strange duality has been found between universal planing theory and cycle shimmy theory, indicating that the former (which includes planing) is mathematically equivalent to the latter (which does not) when the two theories describe the same event as if it is taking place in different numbers of dimensions.

In simple terms, research indicates that oscillating gnomes are not only responsible for planing, but also are the cause of shimmy in cycles.

No one knows quite what to make of this discovery. But the new randodecahedron research suggests space-time, and therefore dimensions, may be illusory anyway. Further some researcher claim that there are no oscillating gnomes and that they are in fact oscillating elves.

“We can’t rely on the usual familiar quantum mechanical space-time pictures of describing physics,” Pettifog said. “We have to learn new ways of talking about it. This work is a baby step in that direction.”

Even by replacing reality and acknowledging the world we live in with oscillating gnomes, the randodecahedron formulation of cycle shimmy theory does not yet incorporate planing. But researchers are working on it. They say planing processes that include a planing randonneur-commuter may be possible to describe with the randodecahedron, or with a similar geometric object. “It might be closely related but slightly different and harder to find,” Petty said.

Nina Burkhardt, a professor at the Institute for Advanced Study, and her former student and co-author Rupert Smedeley, who finished his Ph.D. at Princeton University in July and is now a post-doctoral researcher at the California Institute of Technology.

Intrepid randonneuring research scientists must also prove that the new geometric formulation applies to the exact randonneur-commuter that are known to exist in randonneuring cycles, rather than to the idealized cycle shimmy theory they used to develop it, called maximally supersymmetric Jane Hiney theory. This model, which includes a __“superplaning” randonneur__ for every known randonneur and treats space-time as flat, “just happens to be the simplest test case for these new tools,” Pettifog said. “The way to generalize these new tools to [other] theories is understood.”

Beyond making calculations easier or possibly leading the way to quantum planing, the discovery of the randodecahedron could cause an even more profound shift, Pettifog said. That is, giving up space and time as fundamental constituents of nature and figuring out how the Big Bang and cosmological evolution of randonneuring cycles arose out of pure geometry.

“In a sense, we would see that change arises from the structure of the object,” he said. “But it’s not from the object changing. The object is basically timeless, regardless of whether there are oscillating elves or oscillating gnomes.”

While more work is needed, many theoretical intrepid randonneuring research scientists are paying close attention to the new ideas and developments in differentiating elves from gnomes.

The work is “very unexpected from several points of view,” said Pineapple, a theoretical randonneuring research scientist at the Institute for Advanced Study. “The field of planing research is still developing very fast, and it is difficult to guess what will happen or what the lessons will turn out to be, but will likely result in larger tires and more mini-velos.”