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# Come and take a look at IBM’s Quantum Computer!

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As part of Oxford’s new collaboration with IBM Q, NQIT has arranged to host their Quantum Computer Exhibit in the foyer of the new Beecroft Building in Oxford Physics all of this week.

The exhibit is a prototype of their 50-qubit quantum computer and will be accompanied by a video explaining what it is and how it works.

It’s being set up on Tuesday 19 June and will be there until Monday 25 June, so please do pop along and take a look.

Source: https://www.nqit.ox.ac.uk/news/come-and-take-look-ibms-quantum-computer

# How Mathematicians Use Homology to Make Sense of Topology

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At first, topology can seem like an unusually imprecise branch of mathematics. It’s the study of squishy play-dough shapes capable of bending, stretching and compressing without limit. But topologists do have some restrictions: They cannot create or destroy holes within shapes. (It’s an old joke that topologists can’t tell the difference between a coffee mug and a doughnut, since they both have one hole.) While this might seem like a far cry from the rigors of algebra, a powerful idea called homology helps mathematicians connect these two worlds.

The word “hole” has many meanings in everyday speech — bubbles, rubber bands and bowls all have different kinds of holes. Mathematicians are interested in detecting a specific type of hole, which can be described as a closed and hollow space. A one-dimensional hole looks like a rubber band. The squiggly line that forms a rubber band is closed (unlike a loose piece of string) and hollow (unlike the perimeter of a penny).

Extending this logic, a two-dimensional hole looks like a hollow ball. The kinds of holes mathematicians are looking for — closed and hollow — are found in basketballs, but not bowls or bowling balls.

But mathematics traffics in rigor, and while thinking about holes this way may help point our intuition toward rubber bands and basketballs, it isn’t precise enough to qualify as a mathematical definition. It doesn’t clearly describe holes in higher dimensions, for instance, and you couldn’t program a computer to distinguish closed and hollow spaces.

“There’s not a good definition of a hole,” said Jose Perea of Michigan State University.

So instead, homology infers an object’s holes from its boundaries, a more precise mathematical concept. To study the holes in an object, mathematicians only need information about its boundaries.

The boundary of a shape is the collection of the points on its periphery, and a shape’s boundary is always one dimension lower than the shape itself. For example, the boundary of a one-dimensional line segment consists of the two points on either end. (Points are considered zero-dimensional.) The boundary of a solid triangle is the hollow triangle, which consists of one-dimensional edges. Similarly, the solid pyramid is bounded by a hollow pyramid.

If you stick two line segments together, the boundary points where they meet disappear. The boundary points are like the edge of a cliff — they are close to falling off the line. But when you connect the lines, the points that were on the edges are now securely in the center. Separately, the two lines had four total boundary points, but when they are stuck together, the resulting shape only has two boundary points.

If you can attach a third edge and close off the structure, creating a hollow triangle, then the boundary points disappear entirely. Each boundary point of the component edges cancels with another, and the hollow triangle is left with no boundary. So whenever a collection of lines forms a loop, the boundaries cancel.

Loops circle back on themselves, enclosing a central region. But the loop only forms a hole if the central region is hollow, as with a rubber band. A circle drawn on a paper forms a loop, but it is not a hole because the center is filled in. Loops that enclose a solid region — the non-hole kind — are the boundary of that two-dimensional region.

Therefore, holes have two important rigorous features. First, a hole has no boundary, because it forms a closed shape. And second, a hole is not the boundary of something else, because the hole itself must be hollow.

This definition can extend to higher dimensions. A two-dimensional solid triangle is bounded by three edges. If you attach several triangles together, some boundary edges disappear. When four triangles are arranged into a pyramid, each of the edges cancels with another one. So the walls of a pyramid have no boundary. If that pyramid is hollow — that is, it is not the boundary of a three-dimensional solid block — then it forms a two-dimensional hole.

To find all the types of holes within a particular topological shape, mathematicians build something called a chain complex, which forms the scaffolding of homology.

Many topological shapes can be built by gluing together pieces of different dimensions. The chain complex is a diagram that gives the assembly instructions for a shape. Individual pieces of the shape are grouped by dimension and then arranged hierarchically: The first level contains all the points, the next level contains all the lines, and so on. (There’s also an empty zeroth level, which simply serves as a foundation.) Each level is connected to the one below it by arrows, which indicate how they are glued together. For example, a solid triangle is linked to the three edges that form its boundary.

Mathematicians extract a shape’s homology from its chain complex, which provides structured data about the shape’s component parts and their boundaries — exactly what you need to describe holes in every dimension. When you use the chain complex, the processes for finding a 10-dimensional hole and a one-dimensional hole are nearly identical (except that one is much harder to visualize than the other).

The definition of homology is rigid enough that a computer can use it to find and count holes, which helps establish the rigor typically required in mathematics. It also allows researchers to use homology for an increasingly popular pursuit: analyzing data.

That’s because data can be visualized as points floating in space. These data points can represent the locations of physical objects, such as sensors, or positions in an abstract space, such as a description of food preferences, with nearby points indicating people who have a similar palate.

To form shapes from data, mathematicians draw lines between neighboring points. When three points are close together, they are filled in to form a solid triangle. When larger numbers of points are clustered together, they form more complicated and higher-dimensional shapes. Filling in the data points gives them texture and volume — it creates an image from the dots.

Homology translates this world of vague shapes into the rigorous world of algebra, a branch of mathematics that studies particular numerical structures and symmetries. Mathematicians study the properties of these algebraic structures in a field known as homological algebra. From the algebra they indirectly learn information about the original topological shape of the data. Homology comes in many varieties, all of which connect with algebra.

“Homology is a familiar construction. We have a lot of algebraic things we know about it,” said Maggie Miller of the Massachusetts Institute of Technology.

The information provided by homology even accounts for the imprecision of data: If the data shifts just slightly, the numbers of holes should stay the same. And when large amounts of data are processed, the holes can reveal important features. For example, loops in time-varying data can indicate periodicity. Holes in other dimensions can show clusters and voids in the data.

“There’s a real impetus to have methods that are robust and that are pulling out qualitative features,” said Robert Ghrist of the University of Pennsylvania. “That’s what homology gives you.”

# DNA’s Histone Spools Hint at How Complex Cells Evolved

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Molecular biology has something in common with kite-flying competitions. At the latter, all eyes are on the colorful, elaborate, wildly kinetic constructions darting through the sky. Nobody looks at the humble reels or spools on which the kite strings are wound, even though the aerial performances depend on how skillfully those reels are handled. In the biology of complex cells, or eukaryotes, the ballet of molecules that transcribe and translate genomic DNA into proteins holds centerstage, but that dance would be impossible without the underappreciated work of histone proteins gathering up the DNA into neat bundles and unpacking just enough of it when needed.

Histones, as linchpins of the apparatus for gene regulation, play a role in almost every function of eukaryotic cells. “In order to get complex, you have to have genome complexity, and evolve new gene families, and you have to have a cell cycle,” explained William Martin, an evolutionary biologist and biochemist at Heinrich Heine University in Germany. “And what’s in the middle of all this? Managing your DNA.”

New work on the structure and function of histones in ancient, simple cells has now made the longstanding, central importance of these proteins to gene regulation even clearer. Billions of years ago, the cells called archaea were already using histones much like our own to manage their DNA — but they did so with looser rules and much more variety. From those similarities and differences, researchers are gleaning new insights, not only into how the histones helped to shape the origins of complex life, but also into how variants of histones affect our own health today. At the same time, though, new studies of histones in an unusual group of viruses are complicating the answers about where our histones really came from.

## Dealing With Too Much DNA

Eukaryotes arose about 2 billion years ago, when a bacterium that could metabolize oxygen for energy took up residence inside an archaeal cell. That symbiotic partnership was revolutionary because energy production from that proto-mitochondrion suddenly made expressing genes much more metabolically affordable, Martin argues. The new eukaryotes suddenly had free rein to expand the size and diversity of their genomes and to conduct myriad evolutionary experiments, laying the foundation for the countless eukaryotic innovations seen in life today. “Eukaryotes are an archaeal genetic apparatus that survives with the help of bacterial energy metabolism,” Martin said.

But the early eukaryotes went through serious growing pains as their genomes expanded: The larger genome brought new problems stemming from the need to manage an increasingly unwieldy string of DNA. That DNA had to be accessible to the cell’s machinery for transcribing and replicating it without getting tangled up in a hopeless spaghetti ball.

The DNA also sometimes needed to be compact, both to help regulate transcription and regulation, and to separate the identical copies of DNA during cell division. And one danger of careless compaction is that DNA strands can irreversibly bind together if the backbone of one interacts with the groove of another, rendering the DNA useless.

Bacteria have a solution for this that involves a variety of proteins jointly “supercoiling” the cells’ relatively limited libraries of DNA. But eukaryotes’ DNA management solution is to use histone proteins, which have a unique ability to wrap DNA around themselves rather than just sticking to it. The four primary histones of eukaryotes — H2A, H2B, H3 and H4 — assemble into octamers with two copies of each. These octamers, called nucleosomes, are the basic units of eukaryotic DNA packaging.

By curving the DNA around the nucleosome, the histones prevent it from clumping together and keep it functional. It’s an ingenious solution — but eukaryotes didn’t invent it entirely on their own.

Back in the 1980s, when the cellular and molecular biologist Kathleen Sandman was a postdoc at Ohio State University, she and her adviser, John Reeve, identified and sequenced the first known histones in archaea. They showed how the four principal eukaryotic histones were related to each other and to the archaeal histones. Their work provided the early evidence that in the original endosymbiotic event that led to eukaryotes, the host was likely to have been an archaeal cell.

But it would be a teleological mistake to think that archaeal histones were just waiting for the arrival of eukaryotes and the chance to enlarge their genomes. “A lot of these early hypotheses looked at histones in terms of their ability to allow the cell to expand its genome. But that doesn’t really tell you why they were there in the first place,” said Siavash Kurdistani, a biochemist at the University of California, Los Angeles.

As a first step toward those answers, Sandman joined forces several years ago with the structural biologist Karolin Luger, who solved the structure of the eukaryotic nucleosome in 1997. Together, they worked out the crystallized structure of the archaeal nucleosome, which they published with colleagues in 2017. They found that the archaeal nucleosomes are “uncannily similar” in structure to eukaryotic nucleosomes, Luger said — despite the marked differences in their peptide sequences.

Archaeal nucleosomes had already “figured out how to bind and bend DNA in this beautiful arc,” said Luger, now a Howard Hughes Medical Institute investigator at the University of Colorado, Boulder. But the difference between the eukaryotic and archaeal nucleosomes is that the crystal structure of the archaeal nucleosome seemed to form looser, Slinky-like assemblies of varying sizes.

In a paper in eLife published in March, Luger, her postdoc Samuel Bowerman, and Jeff Wereszczynski of the Illinois Institute of Technology followed up on the 2017 paper. They used cryo-electron microscopy to solve the structure of the archaeal nucleosome in a state more representative of a live cell. Their observations confirmed that the structures of archaeal nucleosomes are less fixed. Eukaryotic nucleosomes are always stably wrapped by about 147 base pairs of DNA, and always consist of just eight histones. (For eukaryotic nucleosomes, “the buck stops at eight,” Luger said.) Their equivalents in archaea wind up between 60 and 600 base pairs. These “archaeasomes” sometimes hold as few as three histone dimers, but the largest ones consist of as many as 15 dimers.

They also found that unlike the tight eukaryotic nucleosomes, the Slinky-like archaeasomes flop open stochastically, like clamshells. The researchers suggested that this arrangement simplifies gene expression for the archaea, because unlike eukaryotes, they don’t need any energetically expensive supplemental proteins to help unwind DNA from the histones to make them available for transcription.

That’s why Tobias Warnecke, who studies archaeal histones at Imperial College London, thinks that “there’s something special that must have happened at the dawn of eukaryotes, where we transition from just having simple histones … to having octameric nucleosomes. And they seem to be doing something qualitatively different.”

What that is, however, is still a mystery. In archaeal species, there are “quite a few that have histones, and there are other species that don’t have histones. And even those that do have histones vary quite a lot,” Warnecke said. Last December, he published a paper showing that there are diverse variants of histone proteins with different functions. The histone-DNA complexes vary in their stability and affinity for DNA. But they are not as stably or regularly organized as eukaryotic nucleosomes.

As puzzling as the diversity of archaeal histones is, it provides an opportunity to understand the different possible ways of building systems of gene expression. That’s something we cannot glean from the relative “boringness” of eukaryotes, Warnecke says: Through understanding the combinatorics of archaeal systems, “we can also figure out what’s special about eukaryotic systems.” The variety of different histone types and configurations in archaea may also help us deduce what they might have been doing before their role in gene regulation solidified.

## A Protective Role for Histones

Because archaea are relatively simple prokaryotes with small genomes, “I don’t think that the original role of histones was to control gene expression, or at least not in a manner that we are used to from eukaryotes,” Warnecke said. Instead, he hypothesizes that histones might have protected the genome from damage.

Archaea often live in extreme environments, like hot springs and volcanic vents on the seafloor, characterized by high temperatures, high pressures, high salinity, high acidity or other threats. Stabilizing their DNA with histones may make it harder for the DNA strands to melt in those extreme conditions. Histones also might protect archaea against invaders, such as phages or transposable elements, which would find it harder to integrate into the genome when it’s wrapped around the proteins.

Kurdistani agrees. “If you were studying archaea 2 billion years ago, genome compaction and gene regulation are not the first things that would come to mind when you are thinking about histones,” he said. In fact, he has tentatively speculated about a different kind of chemical protection that histones might have offered the archaea.

Last July, Kurdistani’s team reported that in yeast nucleosomes, there is a catalytic site at the interface of two histone H3 proteins that can bind and electrochemically reduce copper. To unpack the evolutionary significance of this, Kurdistani goes back to the massive increase in oxygen on Earth, the Great Oxidation Event, that occurred around the time that eukaryotes first evolved more than 2 billion years ago. Higher oxygen levels must have caused a global oxidation of metals like copper and iron, which are critical for biochemistry (although toxic in excess). Once oxidized, the metals would have become less available to cells, so any cells that kept the metals in reduced form would have had an advantage.

During the Great Oxidation Event, the ability to reduce copper would have been “an extremely valuable commodity,” Kurdistani said. It might have been particularly attractive to the bacteria that were forerunners of mitochondria, since cytochrome c oxidase, the last enzyme in the chain of reactions that mitochondria use to produce energy, requires copper to function.

Because archaea live in extreme environments, they might have found ways to generate and handle reduced copper without being killed by it long before the Great Oxidation Event. If so, proto-mitochondria might have invaded archaeal hosts to steal their reduced copper, Kurdistani suggests.

The hypothesis is intriguing because it could explain why the eukaryotes appeared when oxygen levels went up in the atmosphere. “There was 1.5 billion years of life before that, and no sign of eukaryotes,” Kurdistani said. “So the idea that oxygen drove the formation of the first eukaryotic cell, to me, should be central to any hypotheses that try to come up with why these features developed.”

Kurdistani’s conjecture also suggests an alternative hypothesis for why eukaryotic genomes got so big. The histones’ copper-reducing activity only occurs at the interface of the two H3 histones inside an assembled nucleosome wrapped with DNA. “I think there’s a distinct possibility that the cell wanted more histones. And the only way to do that was to expand this DNA repertoire,” Kurdistani said. With more DNA, cells could wrap more nucleosomes and enable the histones to reduce more copper, which would support more mitochondrial activity. “It wasn’t just that histones allowed for more DNA, but more DNA allowed for more histones,” he said.

“One of the neat things about this is that copper is very dangerous because it will break DNA,” said Steven Henikoff, a chromatin biologist and HHMI investigator at the Fred Hutchinson Cancer Research Center in Seattle. “Here’s a place where you have the active form of copper being made, and it’s right next to the DNA, but it doesn’t break the DNA because, presumably, it’s in a tightly packaged form,” he said. By wrapping the DNA, the nucleosomes keep the DNA safely out of the way.

The hypothesis potentially explains aspects of how the architecture of the eukaryotic genome evolved, but it has met with some skepticism. The key outstanding question is whether archaeal histones have the same copper-reducing ability that some eukaryotic ones do. Kurdistani is investigating this now.

The bottom line is that we still don’t definitively know what functions histones served in the archaea. But even so, “the fact that you see them conserved over long distances strongly suggests that they are doing something distinct and important,” Warnecke said. “We just need to find out what it is.”

## Histones Are Still Evolving

Although the complex eukaryotic histone apparatus has not changed much since its origin about a billion years ago, it hasn’t been totally frozen. In 2018, a team at the Fred Hutchinson Cancer Research Center reported that a set of short histone variants called H2A.B is evolving rapidly. The pace of the changes is a sure sign of an “arms race” between genes vying for control over regulatory resources. It wasn’t initially clear to the researchers what the genetic conflict was about, but through a series of elegant crossbreeding experiments in mice, they eventually showed that the H2A.B variants dictated the survival and growth rate of embryos, as reported in December in PLOS Biology.

The findings suggested that paternal and maternal versions of the histone variants are mediating a conflict over how to allocate resources to the offspring during pregnancy. They are rare examples of parental-effect genes — ones that don’t directly affect the individual carrying them, but instead strongly affect the individual’s offspring.

The H2A.B variants arose with the first mammals, when the evolution of in utero development rewrote the “contract” for parental investment. Mothers had always invested a lot of resources in their eggs, but mammalian mothers also suddenly became responsible for the early development of their progeny. That set up a conflict: Paternal genes in the embryo had nothing to lose by demanding resources aggressively, while the maternal genes benefited from moderating the burden to spare the mother and let her live to breed another day.

“That negotiation is still ongoing,” said Harmit Malik, an HHMI investigator at the Fred Hutchinson Cancer Research Center who studies genetic conflicts. Exactly how the histones affect the growth and viability of offspring is still not completely understood, but Antoine Molaro, the postdoctoral fellow who led the work and who now leads his own research group at the University of Clermont Auvergne in France, is investigating it.

Some histone variants may cause health problems, too. In January, Molaro, Malik, Henikoff and their colleagues reported that short H2A histone variants are implicated in some cancers: More than half of diffuse large B cell lymphomas carry mutations in them. Other histone variants are associated with neurodegenerative diseases.

But little is yet understood about how a single copy of a histone variant can produce such dramatic disease effects. The obvious hypothesis is that the variants affect the stability of nucleosomes and disrupt their signaling functions, changing gene expression in a way that alters cell physiology. But if histones can act as enzymes, then Kurdistani suggests another possibility: The variants may alter enzymatic activity inside cells.

## An Alternative Viral Origin?

Despite the decades-old evidence from Sandman and others that eukaryotic histones evolved from archaeal histones, some intriguing recent work has unexpectedly opened the door to an alternative theory about their origins. According to a paper published on April 29 in Nature Structural & Molecular Biology, giant viruses of the Marseilleviridae family have viral histones that are recognizably related to the four main eukaryotic histones. The only difference is that in the viral versions, the histones that routinely pair up within the octamer (H2A with H2B, and H3 with H4) in eukaryotes are already fused into doublets. The fused viral histones form structures that are “virtually identical to canonical eukaryotic nucleosomes,” according to the paper’s authors.

Luger’s team posted a preprint on biorxiv.org about viral histones the same day, showing that in the cytoplasm of infected cells, viral histones stay near the “factories” that produce new viral particles.

“Here’s the thing that is really compelling,” said Henikoff, who was among the authors on the new Nature Structural & Molecular Biology paper. “All of the histone variants turn out to be derived from a common ancestor that was shared between eukaryotes and giant viruses. By standard phylogenetic criteria, these are a sister group to eukaryotes.”

It makes a compelling case that this common ancestor is where the eukaryotic histones came from, he says. A “proto-eukaryote” that had histone doublets might have been ancestral to both the giant viruses and eukaryotes and could have passed the proteins along to both lines of organisms a very long time ago.

Warnecke, however, is skeptical about inferring phylogenetic relationships from viral sequences, which are notoriously mutable. As he explained in an email to Quanta, reasons other than shared ancestry might explain how the histones ended up in both lineages. In addition, the idea would require that the histone doublets later “unfused” into the H2A, H2B, H3 and H4 histones, because there are no doublets of those histones in extant eukaryotes. “How and why that would have happened is unclear,” he wrote.

Although Warnecke is not convinced that the viral histones tell us much about the origin of eukaryotic histones, he is fascinated by their possible functions. One possibility is that they help to compact the viral DNA; another idea is that they could be disguising the viral DNA from the host’s defenses.

Histones have had myriad roles since the dawn of time. But it was really in the eukaryotes that they became the linchpins for complex life and countless evolutionary innovations. That’s why Martin calls the histone “a basic building block that never could realize its full potential without the help of mitochondria.”

# Multi-time correlations in the positive-P, Q, and doubled phase-space representations

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Piotr Deuar

Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warsaw, Poland

### Abstract

A number of physically intuitive results for the calculation of multi-time correlations in phase-space representations of quantum mechanics are obtained. They relate time-dependent stochastic samples to multi-time observables, and rely on the presence of derivative-free operator identities. In particular, expressions for time-ordered normal-ordered observables in the positive-P distribution are derived which replace Heisenberg operators with the bare time-dependent stochastic variables, confirming extension of earlier such results for the Glauber-Sudarshan P. Analogous expressions are found for the anti-normal-ordered case of the doubled phase-space Q representation, along with conversion rules among doubled phase-space s-ordered representations. The latter are then shown to be readily exploited to further calculate anti-normal and mixed-ordered multi-time observables in the positive-P, Wigner, and doubled-Wigner representations. Which mixed-order observables are amenable and which are not is indicated, and explicit tallies are given up to 4th order. Overall, the theory of quantum multi-time observables in phase-space representations is extended, allowing non-perturbative treatment of many cases. The accuracy, usability, and scalability of the results to large systems is demonstrated using stochastic simulations of the unconventional photon blockade system and a related Bose-Hubbard chain. In addition, a robust but simple algorithm for integration of stochastic equations for phase-space samples is provided.

Multi-time correlations are important for answering many physical questions. For example, the determination of lifetimes out-of-time-order correlations which are important indicators of quantum chaos, or finding the time resolution required to observe a transient effect. In general, however, they are more difficult to calculate in a quantum system than instantaneous correlations, and the difficulty grows with system size. Phase-space representations are a formulation of quantum mechanics in which the calculation of multi-time correlations has a particularly intuitive structure, and in which the difficulties of dealing with large systems are often alleviated.
In this work, the framework for calculating multi-time correlations with phase-space representations has been strongly extended to a much wider range of correlations and representations than before, facilitating future studies of large systems, including systems with dissipation.
The paper also describes a robust but simple algorithm for integration of phase space stochastic equations, something that has been difficult to find in the literature to date.

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### Cited by

[1] J. A. Ross, P. Deuar, D. K. Shin, K. F. Thomas, B. M. Henson, S. S. Hodgman, and A. G. Truscott, “Survival of the quantum depletion of a condensate after release from a harmonic trap in theory and experiment”, arXiv:2103.15283.

[2] Piotr Deuar, Alex Ferrier, Michał Matuszewski, Giuliano Orso, and Marzena H. Szymańska, “Fully quantum scalable description of driven dissipative lattice models”, arXiv:2012.02014.

The above citations are from SAO/NASA ADS (last updated successfully 2021-05-10 11:58:52). The list may be incomplete as not all publishers provide suitable and complete citation data.

Could not fetch Crossref cited-by data during last attempt 2021-05-10 11:58:49: Could not fetch cited-by data for 10.22331/q-2021-05-10-455 from Crossref. This is normal if the DOI was registered recently.

# How to Solve Equations That Are Stubborn as a Goat

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If you’ve ever taken a math test, you’ve probably met a grazing goat. Usually it’s tied to a fence post or the side of some barn, left there by an absent-minded farmer to graze on whatever grass it can reach. When you meet a grazing goat, your job is to calculate the total area of the region it can graze on. It’s a math test, after all.

Math teachers have stymied students by sticking goats in strangely shaped fields for hundreds of years, but one particular grazing goat problem has gotten the goat of mathematicians for more than a century. Until last year they were only able to find approximate answers to the problem, and it took a new approach with some very advanced mathematics to finally produce an exact solution. Let’s take a look at how a question you might find on a math test can turn into a problem that stumps mathematicians for over a century.

The simplest kind of grazing goat problem has the hungry animal attached to the side of a long barn by a fixed length of rope.

Usually in these problems we want to find the area of the region the goat has access to. What does that region look like?

With the leash pulled taut the goat can make a semicircle and can reach anything inside it. The area of a circle is Aπr2, so the area of a semicircle is A = \$latex frac{1}{2}\$πr2. If, for example, the rope has length 4, then the goat could graze in a region with area A = \$latex frac{1}{2}\$π × 42 = 8π square units.

This straightforward setup doesn’t pose much of a challenge to the math student or to the goat, so let’s make it more interesting. What if the goat is tied to the side of a square barn?

Let’s say the rope and the side of the barn both have length 4, and that the rope is attached to the middle of one side. What’s the area of the region the goat has access to now?

Well, the goat still has access to the same semicircle as in the first problem.

But the goat can also continue around the corner of the barn. Once it’s at the corner, the goat has two more units of rope to work with, so it can sweep out another quarter circle of radius 2 on either side of the barn.

The goat can access the semicircle of radius 4 plus two quarter circles of radius 2, for a total area of A = \$latex frac{1}{2}\$π × 42 + \$latex frac{1}{4}\$π × 22 + \$latex frac{1}{4}\$π × 22 = 10π square units.

You can make the problem more challenging by changing the shape of the obstruction. I’ve seen goats attached to triangles, hexagons and even concave shapes.

You can also make a new math question from an old one by reversing it: Instead of starting with rope length and finding the area, you can start with the area and find the rope length.

For example, let’s stick with our square barn and ask a new question: How long would the rope have to be for the goat to have access to a total of 50 square units of area? Reversing a math problem can breathe new life into an old idea, but it also makes this problem much more challenging.

First, notice that the shape of the region depends on the length of the rope. For example, if the rope is shorter than 2 units in length, the goat can’t get around the corner of the barn, so the region will only be a semicircle.

If the rope is longer than 2 units, the goat can get around the corner, as we saw above.

And if the rope is longer than 6 units, the goat can get behind the barn, creating another set of quarter circles to consider. (If the rope gets much longer, there will be overlap. See the exercises at the end of the column for an example of this.)

We want to find the rope length that gives us 50 square units of total area. The way to do this mathematically is to set our area formula equal to 50 and solve for r. But each kind of region has a different area formula. Which one do we use?

Figuring this out requires a little casework. If r ≤ 2 the area of the region is A = \$latex frac{1}{2}\$πr2. The biggest area would occur when r = 2, which yields a total area of A = \$latex frac{1}{2}\$π × 22 = 2π ≈ 6.28. This is less than 50, so we know we need more than 2 units of rope.

If 2 < r ≤ 6, this gives us the semicircle plus the two quarter circles we encountered before. The radius of the semicircle is r, and the radius of the quarter circles is r – 2, since two units of rope are needed to get to the corner and whatever rope remains acts like the radius of the quarter circle centered at the corner.

The area of this semicircle is \$latex frac{1}{2}\$πr2, and the area of each quarter circle is \$latex frac{1}{4}\$π(r – 2)2. Adding this up gives us a total area of

\$latex begin{aligned}A&=frac{1}{2} pi r^{2} +frac{1}{4}pi (r-2)^{2} + frac{1}{4}pi (r-2)^{2}\[1pt]
\A &=frac{1}{2} pi r^{2} + frac{1}{2}pi (r-2)^{2}.end{aligned}\$

We get the biggest possible area when r = 6, which gives an area of A = \$latex frac{1}{2}\$π × 62 + \$latex frac{1}{2}\$π × 42 = 26π ≈ 81.68 square units. Since 50 < 26π, that means the r that will give us 50 square units of area must be less than 6.

Knowing that r must be between 2 and 6 units settles the question of which area formula we should use: When 2 < r ≤ 6, the area is A = \$latex frac{1}{2}\$πr2 + \$latex frac{1}{2}\$π(r – 2)2. To find the exact value of r that gives us 50 square units of area, we set up this equation:

50 = \$latex frac{1}{2}\$πr2 + \$latex frac{1}{2}\$π(r – 2)2.

Notice that this is another way in which our reversed question is more complicated than the original: Instead of just computing the area the goat can reach, we need to solve an equation to figure out the length of the rope. To do that, we need to isolate r. We have to use arithmetic and algebra to get r by itself on one side of the equation, and that will tell us exactly what r must be.

Our equation may look a little intimidating at first, but it’s just a quadratic equation in r. There’s a standard procedure for solving such equations: We rearrange it in the form ar2 + br + c = 0 and then use the quadratic formula. A little algebra and arithmetic does the trick.

50 = \$latex frac{1}{2}\$π2 + \$latex frac{1}{2}\$π(r – 2)2

\$latex frac{100}{pi}\$ = r2 + (r – 2)2

\$latex frac{100}{pi}\$ = 2r2 – 4r + 4

0 = 2r2 – 4r + 4 – \$latex frac{100}{pi}\$.

This may not be the most beautiful mathematical expression in the world, but it’s just a quadratic equation, so we can apply the quadratic formula to solve exactly for r. This gives us an answer of

r = 1 + \$latexsqrt{frac{50}{pi} – 1}\$ ≈ 4.86.

Because we were able to isolate r in our equation, we now know exactly how long the rope must be to get an area of 50 square units. (Notice that the value of r we found is between 2 and 6, as expected.)

As challenging as this reversed goat grazing problem was compared to the initial ones we looked at, mathematicians discovered that the problem becomes even more challenging when you stick the goat inside the barn. So challenging, in fact, that they couldn’t solve it exactly.

Let’s put the goat inside our square barn with side length 4 and attach the rope to the middle of a wall. How long does the rope need to be for the goat to have access to half the area inside the barn?

As above, part of the challenge is that the shape of the region depends on the value of r. To get half the area of the square we need r to be longer than half the side of the barn but shorter than the full side, which gives us a region like this.

Finding a formula for the area of this region isn’t so easy. We can imagine the region as one sector of a circle of radius r plus two right triangles, and then use some high school geometry to get a formula. But as we’ll soon see, the mixing of circles and triangles is going to cause some trouble.

Let’s start with the triangles. The Pythagorean theorem tells us that the length of the missing leg in each right triangle is \$latexsqrt{r^{2}-4}\$. This makes the area of one of the triangles \$latex frac{1}{2}\$ × 2 × \$latexsqrt{r^{2} – 4}\$ = \$latexsqrt{r^{2} – 4}\$, so the two triangles together have an area of 2 \$latexsqrt{r^{2}-4}\$.

Now for the circular sector.

The area of a sector is A = \$latexfrac{1}{2}r^{2}\$θ, where θ is the measure of the central angle (in radians, not degrees). We need a formula for the area in terms of r, so we need to express the angle θ in terms of r. To do this, we’ll use the law of cosines, an underappreciated theorem from high school trigonometry.

Applying the law of cosines to the isosceles triangle with sides r, r and 4 gives us

42 = r2 + r2 – 2r2cosθ,

which we can solve for cosθ:

cosθ = \$latex frac{2 r^{2}-16}{2 r^{2}}\$ = \$latex frac{r^{2}-8}{r^{2}}\$.

To isolate θ, we need to take the inverse cosine, or arccosine, of both sides of the equation. This gives us

θ = arccos \$latex left(frac{r^{2}-8}{r^{2}}right)\$.

Now we have the angle θ in terms of r, so we can now express the area of our sector in terms of r and r alone.

A = \$latex frac{1}{2}\$r2θ

A = \$latex frac{1}{2}\$r2arccos \$latex left(frac{r^{2}-8}{r^{2}}right)\$.

Our final area formula is the sum of the sector area and the area of the two triangles, which is

A = \$latex frac{1}{2}\$r2arccos \$latex left(frac{r^{2}-8}{r^{2}}right)\$ + 2 \$latexsqrt{r^{2}-4}\$.

We now have a formula for the area of the region accessible to the goat inside the square entirely in terms of r. Now we just need to find the value of r that gives the goat access to half the square. The entire square has area 16, so all we have to do is plug A = 8 into our equation and solve for r and we’ll be finished.

8 = \$latex frac{1}{2}\$r2arccos \$latex left(frac{r^{2}-8}{r^{2}}right)\$ + 2 \$latexsqrt{r^{2}-4}\$.

There’s just one small problem: It’s not possible to solve for r in this equation.

That is, it’s not possible to solve exactly for r in this equation. We can use a calculator to approximate the value of r that makes this equation true (r ≈ 2.331), but we can’t isolate rin our equation. The mixing of trigonometric functions and polynomial functions in our equation creates obstacles we can’t get around.

We could try to get the r’s out from inside the arccosine function, but to do that we’d have to put the other r’s inside a cosine function. Either way we’d be dealing with an equation that involves a transcendental function, like an exponential or trigonometric function. Transcendental functions can’t be simply expressed in terms of the usual algebraic operations like addition and multiplication, and so in general transcendental equations can’t be solved exactly.

This issue lies at the heart of a famous grazing goat problem posed in the 19th century where the goat was placed inside a circular barn. As in our square barn problem, the goal was to determine how long the rope had to be for the goat to have access to half the region.

The region accessible by the goat takes the shape of a “lens” — two circular segments stacked together.

It’s possible to use high school geometry to find the area of this lens in terms of the rope length r, but the formula is much more complicated than it is for the square. And when you set this equal to half the area of the circular barn, you run into the same problem we ran into inside the square: You just can’t isolate r. You can approximate it, but you can’t solve for r exactly.

This sort of obstinacy is no more appealing in an equation than it is in a goat. For over 100 years, mathematicians tried to find an exact solution to this goat-in-a-circle puzzle, but it wasn’t until last year that a German mathematician finally figured it out. He used complex analysis — mathematics far removed from the geometry of circles and squares most goat problems rely on — to solve explicitly for . And while using something as advanced as a contour integral to find the length of a goat’s leash may seem like overkill, there’s always mathematical satisfaction in doing what couldn’t be done before. And there’s always the possibility that these new methods, even if they arise from studying a silly problem about goats, might lead to insights beyond the barnyard.

## Exercises

1. If the goat is attached to the middle of the side of a square barn with side length 4 by a rope of length 8, outside the barn, what’s the area of the region the goat has access to?

2. If the goat is attached to the corner of a square barn with side length 4 by a rope of length 8, outside the barn, what’s the area of the region the goat has access to?

3. Suppose the goat is inside an equilateral triangle of side 4 attached to a vertex. How long would the rope have to be for the goat to have access to half the triangle?

4. If the goat is attached to the middle of the side of a square barn with side length 4 by a rope of length 10, outside the barn, what’s the area of the region the goat has access to?

The region is comprised of a semicircle of radius 8, two quarter circles of radius 6, and two quarter circles of radius 2. Since 8 is equal to half the perimeter of the barn, the two semicircles behind the barn meet up at the midpoint.

The area of this region is \$latexfrac{1}{2}\$π × 82 + \$latexfrac{1}{2}\$π × 62 + \$latexfrac{1}{2}\$π × 22 = 52π.

This region is comprised of three-quarters of a circle of radius 8 and two quarter circles of radius 4.

This area is \$latexfrac{3}{4}\$π × 82 + \$latexfrac{1}{2}\$π × 42 = 56π. As a challenge, think about what happens if the rope has length 10.

Since the angles of an equilateral triangle are 60 degrees, the region the goat has access to is one-sixth of a circle of radius r, which has area \$latexfrac{1}{6}\$πr2.

The area of an equilateral triangle is \$latexfrac{sqrt{3}}{4}\$s2, so the area of the triangle of side length 4 is \$latexfrac{sqrt{3}}{4}\$ × 42 = 4 \$latex{sqrt{3}}\$. We set the two areas equal, \$latexfrac{1}{6}\$πr2 = \$latexfrac{1}{2}\$ × 4 \$latex{sqrt{3}}\$, and solve for r to get r = \$latexsqrt{frac{12 sqrt{3}}{pi}}\$. Notice how we can solve exactly for r here, unlike when the region mixed circular sectors and triangles.