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A brief illustrated guide to ‘scissors congruence’ − an ancient geometric idea that’s still fueling cutting-edge mathematical research

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A brief illustrated guide to ‘scissors congruence’ − an ancient geometric idea that’s still fueling cutting-edge mathematical research

While scissors congruence accurately captures the modern algebraic notion of 2D area, things get more complicated in higher dimensions.
Maxine Calle, CC BY-ND

Maxine Calle, University of Pennsylvania and Mona Merling, University of Pennsylvania

In math class, you probably learned how to compute the area of lots of different shapes by memorizing algebraic formulas. Remember “base x height” for rectangles and “½ base x height” for triangles? Or “𝜋 x radius²” for circles?

But if you were in math class in ancient Greece, you might have learned something very different. Ancient Greek mathematicians, such as Euclid, thought of area as something geometric, not algebraic. Euclid’s geometric perspective, recorded in his foundational work “Elements,” has influenced research programs across centuries – even the work of mathematicians today, like the two of us.

A gif of a blue blob bouncing between a triangle, circle and square, along with formulas for their areas.
Math today treats area algebraically, using formulas for simple shapes and calculus for more complicated regions.
Maxine Calle, CC BY-ND

Modern mathematicians refer to Euclid’s concept of “having equal area” as “being scissors congruent.” This idea, based on cutting up shapes and pasting them back together in different ways, has inspired interesting mathematics beyond just computing areas of triangles and squares. The story of scissors congruence demonstrates how classical problems in geometry can find new life in the strange world of abstract modern math.

Euclid’s notion of area

Today, people think of the area of a shape as a single number that can be computed using algebraic formulas or calculus. So what does it mean to think of area as something geometric the way the ancient Greeks did?

Imagine you’re back in math class and you have a pair of scissors, some tape and a piece of construction paper. Your teacher instructs you to make a new flat, two-dimensional shape using all of the construction paper and only straight-line cuts. Using your scissors, you cut the paper into a bunch of pieces. You start moving these pieces around – maybe you rotate them or flip them over – and you tape them back together to form a new shape.

A drawing of a student doodling on some paper. Above them, a pentagon is cut into 3 triangles, and those triangles are reassembled into a new polygon.
What kind of shapes can you make from a pentagon using only scissors and tape?
Maxine Calle, CC BY-ND

Using your algebraic formulas for area, you could check that the area of your new shape is equal to the original area of the construction paper. No matter how a 2D shape is cut up – as long as all the pieces are taped back together without overlap – the area of the old and the new shape will always be equal.

A blue pentagon with a central eyeball is cut up into pieces.
You’re not allowed to make curved cuts or tape overlapping pieces.
Maxine Calle, CC BY-ND

For Euclid, area is the measurement that is preserved by this geometric “cutting-and-pasting.” He would say that the new shape you made is “equal” to the original piece of construction paper – mathematicians today would say the two are “scissors congruent.”

What can your new shape look like? Because you’re only allowed to make straight-line cuts, it has to be a polygon, meaning none of the sides can be curved.

A gif of a pentagon splitting up into five triangles
When a polygon is cut up into pieces, its area doesn’t change.
Maxine Calle, CC BY-ND

Could you have made any possible polygon with the same area as your original piece of paper? The answer, amazingly, is yes – there’s even a step-by-step guide from the 1800s that tells you exactly how to do it.

In other words, for polygons, Euclid’s notion of area is exactly the same as the modern one. In fact, you may have even used Euclid’s idea of area before in computations without knowing it.

For example, you can use scissors congruence to compute the area of a pentagon. Since area is preserved if you cut the pentagon up into smaller triangles, you can instead find the area of these triangles (using “½ base x height”) and add them up to get the answer.

A pentagon, the same pentagon split up into five triangles, and the five separate triangles
It can be easier to compute the area of a pentagon by chopping it up into smaller triangles.
Maxine Calle, CC BY-ND

Hilbert’s third problem

Perhaps the most infamous appearance of scissors congruence is on the famous German mathematician David Hilbert’s list of problems, which consisted of some of the most important mathematical questions of the 1900s. Of the original 23 problems Hilbert proposed, some have been solved, some have been shown to be unsolvable and others are still unresolved. The third problem on the list, and the first to be resolved, is about scissors congruence.

Instead of two-dimensional polygons, Hilbert asked about their three-dimensional cousins: polyhedra. Euclid’s notion of scissors congruence was known to be an accurate description of two-dimensional area, but could it be a good notion of three-dimensional volume?

An illustration of the five platonic solids
There are just five Platonic solids – polyhedra, whose faces are all the same polygon.
Maxine Calle, CC BY-ND

The answer came within a year, provided by one of Hilbert’s students, Max Dehn. Dehn’s solution to the problem was very different from the two-dimensional case. He showed that when polyhedra are cut up, volume is not the only thing that is preserved. There is another preserved measurement, now called the Dehn invariant, which is constructed from the lengths of edges and the angles between the faces of the polyhedron.

If two polyhedra are scissors congruent, then they have to have the same Dehn invariant. So, if Dehn could find two polyhedra with the same volume but different values of this invariant, that would prove the answer to Hilbert’s third problem is no – scissors congruence doesn’t precisely capture 3D volume.

A tetrahedron splits into pieces. In its thought bubble is a cube, also splitting into pieces.
There is no way to cut a tetrahedron into pieces and glue them back together to make a cube with the same volume.
Maxine Calle, CC BY-ND

This is exactly what Dehn did, showing that the invariants associated to a cube and tetrahedron with the same volume are different. This means that there’s no possible way to cut up a tetrahedron into a finite number of pieces and reassemble them back into a cube with the same volume.

Are volume and the Dehn invariant all we need to know? If two polyhedra have the same volume and the same Dehn invariant, does that tell us they’re scissors congruent? It took mathematicians another 60 years to answer this question. In 1965, Jean-Pierre Sydler confirmed that the answer is yes, closing this chapter on scissors congruence.

Strange shapes and stranger connections

But the story doesn’t end there. Mathematics is full of shapes living in higher dimensions – like 4D, 100D, 3,485D or any dimension you can imagine – which are impossible to visualize. An active new research area called generalized scissors congruence seeks to uncover whether Hilbert’s question about scissors congruence can also be stated – and maybe even solved – for these strange shapes.

An illustration of various shapes lining a landscape, with arrows pointing between them. Some resemble waves.
Mathematicians study other strange kinds of geometries, like hyperbolic and spherical geometry.
Maxine Calle, CC BY-ND

However, what it means for two things to be scissors congruent is now far more complicated. While Hilbert and Dehn cared about things like volume and angles, other mathematicians could exchange these physical traits for something far less tangible.

A recent research program pioneered by mathematicians Jonathan Campbell and Inna Zakharevich proposes a unifying framework for generalized scissors congruence. This framework is built using a very abstract, seemingly unrelated mathematical toolkit called algebraic K-theory.

A triangle jumps into a machine and splits into three triangles.
Algebraic K-theory was developed in the late 20th century in the field of abstract math known as algebraic topology.
Maxine Calle, CC BY-ND

The big idea of K-theory is that mathematical objects can be understood by how they decompose into fundamental building blocks – much like molecules are broken up into atoms. With a little bit of adjustment, mathematicians can harness the machinery of K-theory and apply it to generalized scissors congruence problems.

This use of K-theory reimagines the problem of scissors congruence and opens the doors for future research. But at the end of the day, scissors congruence is a concrete idea that you don’t need fancy math to understand – just some patience, creativity, a pair of scissors and a lot of tape.The Conversation

Maxine Calle, Ph.D. Candidate in Mathematics, University of Pennsylvania and Mona Merling, Assistant Professor of Mathematics, University of Pennsylvania

This article is republished from The Conversation under a Creative Commons license. Read the original article.

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Colors are objective, according to two philosophers − even though the blue you see doesn’t match what I see

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theconversation.com – Elay Shech, Professor of Philosophy, Auburn University – 2025-04-25 07:55:00

What appear to be blue and green spirals are actually the same color.
Akiyoshi Kitaoka

Elay Shech, Auburn University and Michael Watkins, Auburn University

Is your green my green? Probably not. What appears as pure green to me will likely look a bit yellowish or blueish to you. This is because visual systems vary from person to person. Moreover, an object’s color may appear differently against different backgrounds or under different lighting.

These facts might naturally lead you to think that colors are subjective. That, unlike features such as length and temperature, colors are not objective features. Either nothing has a true color, or colors are relative to observers and their viewing conditions.

But perceptual variation has misled you. We are philosophers who study colors, objectivity and science, and we argue in our book “The Metaphysics of Colors” that colors are as objective as length and temperature.

Perceptual variation

There is a surprising amount of variation in how people perceive the world. If you offer a group of people a spectrum of color chips ranging from chartreuse to purple and asked them to pick the unique green chip – the chip with no yellow or blue in it – their choices would vary considerably. Indeed, there wouldn’t be a single chip that most observers would agree is unique green.

Generally, an object’s background can result in dramatic changes in how you perceive its colors. If you place a gray object against a lighter background, it will appear darker than if you place it against a darker background. This variation in perception is perhaps most striking when viewing an object under different lighting, where a red apple could look green or blue.

Of course, that you experience something differently does not prove that what is experienced is not objective. Water that feels cold to one person may not feel cold to another. And although we do not know who is feeling the water “correctly,” or whether that question even makes sense, we can know the temperature of the water and presume that this temperature is independent of your experience.

Similarly, that you can change the appearance of something’s color is not the same as changing its color. You can make an apple look green or blue, but that is not evidence that the apple is not red.

Apple under a gradient of red to blue light
Under different lighting conditions, objects take on different colors.
Gyozo Vaczi/iStock via Getty Images Plus

For comparison, the Moon appears larger when it’s on the horizon than when it appears near its zenith. But the size of the Moon has not changed, only its appearance. Hence, that the appearance of an object’s color or size varies is, by itself, no reason to think that its color and size are not objective features of the object. In other words, the properties of an object are independent of how they appear to you.

That said, given that there is so much variation in how objects appear, how do you determine what color something actually is? Is there a way to determine the color of something despite the many different experiences you might have of it?

Matching colors

Perhaps determining the color of something is to determine whether it is red or blue. But we suggest a different approach. Notice that squares that appear to be the same shade of pink against different backgrounds look different against the same background.

Green, purple and orange squares with smaller squares in shades of pink placed at their centers and at the bottom of the image
The smaller squares may appear to be the same color, but if you compare them with the strip of squares at the bottom, they’re actually different shades.
Shobdohin/Wikimedia Commons, CC BY-SA

It’s easy to assume that to prove colors are objective would require knowing which observers, lighting conditions and backgrounds are the best, or “normal.” But determining the right observers and viewing conditions is not required for determining the very specific color of an object, regardless of its name. And it is not required to determine whether two objects have the same color.

To determine whether two objects have the same color, an observer would need to view the objects side by side against the same background and under various lighting conditions. If you painted part of a room and find that you don’t have enough paint, for instance, finding a match might be very tricky. A color match requires that no observer under any lighting condition will see a difference between the new paint and the old.

YouTube video
Is the dress yellow and white or black and blue?

That two people can determine whether two objects have the same color even if they don’t agree on exactly what that color is – just as a pool of water can have a particular temperature without feeling the same to me and you – seems like compelling evidence to us that colors are objective features of our world.

Colors, science and indispensability

Everyday interactions with colors – such as matching paint samples, determining whether your shirt and pants clash, and even your ability to interpret works of art – are hard to explain if colors are not objective features of objects. But if you turn to science and look at the many ways that researchers think about colors, it becomes harder still.

For example, in the field of color science, scientific laws are used to explain how objects and light affect perception and the colors of other objects. Such laws, for instance, predict what happens when you mix colored pigments, when you view contrasting colors simultaneously or successively, and when you look at colored objects in various lighting conditions.

The philosophers Hilary Putnam and Willard van Orman Quine made famous what is known as the indispensability argument. The basic idea is that if something is indispensable to science, then it must be real and objective – otherwise, science wouldn’t work as well as it does.

For example, you may wonder whether unobservable entities such as electrons and electromagnetic fields really exist. But, so the argument goes, the best scientific explanations assume the existence of such entities and so they must exist. Similarly, because mathematics is indispensable to contemporary science, some philosophers argue that this means mathematical objects are objective and exist independently of a person’s mind.

Blue damselfish, seeming iridescent against a black background
The color of an animal can exert evolutionary pressure.
Paul Starosta/Stone via Getty Images

Likewise, we suggest that color plays an indispensable role in evolutionary biology. For example, researchers have argued that aposematism – the use of colors to signal a warning for predators – also benefits an animal’s ability to gather resources. Here, an animal’s coloration works directly to expand its food-gathering niche insofar as it informs potential predators that the animal is poisonous or venomous.

In fact, animals can exploit the fact that the same color pattern can be perceived differently by different perceivers. For instance, some damselfish have ultraviolet face patterns that help them be recognized by other members of their species and communicate with potential mates while remaining largely hidden to predators unable to perceive ultraviolet colors.

In sum, our ability to determine whether objects are colored the same or differently and the indispensable roles they play in science suggest that colors are as real and objective as length and temperature.The Conversation

Elay Shech, Professor of Philosophy, Auburn University and Michael Watkins, Professor of Philosophy, Auburn University

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‘Extraordinary claims require extraordinary evidence’ − an astronomer explains how much evidence scientists need to claim discoveries like extraterrestrial life

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theconversation.com – Chris Impey, University Distinguished Professor of Astronomy, University of Arizona – 2025-04-25 07:54:00

The universe is filled with countless galaxies, stars and planets. Astronomers may find life one day, but they will need extraordinary proof.
ESA/Euclid/Euclid Consortium/NASA, image processing by J.-C. Cuillandre (CEA Paris-Saclay), G. Anselmi

Chris Impey, University of Arizona

The detection of life beyond Earth would be one of the most profound discoveries in the history of science. The Milky Way galaxy alone hosts hundreds of millions of potentially habitable planets. Astronomers are using powerful space telescopes to look for molecular indicators of biology in the atmospheres of the most Earth-like of these planets.

But so far, no solid evidence of life has ever been found beyond the Earth. A paper published in April 2025 claimed to detect a signature of life in the atmosphere of the planet K2-18b. And while this discovery is intriguing, most astronomers – including the paper’s authors – aren’t ready to claim that it means extraterrestrial life exists. A detection of life would be a remarkable development.

The astronomer Carl Sagan used the phrase, “Extraordinary claims require extraordinary evidence,” in regard to searching for alien life. It conveys the idea that there should be a high bar for evidence to support a remarkable claim.

I’m an astronomer who has written a book about astrobiology. Over my career, I’ve seen some compelling scientific discoveries. But to reach this threshold of finding life beyond Earth, a result needs to fit several important criteria.

When is a result important and reliable?

There are three criteria for a scientific result to represent a true discovery and not be subject to uncertainty and doubt. How does the claim of life on K2-18b measure up?

First, the experiment needs to measure a meaningful and important quantity. Researchers observed K2-18b’s atmosphere with the James Webb Space Telescope and saw a spectral feature that they identified as dimethyl sulfide.

On Earth, dimethyl sulfide is associated with biology, in particular bacteria and plankton in the oceans. However, it can also arise by other means, so this single molecule is not conclusive proof of life.

Second, the detection needs to be strong. Every detector has some noise from the random motion of electrons. The signal should be strong enough to have a low probability of arising by chance from this noise.

The K2-18b detection has a significance of 3-sigma, which means it has a 0.3% probability of arising by chance.

That sounds low, but most scientists would consider that a weak detection. There are many molecules that could create a feature in the same spectral range.

The “gold standard” for scientific detection is 5-sigma, which means the probability of the finding happening by chance is less than 0.00006%. For example, physicists at CERN gathered data patiently for two years until they had a 5-sigma detection of the Higgs boson particle, leading to a Nobel Prize one year later in 2013.

YouTube video
The announcement of the discovery of the Higgs boson took decades from the time Peter Higgs first predicted the existence of the particle. Scientists, such as Joe Incandela shown here, waited until they’d reached that 5-sigma level to say, ‘I think we have it.’

Third, a result needs to be repeatable. Results are considered reliable when they’ve been repeated – ideally corroborated by other investigators or confirmed using a different instrument. For K2-18b, this might mean detecting other molecules that indicate biology, such as oxygen in the planet’s atmosphere. Without more and better data, most researchers are viewing the claim of life on K2-18b with skepticism.

Claims of life on Mars

In the past, some scientists have claimed to have found life much closer to home, on the planet Mars.

Over a century ago, retired Boston merchant turned astronomer Percival Lowell claimed that linear features he saw on the surface of Mars were canals, constructed by a dying civilization to transport water from the poles to the equator. Artificial waterways on Mars would certainly have been a major discovery, but this example failed the other two criteria: strong evidence and repeatability.

Lowell was misled by his visual observations, and he was engaging in wishful thinking. No other astronomers could confirm his findings.

An image of Mars in space
Mars, as taken by the OSIRIS instrument on the ESA Rosetta spacecraft during its February 2007 flyby of the planet and adjusted to show color.
ESA & MPS for OSIRIS Team MPS/UPD/LAM/IAA/RSSD/INTA/UPM/DASP/IDA, CC BY-SA

In 1996, NASA held a press conference where a team of scientists presented evidence for biology in the Martian meteorite ALH 84001. Their evidence included an evocative image that seemed to show microfossils in the meteorite.

However, scientists have come up with explanations for the meteorite’s unusual features that do not involve biology. That extraordinary claim has dissipated.

More recently, astronomers detected low levels of methane in the atmosphere of Mars. Like dimethyl sulfide and oxygen, methane on Earth is made primarily – but not exclusively – by life. Different spacecraft and rovers on the Martian surface have returned conflicting results, where a detection with one spacecraft was not confirmed by another.

The low level and variability of methane on Mars is still a mystery. And in the absence of definitive evidence that this very low level of methane has a biological origin, nobody is claiming definitive evidence of life on Mars.

Claims of advanced civilizations

Detecting microbial life on Mars or an exoplanet would be dramatic, but the discovery of extraterrestrial civilizations would be truly spectacular.

The search for extraterrestrial intelligence, or SETI, has been underway for 75 years. No messages have ever been received, but in 1977 a radio telescope in Ohio detected a strong signal that lasted only for a minute.

This signal was so unusual that an astronomer working at the telescope wrote “Wow!” on the printout, giving the signal its name. Unfortunately, nothing like it has since been detected from that region of the sky, so the Wow! Signal fails the test of repeatability.

An illustration of a long, thin rock flying through space.
‘Oumuamua is the first object passing through the solar system that astronomers have identified as having interstellar origins.
European Southern Observatory/M. Kornmesser

In 2017, a rocky, cigar-shaped object called ‘Oumuamua was the first known interstellar object to visit the solar system. ‘Oumuamua’s strange shape and trajectory led Harvard astronomer Avi Loeb to argue that it was an alien artifact. However, the object has already left the solar system, so there’s no chance for astronomers to observe it again. And some researchers have gathered evidence suggesting that it’s just a comet.

While many scientists think we aren’t alone, given the enormous amount of habitable real estate beyond Earth, no detection has cleared the threshold enunciated by Carl Sagan.

Claims about the universe

These same criteria apply to research about the entire universe. One particular concern in cosmology is the fact that, unlike the case of planets, there is only one universe to study.

A cautionary tale comes from attempts to show that the universe went through a period of extremely rapid expansion a fraction of a second after the Big Bang. Cosmologists call this event inflation, and it is invoked to explain why the universe is now smooth and flat.

In 2014, astronomers claimed to have found evidence for inflation in a subtle signal from microwaves left over after the Big Bang. Within a year, however, the team retracted the result because the signal had a mundane explanation: They had confused dust in our galaxy with a signature of inflation.

On the other hand, the discovery of the universe’s acceleration shows the success of the scientific method. In 1929, astronomer Edwin Hubble found that the universe was expanding. Then, in 1998, evidence emerged that this cosmic expansion is accelerating. Physicists were startled by this result.

Two research groups used supernovae to separately trace the expansion. In a friendly rivalry, they used different sets of supernovae but got the same result. Independent corroboration increased their confidence that the universe was accelerating. They called the force behind this accelerating expansion dark energy and received a Nobel Prize in 2011 for its discovery.

On scales large and small, astronomers try to set a high bar of evidence before claiming a discovery.The Conversation

Chris Impey, University Distinguished Professor of Astronomy, University of Arizona

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Perfect brownies baked at high altitude are possible thanks to Colorado’s home economics pioneer Inga Allison

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theconversation.com – Tobi Jacobi, Professor of English, Colorado State University – 2025-04-22 07:47:00

Students work in the high-altitude baking laboratory.
Archives and Special Collections, Colorado State University

Tobi Jacobi, Colorado State University and Caitlin Clark, Colorado State University

Many bakers working at high altitudes have carefully followed a standard recipe only to reach into the oven to find a sunken cake, flat cookies or dry muffins.

Experienced mountain bakers know they need a few tricks to achieve the same results as their fellow artisans working at sea level.

These tricks are more than family lore, however. They originated in the early 20th century thanks to research on high-altitude baking done by Inga Allison, then a professor at Colorado State University. It was Allison’s scientific prowess and experimentation that brought us the possibility of perfect high-altitude brownies and other baked goods.

A recipe for brownies at high altitude.
Inga Allison’s high-altitude brownie recipe.
Archives and Special Collections, Colorado State University

We are two current academics at CSU whose work has been touched by Allison’s legacy.

One of us – Caitlin Clark – still relies on Allison’s lessons a century later in her work as a food scientist in Colorado. The other – Tobi Jacobi – is a scholar of women’s rhetoric and community writing, and an enthusiastic home baker in the Rocky Mountains, who learned about Allison while conducting archival research on women’s work and leadership at CSU.

That research developed into “Knowing Her,” an exhibition Jacobi developed with Suzanne Faris, a CSU sculpture professor. The exhibit highlights dozens of women across 100 years of women’s work and leadership at CSU and will be on display through mid-August 2025 in the CSU Fort Collins campus Morgan Library.

A pioneer in home economics

Inga Allison is one of the fascinating and accomplished women who is part of the exhibit.

Allison was born in 1876 in Illinois and attended the University of Chicago, where she completed the prestigious “science course” work that heavily influenced her career trajectory. Her studies and research also set the stage for her belief that women’s education was more than preparation for domestic life.

In 1908, Allison was hired as a faculty member in home economics at Colorado Agricultural College, which is now CSU. She joined a group of faculty who were beginning to study the effects of altitude on baking and crop growth. The department was located inside Guggenheim Hall, a building that was constructed for home economics education but lacked lab equipment or serious research materials.

A sepia-toned photograph of Inga Allison, a white woman in dark clothes with her hair pulled back.
Inga Allison was a professor of home economics at Colorado Agricultural College, where she developed recipes that worked in high altitudes.
Archives and Special Collections, Colorado State University

Allison took both the land grant mission of the university with its focus on teaching, research and extension and her particular charge to prepare women for the future seriously. She urged her students to move beyond simple conceptions of home economics as mere preparation for domestic life. She wanted them to engage with the physical, biological and social sciences to understand the larger context for home economics work.

Such thinking, according to CSU historian James E. Hansen, pushed women college students in the early 20th century to expand the reach of home economics to include “extension and welfare work, dietetics, institutional management, laboratory research work, child development and teaching.”

News articles from the early 1900s track Allison giving lectures like “The Economic Side of Natural Living” to the Colorado Health Club and talks on domestic science to ladies clubs and at schools across Colorado. One of her talks in 1910 focused on the art of dishwashing.

Allison became the home economics department chair in 1910 and eventually dean. In this leadership role, she urged then-CSU President Charles Lory to fund lab materials for the home economics department. It took 19 years for this dream to come to fruition.

In the meantime, Allison collaborated with Lory, who gave her access to lab equipment in the physics department. She pieced together equipment to conduct research on the relationship between cooking foods in water and atmospheric pressure, but systematic control of heat, temperature and pressure was difficult to achieve.

She sought other ways to conduct high-altitude experiments and traveled across Colorado where she worked with students to test baking recipes in varied conditions, including at 11,797 feet in a shelter house on Fall River Road near Estes Park.

Early 1900s car traveling in the Rocky Mountains.
Inga Allison tested her high-altitude baking recipes at 11,797 feet at the shelter house on Fall River Road, near Estes Park, Colorado.
Archives and Special Collections, Colorado State University

But Allison realized that recipes baked at 5,000 feet in Fort Collins and Denver simply didn’t work in higher altitudes. Little advancement in baking methods occurred until 1927, when the first altitude baking lab in the nation was constructed at CSU thanks to Allison’s research. The results were tangible — and tasty — as public dissemination of altitude-specific baking practices began.

A 1932 bulletin on baking at altitude offers hundreds of formulas for success at heights ranging from 4,000 feet to over 11,000 feet. Its author, Marjorie Peterson, a home economics staff person at the Colorado Experiment Station, credits Allison for her constructive suggestions and support in the development of the booklet.

Science of high-altitude baking

As a senior food scientist in a mountain state, one of us – Caitlin Clark – advises bakers on how to adjust their recipes to compensate for altitude. Thanks to Allison’s research, bakers at high altitude today can anticipate how the lower air pressure will affect their recipes and compensate by making small adjustments.

The first thing you have to understand before heading into the kitchen is that the higher the altitude, the lower the air pressure. This lower pressure has chemical and physical effects on baking.

Air pressure is a force that pushes back on all of the molecules in a system and prevents them from venturing off into the environment. Heat plays the opposite role – it adds energy and pushes molecules to escape.

When water is boiled, molecules escape by turning into steam. The less air pressure is pushing back, the less energy is required to make this happen. That’s why water boils at lower temperatures at higher altitudes – around 200 degrees Fahrenheit in Denver compared with 212 F at sea level.

So, when baking is done at high altitude, steam is produced at a lower temperature and earlier in the baking time. Carbon dioxide produced by leavening agents also expands more rapidly in the thinner air. This causes high-altitude baked goods to rise too early, before their structure has fully set, leading to collapsed cakes and flat muffins. Finally, the rapid evaporation of water leads to over-concentration of sugars and fats in the recipe, which can cause pastries to have a gummy, undesirable texture.

Allison learned that high-altitude bakers could adjust to their environment by reducing the amount of sugar or increasing liquids to prevent over-concentration, and using less of leavening agents like baking soda or baking powder to prevent dough from rising too quickly.

Allison was one of many groundbreaking women in the early 20th century who actively supported higher education for women and advanced research in science, politics, humanities and education in Colorado.

Others included Grace Espy-Patton, a professor of English and sociology at CSU from 1885 to 1896 who founded an early feminist journal and was the first woman to register to vote in Fort Collins. Miriam Palmer was an aphid specialist and master illustrator whose work crafting hyper-realistic wax apples in the early 1900s allowed farmers to confirm rediscovery of the lost Colorado Orange apple, a fruit that has been successfully propagated in recent years.

In 1945, Allison retired as both an emerita professor and emerita dean at CSU. She immediately stepped into the role of student and took classes in Russian and biochemistry.

In the fall of 1958, CSU opened a new dormitory for women that was named Allison Hall in her honor.

“I had supposed that such a thing happened only to the very rich or the very dead,” Allison told reporters at the dedication ceremony.

Read more of our stories about Colorado.The Conversation

Tobi Jacobi, Professor of English, Colorado State University and Caitlin Clark, Senior Food Scientist at the CSU Spur Food Innovation Center, Colorado State University

This article is republished from The Conversation under a Creative Commons license. Read the original article.

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