What you’re looking at is a sea slug:

This is one of my favourite stories. From a November 27, 2025 article “A rare photosynthesizing sea slug has been found off N.S. Here’s why scientists are excited” by Frances Willick for the Canadian Broadcasting Corporation (CBC) news online website,

When she made the discovery that would thrill her fellow snorkellers and excite researchers across North America, she didn’t think much of it at first.

“I just thought, oh, that’s a rotten leaf, keep going,” says Elli Ofthenorth.

The avid snorkeller passed by this “black gunk” once, twice, but it wasn’t until her third pass that something caught her eye enough to take a closer look, and she realized it was a living creature.

“I just started yelling, there’s a sea slug here!”

Ofthenorth’s mother, who was on the shore at Rainbow Haven Provincial Park near Dartmouth, N.S., lit up the snorkel group chat, and within minutes, members identified it as Elysia chlorotica, or Eastern emerald elysia.

…

This unassuming creature could almost pass for your garden-variety slug — the kind that decimates your lettuce every summer. That is, until its crinkly-looking back unfurls a stunning, emerald green “leaf,” complete with pale “veins” branching outward from the centre.

It’s this “leaf,” and what it does for the sea slug, that holds so much promise for research in medicine, clean energy and other fields. 

But it’s so elusive that researchers are having a hard time studying it.

E. chlorotica can photosynthesize, stealing the chloroplasts — the photosynthesizing organs — of the algae it eats, keeping them alive in its body, and using them to get energy from the sun. The sea slugs can then subsist for months at a time without consuming food.

“It’s like if I ate a whole bunch of spinach and then I just woke up this morning and I just sunbathed for an hour and then I wouldn’t need to eat for the rest of the week,” says Hunter Stevens, a biologist with the Canadian Parks and Wilderness Society’s Nova Scotia chapter. “These slugs are essentially doing the same thing.”

…

Elusive and ephemeral

However, this coveted slug excels at eluding researchers.

Historically, known populations have existed in the Minas Basin area of Nova Scotia and in Martha’s Vineyard in Massachusetts — and theoretically their habitat exists all along the Eastern Seaboard of the U.S. — but recent efforts to find them have been unsuccessful.

“For so long it just seemed like nobody had seen them,” says Krug [Patrick Krug, a professor of biological sciences at California State University, Los Angeles]. “It was such a shot in the dark, it wasn’t even worth going to look.”

…

Dylan Gagler, a PhD student at the American Museum of Natural History in New York, has searched the slug’s favoured habitat off Martha’s Vineyard repeatedly this year, but without luck so far.

When Stevens’s Instagram post about the Rainbow Haven discovery popped up in Gagler’s feed, he says he was “having like a freak-out, FOMO [fear of missing out] moment of, like, I got to get up to Nova Scotia. Like, this is clearly where all the action is.”

Gagler contacted Stevens to get information about the conditions at the Rainbow Haven location, such as the air and water temperature and the depth they were found at, in order to fine-tune his own searches. He’s also exploring the permitting process to collect specimens from Nova Scotia to raise in a lab.

Though E. chlorotica has been hard to find, Krug says there have been a few sightings in recent months, including the one in Nova Scotia, as well as in the Carolinas and in Tampa Bay, Fla.

He says the populations are “ephemeral,” seeming to go through cycles of boom and bust — sometimes abundant, but then vanishing suddenly.

…

The fact that the recent discovery of this thriving population was made within the bounds of a provincial park in Nova Scotia underscores how important protected areas are to biodiversity, Stevens says.

“As coastal development proliferates and continues to advance, some of these populations, we may not even know about them, and they’ll disappear,” he says. “And so these slugs will probably get rarer as time goes on.”

Stevens says Ofthenorth’s discovery highlights the importance of citizen science.

“It just shows the power of curiosity and how anybody here can go into the water and there’s still that potential to find this really scientifically significant observation.”

…

If you have the time, Willick’s November 27, 2025 article “A rare photosynthesizing sea slug has been found off N.S. Here’s why scientists are excited” includes pictures and a video.

Introduction

I always look forward to this Christmas posting as it’s an opportunity to publish some stories that wouldn’t ordinarily be featured here like the sea slug/leaf that opened this post.The focus will be (mostly) on animals. Note: I will not be removing links from the news/press releases nor will I be providing separate citations and links.

On to the rest of the programme, there should always be a little dance in one’s life.

Nicholas Brothers

[downloaded from https://mymodernmet.com/cab-calloway-jumpin-jive-nicholas-brothers/]

Watch their feet. There’s a reason dancers still talk about those two; the synchronization is something. There was link to the Nicholas Brothers’ routine in Stormy Weather (1943 film) was in my December 24, 2025 posting but …

Given the past year, I think it’s time to revisit the brothers and Emma Taggart’s October 4, 2019 article on My Modern Met (scroll down to see Cab Calloway and his band performing ‘Jumpin Jive’ as an introduction to what is one of the most lauded tap routines ever recorded on film).

What makes it jaw-dropping is that it was done in one take and with no rehearsal .Do read Taggart’s October 4, 2019 article for a detail I found a little mystifying (what was the director thinking?).

I have a little more about ‘Stormy Weather’ and the dance. Nick Castle was the dance director (uncredited) for the movie and the Nicholas Brothers’ routine was something he worked out with the brothers..I found that bit of information in “C’mon, Get Happy; the Making of Summer Stock,” a 2023 book by David Frantle and Tom Johnson. The book has a number of dancers and choreographers commenting on the movie’s (Summer Stock) dance routines (Gene Kelly was in the movie) in detail. Castle was the ‘dance stager’ for “Summer Stock,” which is why the routine and the Nicholas Brothers are mentioned.

8,000 years ago, before numbers existed, art demonstrated early mathematical thinking (ethnomathematics)

A December 16, 2025 news item on ScienceDaily makes an extraordinary statement about art, mathematics, and prehistoric civilization,

A study published in the Journal of World Prehistory suggests that some of the earliest known images of plants created by humans served a deeper purpose than decoration. According to the researchers, these ancient designs also reveal early mathematical thinking.

By closely examining prehistoric pottery, Prof. Yosef Garfinkel and Sarah Krulwich of the Hebrew University traced the oldest consistent use of plant imagery in human art to more than 8,000 years ago. The pottery comes from the Halafian culture of northern Mesopotamia (c. 6200-5500 BCE). Their findings show that early farming communities carefully painted flowers, shrubs, branches, and trees, arranging them in ways that reflect deliberate geometric structure and numerical order.

…

This undated news release on the Canadian Friends of the Hebrew University of Jerusalem website provides more detail about the work, Note: There are lots of images accompanying this story that are not included here,

A new study reveals that the Halafian culture of northern Mesopotamia (c. 6200–5500 BCE) produced the earliest systematic plant imagery in prehistoric art, flowers, shrubs, branches, and trees painted on fine pottery, arranged with precise symmetry and numerical sequences, especially petal and flower counts of 4, 8, 16, 32, and 64. This suggests that early farming villages in the Near East already possessed sophisticated, practical mathematical thinking about dividing space and quantities, likely tied to everyday needs such as fairly sharing crops from collectively worked fields, long before writing or formal number systems existed.

A new study published in the Journal of World Prehistory reveals that some of humanity’s earliest artistic representations of botanical figures were far more than decorative, they were mathematical.

…

In an extensive analysis of ancient pottery, Prof. Yosef Garfinkel and Sarah Krulwich of the Hebrew University have identified the earliest systematic depictions of vegetal motifs in human history, dating back over 8,000 years to the Halafian culture of northern Mesopotamia (c. 6200–5500 BCE). Their research shows that these early agricultural communities painted flowers, shrubs, branches, and trees with remarkable care, and embedded within them evidence of complex geometric and arithmetic thinking. 

A New Understanding of Prehistoric Art

Earlier prehistoric art focused primarily on humans and animals. Halafian pottery, however, marks the moment when the plant world entered human artistic expression in a systematic and visually sophisticated way.

Across 29 archaeological sites, Garfinkel and Krulwich documented hundreds of carefully rendered vegetal motifs, some naturalistic, others abstract, all reflecting conscious artistic choice.

…

“These vessels represent the first moment in history when people chose to portray the botanical world as a subject worthy of artistic attention,” the authors note. “It reflects a cognitive shift tied to village life and a growing awareness of symmetry and aesthetics.” 

Among the study’s most striking insights is the precise numerical patterning in Halafian floral designs. Many bowls feature flowers with petal counts that follow geometric progression: 4, 8, 16, 32, and even arrangements of 64 flowers.

…

These sequences, the researchers argue, are intentional and demonstrate a sophisticated grasp of spatial division long before the appearance of written numbers.

“The ability to divide space evenly, reflected in these floral motifs, likely had practical roots in daily life, such as sharing harvests or allocating communal fields,” Garfinkel explains. 

This work contributes to the field of ethnomathematics, which identifies mathematical knowledge embedded in cultural expression.

…

The motifs documented span the full botanical spectrum:

• Flowers with meticulously balanced petals
• Seedlings and shrubs, rendered with botanical clarity
• Branches, arranged in rhythmic, repeating bands
• Tall, imposing trees, sometimes shown alongside animals or architecture

Notably, none of the images depict edible crops, suggesting that the purpose was aesthetic rather than agricultural or ritualistic. Flowers, the authors note, are associated with positive emotional responses, which may explain their prominence. 

Revising the History of Mathematics

While written mathematical texts appear millennia later in Sumer, Halafian pottery reveals an earlier, intuitive form of mathematical reasoning, rooted in symmetry, repetition, and geometric organization.

“These patterns show that mathematical thinking began long before writing,” Krulwich says. “People visualized divisions, sequences, and balance through their art.”

…

By cataloguing these vegetal motifs and revealing their mathematical foundations, the study offers a new perspective on how early communities understood the natural world, organized their environments, and expressed cognitive complexity.

The research paper titled “The Earliest Vegetal Motifs in Prehistoric Art: Painted Halafian Pottery of Mesopotamia and Prehistoric Mathematical Thinking” is now available in Journal of World Prehistory and can be accessed at https://doi.org/10.1007/s10963-025-09200-9

And now, back to the animal stories

Frogs! There must always be at least one story here at Christmas about these critters.

According to a June 25, 2025 Pensoft (publisher) news release on EurekAlert, three new species have been round in Northern Peru, Note: The link to the study has been retained while one other link has been removed,

High in the cloud-wrapped peaks of the Cordillera de Huancabamba, where the Andes dip and twist into isolated ridges, a team of Peruvian scientists has brought three secretive frogs out of obscurity and into the scientific record. The study [appears to be open access], led by herpetologist Germán Chávez and published in Evolutionary Systematics, describes Pristimantis chinguelas, P. nunezcortezi, and P. yonke—three new species discovered in the rugged, misty highlands of northwestern Peru.

“They’re small and unassuming,” Chávez says, “but these frogs are powerful reminders of how much we still don’t know about the Andes.”

Between 2021 and 2024, the team carried out a series of tough expeditions, hiking steep trails and combing mossy forests and wet páramo for signs of amphibian life. It was in this setting—both harsh and enchanting—that they encountered the new species.

Each frog tells a different story:

P. chinguelas, discovered on a cliffside of Cerro Chinguelas, has a body dotted with prominent large tubercles on both sides. Its high-pitched “peep” can be heard on humid nights.

P. nunezcortezi lives near a cool mountain stream in a regenerating forest. With large black blotches on axillae and groins, it was named in honour of ornithologist Elio Nuñez-Cortez, a conservation trailblazer in the region.

P. yonke, the smallest of the three, was found nestled in bromeliads at nearly 3,000 meters. Its name nods to “yonque,” a sugarcane spirit consumed by locals to brave the highland chill.

“Exploring this area is more than fieldwork—it’s an immersion into wilderness, culture, and resilience,” says co-author Karen Victoriano-Cigüeñas.

“Many of these mountain ridges are isolated, with no roads and extreme terrain,” adds Ivan Wong. “The weather shifts within minutes, and the steep cliffs make every step a challenge. It’s no wonder so few scientists have worked here before. But that’s exactly why there’s still so much to find.”

Despite the thrill of discovery, the frogs’ future is uncertain. The team observed signs of habitat degradation, fire damage, and expanding farmland. For now, the species are listed as Data Deficient under IUCN criteria, but the call to action is clear.

“The Cordillera de Huancabamba is not just a remote range—it’s a living archive of biodiversity and cultural legacy,” says co-author Wilmar Aznaran. “And we’ve barely scratched the surface.”

…

Hungarian scientists have furthered the research into dogs and learning. From a September 18, 2025 Eötvös Loránd University press release (also on EurekAlert),

BUDAPEST, Hungary — A new study publishing in Current Biology on September 18 by the Department of Ethology at Eötvös Loránd University reveals that dogs with a vocabulary of toy names—known as Gifted Word Learners—can extend learned labels to entirely new objects, not because the objects look similar, but because they are used in the same way.

In humans, “label extension” is a cornerstone of early language development. In non-humans, until now, it had only been documented in few so-called language-trained individual animals, after years of intensive training in captivity.

But learning to extend labels to objects that share the same function, rather than visual similarities, is considered an even more complex skill. A toddler learns that the word “cup” can apply to mugs, tumblers, and sippy cups, or that both a spoon and a ladle are “for scooping.” While individuals of many animal species can group items by appearance, extending a learned label to a functionally similar but visually different object has long been considered an advanced skill.

Video abstract at this link: https://youtu.be/8_NbCYAWSfU

The time and efforts needed to train animals in captivity to learn verbal labels, as well as the very limited number of subjects that successfully acquired such vocabulary, have until now limited the feasibility of this type of research.

But here comes the twist! “Gifted Word Learner dogs offer a unique possibility to study this phenomenon because they rapidly learn verbal labels – the names of toys – during natural interactions in their human families” said Dr. Claudia Fugazza, lead author of the study.

“Our results show that these dogs do not just memorize object names,” continues Dr. Fugazza. “They understand the meaning behind those labels well enough to apply them to new, very different-looking toys— by recognizing what the toys were for.”

Link to the social media of the Gifted Word Learner dogs project: https://linktr.ee/geniusdogchallenge

A Play-Based Experiment

Researchers of the Department of Ethology, at Eötvös Lorand University tested 7 Gifted Word Learner dogs—(six Border collies and a Blue heeler)—known for their unusual ability to learn the names of dozens of toys naturally, through everyday play.

The experiment had four stages, all of them conducted in a natural setup, at the house of each dog owner, during playful interactions:

1. Fist, in the Learning Phase, Dogs learned two new labels, such as “Pull” and “Fetch,” each referring not to a single item, but to a group of toys that looked completely different but were used in the same way during play (tug or retrieve).
2. Second, during a formal Assessment, the dogs showed that they had successfully learned those labels and could appropriately choose the “Pulls” and “Fetches” when asked.
3. The crucial part of the experiment was carried out after this Assessment: in the Generalization Phase, the dogs were introduced to new toys, also with diverse physical features, and the owner played in the same two ways as before, but this time saying no labels.
4. Test – When asked for a “Pull” or “Fetch,” the dogs selected the correct unlabelled toy significantly above chance, indicating they had generalized the labels to a functional category.

Why This Matters

The study provides the first evidence that dogs can generalize verbal labels to functional categories during natural-like playful interactions in their human families—mirroring, in functional terms, the natural context of human language development.

“This ability shows that classification linked to verbal labels can emerge in non-human, non-linguistic species living in natural settings,” said Dr. Adam Miklosi, coauthor of the study. “It opens exciting new avenues for studying how language-related skills may evolve and function beyond our own species.”

Key Points

• Dogs extended verbal labels to objects that shared only functional properties, not appearance.
• The skill emerged naturally through play with owners—no formal training required.
• While the mechanisms of such learning are not known, the context in which it happens present a striking parallel with that of human infants: daily life in a human family.
• The study of these skills in a non-human species in its natural environment paves the way for understanding the how language-related skills evolved and function.

---

Journal

Current Biology

DOI

10.1016/j.cub.2025.08.013

Article Title

Dogs extend verbal labels for functional classification of objects

A September 30, 2025 University of Utah news release (also on EurekAlert) announced the BIRDBASE dataset has tracked (and continues to track) ecological traits for over 11,000 birds,

Çağan Şekercioğlu was an ambitious, but perhaps naive graduate student when, 26 years ago, he embarked on a simple data-compilation project that would soon evolve into a massive career-defining achievement.

With the help of countless students and volunteers, the University of Utah conservation biologist has finally released BIRDBASE, an encyclopedic dataset of traits covering all the bird species recognized by the world’s four major avian taxonomies.

Described this week in a study published in the journal Scientific Data, the dataset covers 78 ecological traits, including conservation status, for 11,589 species of birds in 254 families. The main trait categories tracked are body mass; habitat; diet; nest type; clutch size; life history; elevational range; and movement strategy, that is whether and how they migrate.

While some little-known species still have incomplete data, the dataset provides a foundation for ornithologists around the world to conduct new global analyses in ornithology, conservation biology and macroecology, including the links between bird species’ ecological traits and their risk of extinction, according to Şekercioğlu, a professor in the university’s School of Biological Sciences. He also hopes BIRDBASE will help other biologists win support for studying avian conservation.

“To get funding you have to have a big question, but without data, how are you going to answer those big questions?” Şekercioğlu posed. “It also shows we still have ways to go. Birds are the best-known class of organism, but even though they are the best known, we still have big data gaps.”

BIRDBASE’s public launch coincides with the release of the first unified global checklist for birds, known as AviList, a grand taxonomy under one cover.

The BIRDBASE project started in 1999 when Şekercioğlu was a graduate student at Stanford University, spending field seasons in Costa Rica. While writing the first chapter of his Ph.D. thesis, he needed to know the percentage of tropical forest understory bug-eating birds, technically known as insectivores, that are threatened with extinction. He was perplexed to discover that information had yet to be determined.

“I realized that statistic doesn’t exist because nobody had analyzed all the birds of the world and their threat status based on diet,” he said. “I’m like, this is unbelievable. There’s no global database on birds. I’m lucky that I was in grad school because I was naive and I love birds.”

In other words, he set out to figure it out himself. That meant gathering and organizing life history traits for all such bird species, including their diets, habitats and conservation status. For a keen birder like Şekercioğlu, it seemed like a simple task that would be fun, compiling data found on thousands of bird species published in huge beautifully illustrated volumes. It turned out to be tedious and seriously time consuming, but worthwhile.

Thanks to a cadre of volunteers in the Stanford Volunteer Program and undergraduates, whose labors were compensated by the Stanford Center for Conservation Biology, Şekercioğlu answered his question within a couple years. Twenty-seven percent of tropical understory insectivores were threatened or near threatened with extinction. This finding wound up not supporting the hypothesis of his research, but that’s science.

Yet the dataset was so helpful that he labored on with the data-compiling project to eventually cover all bird species and expanded the number of traits included. “What started as this little specialized question turned into this global database, the first of its kind” he said.

BIRDBASE has proven a boon to many other avian researchers who have tapped it to support dozens of papers, most of them listing Şekercioğlu as co-author. The tally of Şekercioğlu’s papers that have used BIRDBASE currently stands at 98, accounting for 14,000 of Şekercioğlu’s 24,000-plus citations.

Among the conclusions the dataset has enabled is that a majority of the world’s bird species, or 54%, are insectivores, and many species in this group are under pressure.

“Most of them are tropical forest species. It is a very important group and they’re declining,” he said. “They’re sensitive even though they’re not hunted. They are small, so they don’t need a big area. You wouldn’t expect them to be the most sensitive group to habitat fragmentation but they are highly specialized.”

The dataset also showed that fish-eating seabirds are at elevated risk of extinction as well, and fruit-eating birds are vital to the survival of tropical rain forests.

“The most important seed dispersers in the tropics are frugivorous birds,” Şekercioğlu said. “In some tropical forests, over 90% of all woody plants’ seeds are dispersed by fruit-eating birds who eat them and then defecate the seeds somewhere else and they germinate.”

Now for the first time BIRDBASE is publicly available to all researchers online, “no strings attached.” It can be found as an Excel spreadsheet on a site hosted by Figshare, with separate worksheet tabs for trait values, trait definitions, nest details and data sources, packaged on one row per species.

Şekercioğlu emphasized that BIRDBASE remains a work in progress that will be continuously updated. Kind of like a medieval cathedral that is open for worship, but never really finished. He estimated that nearly 30 person-years of labor have gone into the project, work that entails entering data collected from various authoritative sources, such as BirdLife International, Birds of the World, hundreds of bird books and ornithological papers, and Şekercioğlu’s field observations of more than 9,400 bird species.

“Thanks to my being naïve, something that started with just a little question in grad school led to the foundation of my career. Right now, if one of my students came to me and said, ‘Hey, as part of my PhD I want to enter the world’s birds into a dataset,’ I’m like, ‘No, you’re not doing that. You’ll never finish your Ph.D.’ Fortunately I finished my Ph.D., but think about it, 1999 is when I had the idea and we are still putting finishing touches in 2025.”

Bumlebees can read Morse code? Apparently, the answer is yes. From a November 13, 2025 Queen Mary University of London (QMUL) press release (also on EurekAlert but published on November 12, 2025),

Researchers at Queen Mary University of London have shown for the first time that an insect – the bumblebee Bombus terrestris – can decide where to forage for food based on different durations of visual cues.  

In Morse code, a short duration flash or ‘dot’ denotes a letter ‘E’ and a long duration flash, or ‘dash’, means letter ‘T’. Until now, the ability to discriminate between ‘dot’ and ‘dash’ has been seen only in humans and other vertebrates such as macaques or pigeons.  

PhD student Alex Davidson and his supervisor Dr Elisabetta Versace, Senior Lecturer in Psychology at Queen Mary, led a team that studied this ability in bees. They built a special maze to train individual bees to find a sugar reward at one of two flashing circles, shown with either a long or short flash duration. For instance, when the short flash, or ‘dot’, was associated with sugar, then the long flash, or ‘dash’, was instead associated with a bitter substance that bees dislike.  

At each room in the maze, the position of the ‘dot’ and ‘dash’ stimulus was changed, so that bees could not rely on spatial cues to orient their choices. After bees learned to go straight to the flashing circle paired with the sugar, they were tested with flashing lights but no sugar present, to check whether bees’ choices were driven by the flashing light, rather than by olfactory or visual cues present in the sugar.   

It was clear the bees had learnt to tell the light apart based on their duration, as most of them went straight to the ‘correct’ flashing light duration previously associated with sugar, irrespective of spatial location of the stimulus. 

Alex Davidson said: “We wanted to find out if bumblebees could learn to the difference between these different durations, and it was so exciting to see them do it”. 

“Since bees don’t encounter flashing stimuli in their natural environment, it’s remarkable that they could succeed at this task. The fact that they could track the duration of visual stimuli might suggest an extension of a time processing capacity that has evolved for different purposes, such as keeping track of movement in space or communication”. 

“Alternatively, this surprising ability to encode and process time duration might be a fundamental component of the nervous system that is intrinsic in the properties of neurons. Only further research will be able to address this issue.” 

The neural mechanisms involved in the ability to keep track of time for these durations remain mostly unknown, as the mechanisms discovered for entraining with the daylight cycle (circadian rhythms) and seasonal changes are too slow to explain the ability to differentiate between a ‘dash’ and a ‘dot’ with different duration.  

Various theories have been put forward, suggesting the presence of a single or multiple internal clocks. Now that the ability to differentiate between durations of flashing lights has been discovered in insects, researchers will be able to test different models in these ‘miniature brains’ smaller than one cubic millimetre. 

Elisabetta Versace continued: “Many complex animal behaviours, such as navigation and communication, depend on time processing abilities. It will be important to use a broad comparative approach across different species, including insects, to shed light on the evolution of those abilities. Processing durations in insects is evidence of a complex task solution using minimal neural substrate. This has implications for complex cognitive-like traits in artificial neural networks, which should seek to be as efficient as possible to be scalable, taking inspiration from biological intelligence.” 

---

Journal

Biology Letters

DOI

10.1098/rsbl.2025.0440

Article Title

Duration discrimination in the bumblebee Bombus terrestris

The bumblebee image at the start of this news bit is from Bumblebee Conservation Trust in the UK; their website can be found here.

Joyeux Noël!

We live in such an extraordinary world: able to watch the Nicholas Brothers give a performance that is decades old, observe a leaf that’s really a sea slug, discover that bumblebees can learn Morse code, etc.

I’m ‘wrapping’ this up with two more items.

The mathematics of gift wrapping

A December 18, 2025 University of British Columbia (UBC) Question & Answer (Q&A) interview (also received via email) features mathematician Adam Martens,

UBC Mathematics postdoctoral fellow Adam Martens talks about the geometry of gift wrapping—and why you can’t wrap a ball perfectly (so don’t even bother!). 

From Christmas to Hanukkah to Kwanzaa, the gift-giving season is upon us. After we track down the perfect items for our favourite people, another task awaits us: gift wrapping. It’s not just an art—it’s math in disguise. 

We spoke to Dr. Adam Martens, UBC mathematics postdoctoral fellow and differential geometer about the best shapes to reduce waste—and why a donut-shaped object can be wrapped perfectly, but only if you work in four dimensions. 

What is a differential geometer? 

 A geometer is a specialist in geometry, or the study of points, lines, angles, surfaces, and solids. A differential mathematician studies smooth objects called ‘manifolds’, for example, a flat piece of paper or the surface of a ball. We also think about higher-dimensional objects, like the space-time of the universe. 

What is the easiest shape to wrap? 

No surprises here, but a box. The nice thing about wrapping a box is that each side is flat, and the flat edges meet at simple creases. Wrapping paper can be easily folded over the edges—mathematicians call this a manifold with corners. 

Wrapping paper is inherently flat and rigid. It can be folded, but from a mathematical point of view, it cannot be warped so that it lies flat on a curved surface. 

This means it’s mathematically impossible to wrap a sphere perfectly i.e. without any creases or folds. The only way to effectively wrap a ball is to put the ball in a box. 
A closely related theorem in calculus is the “hairy ball theorem,” which says you can’t comb a hairy ball flat without creating a cowlick or hair swirl. 

What is the most difficult shape to wrap? 

Technically, any shape that is not flat is equally difficult because they are all impossible. You cannot bend the wrapping paper to fit non-flat shapes. You could work around this by cutting and taping, but if any point is not flat, it’s impossible – at least not without creasing the wrapping paper. 

That being said, there are shapes that seem impossible to wrap but are actually technically doable. Take a donut shape, what we call a “torus” in math. This object sits inside four-dimensional space where, if you were a 4D creature, you could make a torus flat and wrap it— so potentially not very helpful for your holiday shopping since we’re 3D beings and can’t visualize what is going on.  

We can see this by taking a flat piece of paper. If you glued the long sides together, you would get a cylinder. You can’t do this in 3D because the paper would crinkle, but if you bend the paper and glue the short ends together, you’re able to take a flat piece of paper and bend it into a torus. 

What gift-container shape minimizes the amount of wrapping needed? 

In geometry, the isoperimetric inequality is a principle that tells us that a sphere is the most efficient shape for enclosing an item. An example of this is when we blow bubbles in a glass of water—the bubbles form as spheres because the air inside of them wants to take up as little space as possible due to the air pressure they face on the outside. In this sense, a sphere would be your most optimal shape for minimizing wrapping, except it wouldn’t really because, as we know, you can’t really wrap a sphere very well. 

The next best option would be a cube—not an arbitrary rectangular box—where all sides are equal in length. For a fixed volume, a cube minimizes the surface area that needs to be covered in wrapping paper. 

How about gift bags? 

It’s not always about optimization. As human beings, we tend to find things aesthetically pleasing when they’re not square. Gift bags, for example, are elongated in one direction. We like the look of this. A lot of it has to do with the golden ratio—1.618, also known as Phi—which we can find in nature, including in the radial spiral of pinecones or sunflower seeds, in art in the proportions of the Mona Lisa’s face and torso, and architecture, in the proportions of the Parthenon. I even have it tattooed on my arm. Many people think that some of these appearances in nature are just a coincidence or selection bias, but something about this ratio is very pleasing to the eye. 

A tree made entirely of ice with not a freezer nor piece of refrigeration equipment nor chainsaw and ice block in sight. You might call if a physics miracle.

A thank you to Nanowerk where I found the December 17, 2025 news item.

You can also read more about the icy Christmas tree in a December 17, 2025 University of Amsterdam (Netherlands) press release or in a December 19, 2025 article by Happy Jasta for homecrux.com

I wish you all the best of celebrations.