Author: IMC

From the Fibonacci sequence to the Bell numbers, there is more overlap between mathematics and poetry than you might think, says Peter Rowlett, who has found his inner poet.

People like to position maths as cold, hard logic, quite distinct from creative pursuits. Actually, maths often involves a great deal of creativity. As mathematician Sofya Kovalevskaya wrote, “It is impossible to be a mathematician without being a poet in soul.” Poetry is often constrained by rules, and these add to, rather than detract from, its creativity.

Rhyming poems generally follow a scheme formed by giving each line a letter, so that lines with matching letters rhyme. This verse from a poem by A. A. Milne uses an ABAB scheme:

What shall I call
My dear little dormouse?
His eyes are small,
But his tail is e-nor-mouse.

In poetry, as in maths, it is important to understand the rules well enough to know when it is okay to break them. “Enormous” doesn’t rhyme with “dormouse”, but using a nonsense word preserves the rhyme while enhancing the playfulness.

There are lots of rhyme schemes. We can count up all the possibilities for any number of lines using what are known as the Bell numbers. These count the ways of dividing up a set of objects into smaller groupings. Two lines can either rhyme or not, so AA and AB are the only two possibilities. With three lines, we have five: AAA, ABB, ABA, AAB, ABC. With four, there are 15 schemes. And for five lines there are 52 possible rhyme schemes!

Maths is also at play in Sanskrit poetry, in which syllables have different weights. “Laghu” (light) syllables take one unit of metre to pronounce, and “guru” (heavy) syllables take two units. There are two ways to arrange a line of two units: laghu-laghu, or guru. There are three ways for a line of three units: laghu-laghu-laghu; laghu-guru; and guru-laghu. For a line of four units, we can add guru to all the ways to arrange two units or add laghu to all the ways to arrange three units, yielding five possibilities in total. As the number of arrangements for each length is counted by adding those of the previous two, these schemes correspond with Fibonacci numbers.

Not all poetry rhymes, and there are many ways to constrain writing. The haiku is a poem of three lines with five, seven and five syllables, respectively – as seen in an innovative street safety campaign in New York City, above.

Some creative mathematicians have come up with the idea of a π-ku (pi-ku) based on π, which can be approximated as 3.14. This is a three-line poem with three syllables on the first line, one on the second and four on the third. Perhaps you can come up with your own π-ku – here is my attempt, dreamt up in the garden:

White seeds float,
dance,
spinning around.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Peter Rowlett

Math education outcomes in the United States have been unequal for decades. Learners in the top 10% socioeconomically tend to be about four grade levels ahead of learners in the bottom 10%—a statistic that has remained stubbornly persistent for 50 years.

To advance equity, policymakers and educators often focus on boosting test scores and grades and making advanced courses more widely available. Through this lens, equity means all students earn similar grades and progress to similar levels of math.

With more than three decades of experience as a researcher, math teacher and teacher educator, we advocate for expanding what equity means in mathematics education. We believe policymakers and educators should focus less on test scores and grades and more on developing students’ confidence and ability to use math to make smart personal and professional decisions. This is mathematical power—and true equity.

What is ‘equity’ in math?

To understand the limitations of thinking about equity solely in terms of academic achievements, consider a student whom We interviewed during her freshman year of college.

Jasmine took Algebra 1 in ninth grade, followed by a summer online geometry course. This put her on a pathway to study calculus during her senior year in an AP class in which she earned an A. She graduated high school in the top 20% of her class and went to a highly selective liberal arts college. Now in her first year, she plans to study psychology.

Did Jasmine receive an equitable mathematics education? From an equity-as-achievement perspective, yes. But let’s take a closer look.

Jasmine experienced anxiety in her math classes during her junior and senior years in high school. Despite strong grades, she found herself “in a little bit of a panic” when faced with situations that require mathematical analysis. This included deciding the best loan options.

In college, Jasmine’s major required statistics. Her counsellor and family encouraged her to take calculus over statistics in high school because calculus “looked better” for college applications. She wishes now she had studied statistics as a foundation for her major and for its usefulness outside of school. In her psychology classes, knowledge of statistics helps her better understand the landscape of disorders and to ask questions like, “How does gender impact this disorder?”

These outcomes suggest Jasmine did not receive an equitable mathematics education, because she did not develop mathematical power. Mathematical power is the know-how and confidence to use math to inform decisions and navigate the demands of daily life—whether personal, professional or civic. An equitable education would help her develop the confidence to use mathematics to make decisions in her personal life and realize her professional goals. Jasmine deserved more from her mathematics education.

The prevalence of inequitable math education

Experiences like Jasmine’s are unfortunately common. According to one large-scale study, only 37% of U.S. adults have mathematical skills that are useful for making routine financial and medical decisions.

A National Council on Education and the Economy report found that coursework for nine common majors, including nursing, required relatively few of the mainstream math topics taught in most high schools. A recent study found that teachers and parents perceive math education as “unengaging, outdated and disconnected from the real world.”

Looking at student experiences, national survey results show that large proportions of students experience anxiety about math class, low levels of confidence in math, or both. Students from historically marginalized groups experience this anxiety at higher rates than their peers. This can frustrate their postsecondary pursuits and negatively affect their lives.

How to make math education more equitable

In 2023, We collaborated with other educators from Connecticut’s professional math education associations to author an equity position statement. The position statement, which was endorsed by the Connecticut State Board of Education, outlines three commitments to transform mathematics education.

1. Foster positive math identities: The first commitment is to foster positive math identities, which includes students’ confidence levels and their beliefs about math and their ability to learn it. Many students have a very negative relationship with mathematics. This commitment is particularly important for students of colour and language learners to counteract the impact of stereotypes about who can be successful in mathematics.

A growing body of material exists to help teachers and schools promote positive math identities. For example, writing a math autobiography can help students see the role of math in their lives. They can also reflect on their identity as a “math person.” Teachers should also acknowledge students’ strengths and encourage them to share their own ideas as a way to empower them.

2. Modernize math content: The second commitment is to modernize the mathematical content that school districts offer to students. For example, a high school mathematics pathway for students interested in health care professions might include algebra, math for medical professionals and advanced statistics. With these skills, students will be better prepared to calculate drug dosages, communicate results and risk factors to patients, interpret reports and research, and catch potentially life-threatening errors.
3. Align state policies and requirements:The third commitment is to align state policies and school districts in their definition of mathematical proficiency and the requirements for achieving it. In 2018, for instance, eight states had a high school math graduation requirement insufficient for admission to the public universities in the same state. Other states’ requirements exceed the admission requirements. Aligning state and district definitions of math proficiency clears up confusion for students and eliminates unnecessary barriers.

What’s next?

As long as educators and policymakers focus solely on equalizing test scores and enrolment in advanced courses, we believe true equity will remain elusive. Mathematical power—the ability and confidence to use math to make smart personal and professional decisions—needs to be the goal.

No one adjustment to the U.S. math education system will immediately result in students gaining mathematical power. But by focusing on students’ identities and designing math courses that align with their career and life goals, we believe schools, universities and state leaders can create a more expansive and equitable math education system.

For more insights like this, visit our website at www.international-maths-challenge.com.

Credit of the article given to Megan Staples, The Conversation

Statisticians from the National University of Singapore (NUS) have introduced a new technique that accurately describes high-dimensional data using lower-dimensional smooth structures. This innovation marks a significant step forward in addressing the challenges of complex nonlinear dimension reduction.

Traditional data analysis methods often rely on Euclidean (linear) dependencies among features. While this approach simplifies data representation, it struggles to capture the underlying complex patterns in high-dimensional data, typically located close to low-dimensional manifolds.

To bridge this gap, manifold-learning techniques have emerged as a promising solution. However, existing methods, such as manifold embedding and denoising, have been limited by a lack of detailed geometric understanding and robust theoretical underpinnings.

The team, led by Associate Professor Zhigang Yao from the Department of Statistics and Data Science, NUS with his Ph.D. student Jiaji Su pioneered a novel method for effectively estimating low-dimensional manifolds hidden within high-dimensional data. This approach not only achieves cutting-edge estimation accuracy and convergence rates but also enhances computational efficiency through the utilization of deep Generative Adversarial Networks (GANs).

This work was conducted in collaboration with Professor Shing-Tung Yau from the Yau Mathematical Sciences Center (YMSC) at Tsinghua University. Part of the work comes from Prof. Yao’s collaboration with Prof. Yau during his sabbatical visit to the Center of Mathematical Sciences and Applications (CMSA) at Harvard University.

Their findings have been published as a methodology paper in the Proceedings of the National Academy of Sciences.

Prof. Yao delivered a 45-minute invited lecture on this research at the recent International Congress of Chinese Mathematicians (ICCM) held in Shanghai, Jan. 2–5, 2024.

Highlighting the significance of the work, Prof. Yao said, “By accurately fitting manifolds, we can reduce data dimensionality while preserving crucial information, including the underlying geometric structure. This represents a major leap in data analysis, enhancing both accuracy and efficiency. By providing a solution that overcomes the limitations of previous methods, our research paves the way for enhanced data analysis and offers valuable insights for diverse applications in the scientific community.”

Looking ahead, Yao’s research team is developing a new framework to process even more complex data, such as single-cell RNA sequence data, while continuing to collaborate with the YMSC team. This ongoing work promises to revolutionize the approach for the reduction and processing of complex datasets, potentially offering new insights into a range of scientific fields.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to National University of Singapore

Math education outcomes in the United States have been unequal for decades. Learners in the top 10% socioeconomically tend to be about four grade levels ahead of learners in the bottom 10% – a statistic that has remained stubbornly persistent for 50 years.

To advance equity, policymakers and educators often focus on boosting test scores and grades and making advanced courses more widely available. Through this lens, equity means all students earn similar grades and progress to similar levels of math.

With more than three decades of experience as a researcher, math teacher and teacher educator, WEadvocate for expanding what equity means in mathematics education. WEbelieve policymakers and educators should focus less on test scores and grades and more on developing students’ confidence and ability to use math to make smart personal and professional decisions. This is mathematical power – and true equity.

What is ‘equity’ in math?

To understand the limitations of thinking about equity solely in terms of academic achievements, consider a student whom WEinterviewed during her freshman year of college.

Jasmine took Algebra 1 in ninth grade, followed by a summer online geometry course. This put her on a pathway to study calculus during her senior year in an AP class in which she earned an A. She graduated high school in the top 20% of her class and went to a highly selective liberal arts college. Now in her first year, she plans to study psychology.

Did Jasmine receive an equitable mathematics education? From an equity-as-achievement perspective, yes. But let’s take a closer look.

Jasmine experienced anxiety in her math classes during her junior and senior years in high school. Despite strong grades, she found herself “in a little bit of a panic” when faced with situations that require mathematical analysis. This included deciding the best loan options.

In college, Jasmine’s major required statistics. Her counselor and family encouraged her to take calculus over statistics in high school because calculus “looked better” for college applications. She wishes now she had studied statistics as a foundation for her major and for its usefulness outside of school. In her psychology classes, knowledge of statistics helps her better understand the landscape of disorders and to ask questions like, “How does gender impact this disorder?”

These outcomes suggest Jasmine did not receive an equitable mathematics education, because she did not develop mathematical power. Mathematical power is the know-how and confidence to use math to inform decisions and navigate the demands of daily life – whether personal, professional or civic. An equitable education would help her develop the confidence to use mathematics to make decisions in her personal life and realize her professional goals. Jasmine deserved more from her mathematics education.

The prevalence of inequitable math education

Experiences like Jasmine’s are unfortunately common. According to one large-scale study, only 37% of U.S. adults have mathematical skills that are useful for making routine financial and medical decisions.

A National Council on Education and the Economy report found that coursework for nine common majors, including nursing, required relatively few of the mainstream math topics taught in most high schools. A recent study found that teachers and parents perceive math education as “unengaging, outdated and disconnected from the real world.”

Looking at student experiences, national survey results show that large proportions of students experience anxiety about math class, low levels of confidence in math, or both. Students from historically marginalized groups experience this anxiety at higher rates than their peers. This can frustrate their postsecondary pursuits and negatively affect their lives.

How to make math education more equitable

In 2023, WEcollaborated with other educators from Connecticut’s professional math education associations to author an equity position statement. The position statement, which was endorsed by the Connecticut State Board of Education, outlines three commitments to transform mathematics education.

1. Foster positive math identities: The first commitment is to foster positive math identities, which includes students’ confidence levels and their beliefs about math and their ability to learn it. Many students have a very negative relationship with mathematics. This commitment is particularly important for students of color and language learners to counteract the impact of stereotypes about who can be successful in mathematics.

A growing body of material exists to help teachers and schools promote positive math identities. For example, writing a math autobiography can help students see the role of math in their lives. They can also reflect on their identity as a “math person.” Teachers should also acknowledge students’ strengths and encourage them to share their own ideas as a way to empower them.

2. Modernize math content: The second commitment is to modernize the mathematical content that school districts offer to students. For example, a high school mathematics pathway for students interested in health care professions might include algebra, math for medical professionals and advanced statistics. With these skills, students will be better prepared to calculate drug dosages, communicate results and risk factors to patients, interpret reports and research, and catch potentially life-threatening errors.
3. Align state policies and requirements:The third commitment is to align state policies and school districts in their definition of mathematical proficiency and the requirements for achieving it. In 2018, for instance, eight states had a high school math graduation requirement insufficient for admission to the public universities in the same state. Other states’ requirements exceed the admission requirements. Aligning state and district definitions of math proficiency clears up confusion for students and eliminates unnecessary barriers.

What’s next?

As long as educators and policymakers focus solely on equalizing test scores and enrollment in advanced courses, WEbelieve true equity will remain elusive. Mathematical power – the ability and confidence to use math to make smart personal and professional decisions – needs to be the goal.

No one adjustment to the U.S. math education system will immediately result in students gaining mathematical power. But by focusing on students’ identities and designing math courses that align with their career and life goals, WEbelieve schools, universities and state leaders can create a more expansive and equitable math education system.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Megan Staples, The Conversation

A simple experiment and mathematical model suggest that when you snack on pistachios, you may need a surprisingly large bowl to accommodate the discarded shells.

Shelling your favourite snack nuts just got a lot easier: physicists have worked out the exact size of bowl to best fit discarded pistachio shells.

Ruben Zakine and Michael Benzaquen at École Polytechnique in Paris often find themselves discussing science in the cafeteria while eating pistachios. Naturally, they began wondering about the mathematics behind storing their snack refuse.

The researchers stuffed 613 pistachios into a cylindrical container to determine “packing density”, or the fraction of space taken up by whole nuts in their shells. Separately, they measured the packing density of the shells alone. In one experiment setup, the researchers poured the shells into a container and let them fall as they may, and in another they shook them into a denser, more efficient configuration.

Without shaking, the shells had about 73 per cent of the original packing density. Shaking decreased this number to 57 per cent. This suggests that, with any pistachio container, an additional half-sized container will hold shell refuse as long you occasionally shake the container while eating.

Zakine and Benzaquen backed up their findings by modelling pistachios as ellipsoids – three-dimensional shapes resembling squashed spheres – and their shells as hollow half-spheres and calculated their packing densities based on mathematical rules. These results confirmed the real-life experiments and suggested that the same ratios would work for other container shapes.

Despite these similarities, the researchers found about a 10 per cent discrepancy between the calculations and the real-life measurements. Zakine says that this is not surprising because pistachios are not perfect ellipsoids and have natural variations in shape. More broadly, it is tricky to calculate how best to pack objects into containers. So far, mathematics researchers have only had luck with doing calculations for spheres, like marbles, and uniform shapes like M&M’s, he says.

Going forward, the researchers want to run more complex calculations on a computer. But for now, they are looking forward to fielding mathematical questions whenever they serve pistachios at dinner parties.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Karmela Padavic-Callaghan*

Catering for students’ learning needs is something we all aim to do. But it can be challenging. Is it just about differentiation? What is the best way to differentiate? How do we put it into practice? Let’s explore some ideas, strategies and tips.

Differentiation

When you hear the word differentiation, what do you think of? Ability groupings? Open-ended tasks? Educational consultant Jennifer Bowden from the Mathematical Association of Victoria believes differentiation involves teachers considering “a whole range of different pedagogies … and making choices about pedagogical approaches based on the students that they teach”. In a nutshell it comes down to knowing your students and how they learn, so you can cater for their needs.

Find out what students know

Assessment is key to discovering what your students know – and don’t know! You can assess students to find out what knowledge they have, the concepts they understand and the skills they can apply to tasks.

Data from this assessment can then be used as a starting point to plan what you will teach.

Find out how students learn

You can go further than just understanding what your students know. Delve deeper and think about; what are your students’ learning behaviours? What are their attitudes towards learning maths? How do they learn best?

It’s important to note that this Is not about learning styles. It’s about knowing how a student:

• thinks and feels about maths
• becomes engaged in a topic, or problem
• responds to certain scaffolds
• makes connections between concepts
• applies what they have learnt.

When you understand your students on this level you have a greater insight into knowing how to best build their knowledge and skills.

Putting it into practice

Once you know your students well you are better prepared to meet their learning needs, but there are still many aspects to think about. Let’s unpack this further.

Planning for instruction

Maths expert Jennifer Bowden promotes the use of the instructional model known as launch, explore, summarise.

• Launch– begin with a question or a task for students to complete or explore.
• Explore– during this stage the teacher supports students at their different levels. Students can work on the same task, but it can be differentiated to extend or give extra support where needed by scaffolding. You can plan for the learning to be done independently, or in small groups.
• Summarise– upon competition of the lesson or task the students come together to share what they have learnt.

In an excellent podcast on the Maths Hub, Jennifer explains this model in greater detail.

Open-ended tasks

These rich tasks provide differentiation by output. Essentially all students are working on the same, or similar task, and students reach various outcomes, according to their individual knowledge and skill application.

Grouping students

There are times when you can best meet students’ needs through grouping them in certain ways. When doing so, consider the purpose of the groupings, and ensure the groups are flexible.

• You should be clear about the specific purpose of your groupings. What needs are you addressing by grouping students together? Are you extending them? Providing consolidation? Are you supporting them to ‘catch up’ on learning they have missed? Or providing intervention?
• Student groupings should beflexible and change according to their purpose. Sometimes groups are ability based, so students can complete different tasks, at different levels. Sometimes groups have mixed abilities so that students can use their various skills and levels of knowledge to problem solve and use their reasoning skills.

Student agency

Giving students a voice by encouraging them to discuss their learning can help you to understand their individual needs. Ask students about their learning; what they know and want to know, if they are feeling challenged and what helps them to learn. This feedback can help you plan and deliver lessons that cater for all student needs.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to The Mathematics Hub

왼쪽에 있는 14면체 다각형 Tile(1,1)은 약한 카이랄 비주기적 모노타일로 알려져 있습니다. 즉, 반사되지 않은 타일과 반사된 타일이 혼합된 타일링이 금지된다면, 이 도형은 비주기적으로만 타일링됩니다. 하지만 중앙과 오른쪽에 보이는 것처럼 가장자리를 수정함으로써, 비주기적 타일링만 허용하는 “스펙터” 라고 불리는 엄격한 카이랄 비주기적 모노타일이 생성됩니다. 출처: 워털루 대학교

최근 체리턴 컴퓨터 과학 대학의 크레이그 카플란 교수도 포함된 4명으로 구성된 국제 연구팀은 무한한 2차원 표면인 평면을 결코 반복되지 않는 패턴으로 타일링하는 단 하나의 도형을 발견했습니다.

이 발견은 수학자, 타일링 애호가, 그리고 일반 대중 모두를 매료시켰습니다.

그들이 “모자” 라고 부르는 이 13면체 다각형은 수학자들에게 비주기적 모노타일 또는 “아인슈타인” 으로 알려져 있는데, 이는 “하나의 돌” 을 의미하는 독일어 단어입니다.

하지만 연구팀의 가장 최근 발견은 다시 한번 수준을 한 단계 더 끌어올렸습니다. 그들은 첫 번째 도형과 관련이 있으면서 훨씬 더 엄격한 정의를 충족하는 또 다른 도형을 발견했습니다. “스펙터” 라고 불리는 이 새로운 도형은 도형의 거울상을 사용하지 않고도 결코 반복되지 않는 패턴으로 평면을 타일링합니다. 이러한 이유로 이 도형은 “뱀파이어 아인슈타인”—즉, 반사를 요구하지 않고 비주기적으로 타일링하는 도형—이라고도 불립니다.

카플란 교수는 설명합니다. “저희의 첫 번째 논문은 아인슈타인 문제를 해결했지만, 그 도형이 비주기적으로 타일링하기 위해 반사를 필요로 했기 때문에 사람들은 정당한 질문을 제기했습니다. ‘모자가 하는 일을 반사 없이 할 수 있는 도형이 있을까?’” “저희는 이 하위 문제를 해결할 뿐만 아니라, 첫 번째 논문이 나온 지 얼마 되지 않아 이 문제를 해결하는 도형을 찾게 된 것이 큰 행운이었습니다.”

수학자들에게는 모자와 그 거울상은 하나의 도형이지만, 물리적인 세계에서는 왼손잡이 도형과 오른손잡이 도형은 다르게 행동할 수 있습니다. 예를 들어, 오른손 장갑을 왼손에 낄 수는 없습니다.

카플란 교수는 말합니다. “만약 한쪽 면이 유약 처리된 모자 모양의 타일로 넓은 욕실 바닥을 비주기적으로 타일링한다면, 모자와 모자의 거울상이 모두 필요할 것입니다.”

하지만 최근 발견을 촉발한 것은 이러한 사소한 논쟁이 아니었습니다.

뱀파이어 아인슈타인의 발견은 몇 달 전 그의 호기심이 원래의 아인슈타인 발견으로 이어졌던 영국 요크셔 출신의 은퇴한 인쇄 기술자이자 스스로를 도형 애호가라고 칭하는 데이비드 스미스의 생각에서 시작되었습니다.

카플란 교수는 말합니다. “저희의 모자 논문이 온라인에 공개된 지 며칠 후, 데이브는 관련 도형을 가지고 놀다가 이상하게 행동하는 것처럼 보였다고 이메일을 보냈습니다.” “MC 에셔의 정신을 이어받은 작품으로 유명한 일본 수학자이자 예술가인 아라키 요시아키가 Tile(1,1)의 사진을 올렸는데, 데이브가 그것을 더 자세히 살펴보는 데 관심을 갖게 되었습니다.”

요시아키는 트위터에 흥미로운 질문을 올렸습니다. “새로운 비주기적 모노타일 Tile(1, 1.1)을 기반으로 한 비주기적 거북이 테셀레이션. 이 타일링에서 약 12.7%의 타일이 반사된다고 합니다. 녹색 타일이 한 예시입니다. 타일링 안에 반사된 거북이가 하나 더 숨어 있습니다. 반사된 것은 누구일까요?”

“요시가 Tile(1,1)을 거북이로 만들었는데, 그 사진에서는 다른 반사된 거북이를 찾기가 좀 어렵습니다. 하지만 데이브는 호기심을 갖게 되었습니다. ‘이 도형으로 타일링하되 반사 없이 하면 어떨까?’ 그렇게 하면서 데이브는 멈추지 않고 반복되지 않는 패턴으로 타일링을 점진적으로 바깥쪽으로 확장해 나갈 수 있다는 것을 발견했습니다.”

하지만 그때 이 도형은 또 다른 문제점을 안고 있었습니다. 카플란 교수가 설명하듯이, Tile(1,1)의 반사된 형태를 사용하면 패턴은 반복됩니다. 다시 말해, 그것은 주기적입니다. 하지만 Tile(1,1)의 직선 가장자리를 곡선으로 대체하여 수정하면, 그것은 뱀파이어 아인슈타인—즉, 반사 없이 무한 평면을 결코 반복되지 않는 패턴으로 타일링하는 단 하나의 도형—이 됩니다.

수학자와 타일링 애호가들에게 명백한 질문은 ‘다음은 무엇인가?’입니다.

카플란 교수는 말합니다. “우리는 이 문제의 다양한 변형을 제시할 수 있습니다.” “적어도 저에게는 가장 흥미로운 점은 이것이 3D로도 가능한지 여부입니다. 3차원에서 비주기적으로 반복되는 도형이 있다면 좋을 것입니다. 그러한 구성은 시각화하기 훨씬 어렵지만, 만약 모자처럼 비주기적으로만 타일링되는 3차원 도형—폴리폼—을 찾을 수 있다면, 계산적으로 증명하는 것이 그렇게 훨씬 더 어렵지는 않을 것입니다.”

“수학의 한 분야로서 타일링 이론은 아름답고, 실질적이며, 해결해야 할 매혹적인 문제들이 많습니다. 앞으로 해야 할 후속 연구는 부족하지 않습니다.”

“모자” 의 발견을 기념하는 햇페스트(Hatfest)는 7월 20일부터 21일까지 옥스퍼드 대학교 수학 연구소에서 개최될 예정입니다. 행사 첫째 날에는 일반 대중을 대상으로 한 타일링 강연과 워크숍이 진행될 예정이며, 둘째 날에는 물리학자와 수학자 등 폭넓은 청중을 대상으로 한 발표가 있을 예정입니다.

더 많은 통찰력을 얻으려면 저희 웹사이트 https://international-maths-challenge.com에 접속하세요.

기사 제공: 워털루 대학교 조 페트릭

المضلع ذو الـ14 ضلعاً Tile(1,1)، على اليسار، معروف بأنه بلاطة أحادية لا دورية ضعيفة التماثل – بمعنى آخر، إذا كانت التبليطات التي تمزج بين البلاطات غير المنعكسة والمنعكسة محظورة، فإنها تبلط فقط بشكل لا دوري. ومع ذلك، من خلال تعديل حوافها، كما هو موضح في الوسط واليمين، تم إنشاء بلاطات أحادية لا دورية صارمة التماثل تسمى ””أطياف”“ لا تسمح إلا بتبليطات غير دورية. المصدر: جامعة واترلو

اكتشف مؤخراً فريق دولي مكون من أربعة أشخاص، من بينهم الدكتور كريج كابلان أستاذ مدرسة تشيريتون لعلوم الحاسوب، شكلاً واحداً يبلط المستوى – وهو سطح ثنائي الأبعاد لا نهائي – في نمط لا يمكن أبداً جعله يتكرر.

أذهل هذا الاكتشاف علماء الرياضيات وهواة التبليط والجمهور على حد سواء.

الشكل، وهو مضلع ذو 13 ضلعاً أطلقوا عليه اسم ””القبعة”“، معروف لدى علماء الرياضيات باسم البلاطة الأحادية اللادورية أو ””آينشتاين”“، وهي كلمة ألمانية تعني ””حجر واحد”“.

لكن اكتشاف الفريق الأخير رفع المستوى مرة أخرى. لقد وجدوا شكلاً آخر، مرتبطاً بالأول، يلبي تعريفاً أكثر صرامة. أُطلق على الشكل الجديد اسم ””الطيف”“، وهو يبلط المستوى في نمط لا يتكرر أبداً دون استخدام صور مرآوية للشكل. لهذا السبب، أُطلق عليه أيضاً اسم ””آينشتاين مصاص الدماء”“ – وهو شكل يبلط بشكل لا دوري دون الحاجة إلى انعكاسه.

””حلت ورقتنا الأولى مشكلة آينشتاين، ولكن نظراً لأن الشكل يتطلب انعكاساً للتبليط بشكل لا دوري، طرح الناس سؤالاً مشروعاً: هل هناك شكل يمكنه فعل ما تفعله القبعة ولكن بدون انعكاس،”“ يشرح كابلان. ””كان من حسن حظنا أننا وجدنا شكلاً لا يحل هذه المشكلة الفرعية فحسب، بل حلها أيضاً بعد وقت قصير من نشر الورقة الأولى.”“

بالنسبة لعلماء الرياضيات، تعتبر القبعة وصورتها المرآوية شكلاً واحداً، ولكن في العالم المادي يمكن أن تتصرف الأشكال اليسرى واليمنى بشكل مختلف. لا يمكنك، على سبيل المثال، ارتداء قفاز أيمن على يدك اليسرى.

””إذا قمت بتبليط أرضية حمام كبيرة بشكل لا دوري باستخدام بلاط على شكل قبعة تم تزجيجه من جانب واحد، فستحتاج إلى قبعات وصور مرآوية للقبعات،”“ يقول كابلان.

لكن لم يكن هذا الجدل هو الذي حفز الاكتشاف الأخير.

بدأ اكتشاف آينشتاين مصاص الدماء بتأملات ديفيد سميث، وهو فني طباعة متقاعد وهاوي أشكال كما يصف نفسه من يوركشاير، إنجلترا، الذي أدى فضوله قبل أشهر إلى اكتشاف آينشتاين الأصلي.

””أرسل ديف لنا بريداً إلكترونياً بعد يومين من نشر ورقة القبعة عبر الإنترنت ليقول إنه كان يعبث بشكل مرتبط بها وبدا أنه يتصرف بشكل غريب،”“ يقول كابلان. ””يوشياكي أراكي، وهو عالم رياضيات ياباني وفنان معروف يعمل بروح إم سي إيشر، نشر صوراً لـ Tile(1,1) جعلت ديف مهتماً بالنظر إليها بشكل أعمق.”“

نشر يوشياكي سؤالاً مثيراً للاهتمام على تويتر: “تبليط سلحفاة لا دوري مبني على بلاطة أحادية لا دورية جديدة Tile(1, 1.1). في التبليط، يُقال إن حوالي 12.7% من البلاطات منعكسة. اللون الأخضر هو مثال. هناك سلحفاة منعكسة أخرى مخفية في التبليط. من هي المنعكسة؟'”

””حوّل يوشي Tile(1,1) إلى سلاحف ومن الصعب بعض الشيء رؤية السلحفاة المنعكسة الأخرى في تلك الصورة. لكن هذا أثار فضول ديف. ماذا لو قمنا بالتبليط بهذا الشكل ولكن بدون انعكاسات؟ وبينما كان يفعل ذلك، وجد ديف أنه يمكنه بناء تبليطات تتوسع تدريجياً للخارج في نمط لا يتوقف ولا يتكرر.”“

لكن هذا الشكل جاء مع جدل مختلف. كما يشرح كابلان، إذا استخدمت انعكاسات Tile(1,1) فإن النمط يتكرر. بمعنى آخر، إنه دوري. ولكن إذا تم تعديل Tile(1,1) عن طريق استبدال حوافه المستقيمة بمنحنيات، فإنه يصبح آينشتاين مصاص دماء – شكل واحد بدون انعكاس يبلط المستوى اللانهائي في نمط لا يمكن أبداً جعله يتكرر.

السؤال الواضح لعلماء الرياضيات وهواة التبليط هو ما الخطوة التالية؟

””يمكننا طرح العديد من الاختلافات للمشكلة،”“ يقول كابلان. ””الأكثر إثارة للاهتمام، بالنسبة لي على الأقل، هو ما إذا كان يمكن القيام بذلك في الأبعاد الثلاثية. سيكون من الجميل أن يكون لدينا شكل يتكرر بشكل غير دوري في ثلاثة أبعاد. مثل هذه التركيبات أصعب بكثير في التصور، ولكن من الناحية الحسابية ليس من الصعب إثباتها إذا كنا محظوظين بما يكفي للعثور على شكل ثلاثي الأبعاد – متعدد الأشكال – الذي مثل القبعة يبلط فقط بشكل لا دوري.”“

””نظرية التبليط كفرع من الرياضيات جميلة وملموسة وتحتوي على الكثير من المشاكل المثيرة للاهتمام التي يجب حلها. لا يوجد نقص في العمل المتابع الذي يجب القيام به.”“

سيقام مهرجان القبعة، احتفالاً باكتشاف ””القبعة”“، في معهد الرياضيات بجامعة أكسفورد من 20 إلى 21 يوليو. سيتضمن اليوم الأول من الحدث محادثات وورش عمل حول التبليط موجهة للجمهور العام، بينما سيتضمن اليوم الثاني عروضاً تقديمية موجهة لجمهور واسع من الفيزيائيين وعلماء الرياضيات.

لمزيد من هذه الرؤى، قم بتسجيل الدخول إلى موقعنا https://international-maths-challenge.com

تُنسب المقالة إلى جو بيتريك، جامعة واترلو

左側的 14 邊形 Tile(1,1) 被稱為弱手性非週期單瓦 — 換句話說，如果禁止混合未反射和反射瓦片的鋪砌，則它僅以非週期性方式鋪砌。 然而，透過修改其邊緣，如中心和右側所示，可以創建稱為「幽靈」的嚴格手性非週期單瓦，這些單瓦僅允許非週期性鋪砌。 圖片來源：滑鐵盧大學

最近，一個由四人組成的國際團隊（包括 Cheriton 電腦科學學院教授 Craig Kaplan 博士）發現了一種單一形狀，可以鋪滿平面（一個無限的二維表面），其圖案永遠無法重複。

這一發現讓數學家、鋪砌愛好者和公眾都為之著迷。

這種形狀，一個 13 邊形，他們稱之為「帽子」，數學家稱之為非週期單瓦或「愛因斯坦」，這個德語詞的意思是「一塊石頭」。

但該團隊最近的發現再次提高了標準。 他們發現了另一種與第一種形狀相關的形狀，它符合更嚴格的定義。 這種被稱為「幽靈」的新形狀，以一種永遠不會重複的模式鋪滿平面，而無需使用該形狀的鏡像。 因此，它也被稱為「吸血鬼愛因斯坦」— 一種無需反射即可非週期性鋪砌的形狀。

「我們的第一篇論文解決了愛因斯坦問題，但由於該形狀需要反射才能進行非週期性鋪砌，因此人們提出了一個合理的問題：是否有一種形狀可以像帽子一樣，但無需反射，」Kaplan 解釋說。 「我們很幸運地找到了一種形狀，它不僅解決了這個子問題，而且在第一篇論文發表後不久就解決了它。」

對於數學家來說，帽子及其鏡像是一個單一的形狀，但在物理世界中，左手和右手的形狀表現可能不同。 例如，你不能在左手上戴右手手套。

「如果你用帽子形狀的瓷磚以非週期性方式鋪砌一個大型浴室地板，這些瓷磚的一面已經上釉，那麼你需要帽子和帽子的鏡像，」Kaplan 說。

但並非這種吹毛求疵促成了最近的發現。

吸血鬼愛因斯坦的發現始於 David Smith 的沉思，他是一位退休的印刷技術員，也是一位來自英國約克郡的自稱形狀愛好者，幾個月前他的好奇心促成了最初的愛因斯坦發現。

「在我們的帽子論文上線幾天后，Dave 給我們發了一封電子郵件，說他一直在玩一種相關的形狀，這種形狀的表現似乎很奇怪，」Kaplan 說。 「日本數學家兼著名藝術家 Yoshiaki Araki 的作品具有 MC Escher 的精神，他發布了 Tile(1,1) 的圖片，這引起了 Dave 的興趣，並進一步研究它。」

Yoshiaki 在 Twitter 上發布了一個有趣的問題：「基於新的非週期單瓦 Tile(1, 1.1) 的非週期海龜鑲嵌。在鑲嵌中，據說大約 12.7% 的瓷磚被反射。綠色的是一個例子。在鑲嵌中還隱藏著另一隻被反射的海龜。誰是被反射的？」

「Yoshi 將 Tile(1,1) 變成了海龜，很難在圖片中看到另一隻被反射的海龜。但這引起了 Dave 的好奇。如果我們用這種形狀鋪砌，但不進行反射呢？當他這樣做時，Dave 發現他可以逐步向外構建鋪砌，其模式不會停止也不會重複。」

但這種形狀帶有不同的吹毛求疵。 正如 Kaplan 解釋的那樣，如果你使用 Tile(1,1) 的反射，則圖案會重複。 換句話說，它是週期性的。 但是，如果通過用曲線替換其直線邊緣來修改 Tile(1,1)，它就會變成吸血鬼愛因斯坦 — 一種單一形狀，無需反射即可在永不重複的模式中鋪砌無限平面。

對於數學家和鋪砌愛好者來說，顯而易見的問題是下一步是什麼？

「我們可以提出這個問題的許多變體，」Kaplan 說。 「至少對我來說，最有趣的是這是否可以在 3D 中完成。如果有一種形狀可以在三個維度中非週期性地重複，那就太好了。這樣的結構很難視覺化，但如果我們有幸找到一種三維形狀（一種多形體），像帽子一樣僅以非週期性方式鋪砌，那麼在計算上證明它並不會困難得多。」

「作為數學的一個分支，鋪砌理論是美麗的、有形的，並且有很多有趣的問題需要解決。不乏後續工作要做。」

Hatfest，一個慶祝「帽子」發現的活動，將於 7 月 20 日至 21 日在牛津大學數學研究所舉行。 該活動的第一天將以針對普通觀眾的關於鋪砌的講座和研討會為特色，而第二天將以針對廣泛的物理學家和數學家觀眾的演示為特色。

如需更多此類見解，請登入我們的網站 https://international-maths-challenge.com

本文的作者為滑鐵盧大學的 Joe Petrik

左侧的 14 边形 Tile(1,1) 被称为弱手性非周期性单瓦——换句话说，如果禁止混合未反射和反射瓦片的平铺，那么它只能非周期性地平铺。然而，通过修改其边缘，如中心和右侧所示，创建了称为 “spectres” 的严格手性非周期性单瓦，它们只允许非周期性平铺。图片来源：滑铁卢大学

最近，一个由四人组成的国际团队，其中包括切里顿计算机科学学院的 Craig Kaplan 博士，发现了一种可以平铺平面的单一形状——一个无限的二维表面——其图案永远无法重复。

这一发现让数学家、平铺爱好者和公众都为之着迷。

这种形状，一个 13 边形，他们称之为 “the hat”，在数学家眼中被称为非周期性单瓦或 “einstein”，这个德语单词的意思是 “one stone”。

但该团队最近的发现再次提高了标准。他们发现了另一种形状，与第一种形状相关，它符合更严格的定义。这种新形状被称为 “specter”，它以一种永不重复的模式平铺一个平面，而无需使用该形状的镜像。因此，它也被称为 “vampire einstein”——一种无需反射即可非周期性平铺的形状。

“我们的第一篇论文解决了 einstein 问题，但由于该形状需要反射才能非周期性地平铺，人们提出了一个合理的问题：是否存在一种形状可以像 hat 一样做到这一点，但无需反射，” Kaplan 解释说。“我们很幸运地找到了一个不仅解决了这个子问题，而且在第一篇论文之后不久就解决了它的形状。”

对于数学家来说，hat 及其镜像是一个单一的形状，但在物理世界中，左手和右手的形状表现可能不同。例如，你不能在左手上戴右手手套。

“如果你用一面上了釉的 hat 形瓷砖非周期性地铺设一个大型浴室地板，你需要 hat 和 hat 的镜像，” Kaplan 说。

但并非是这种吹毛求疵促成了最近的发现。

vampire einstein 的发现始于 David Smith 的沉思，他是一位退休的印刷技术员，也是一位来自英国约克郡的自称形状爱好者，几个月前他的好奇心促成了最初的 einstein 发现。

“在我们的 hat 论文上线几天后，Dave 给我们发了一封电子邮件，说他一直在摆弄一个相关的形状，这个形状的行为似乎很奇怪，” Kaplan 说。“日本数学家兼著名艺术家 Yoshiaki Araki 的作品具有 MC Escher 的精神，他发布了 Tile(1,1) 的图片，这引起了 Dave 对它的进一步关注。”

Yoshiaki 在 Twitter 上发布了一个有趣的问题：“基于新的非周期性单瓦 Tile(1, 1.1) 的非周期性海龟镶嵌。在镶嵌中，据说大约 12.7% 的瓷砖被反射。绿色的是一个例子。还有一个反射的海龟隐藏在镶嵌中。谁被反射了？”

“Yoshi 将 Tile(1,1) 变成了海龟，并且很难在该图片中看到另一个反射的海龟。但这引起了 Dave 的好奇。如果我们用这种形状平铺，但不进行反射呢？当他这样做时，Dave 发现他可以逐步向外构建平铺，其模式不会停止也不会重复。”

但随后这种形状出现了一个不同的问题。正如 Kaplan 解释的那样，如果你使用 Tile(1,1) 的反射，则该模式会重复。换句话说，它是周期性的。但是，如果通过用曲线替换其直线边缘来修改 Tile(1,1)，它就会变成 vampire einstein——一种无需反射即可在永不重复的模式中平铺无限平面的单一形状。

对于数学家和平铺爱好者来说，显而易见的问题是接下来会发生什么？

“我们可以提出许多问题的变体，” Kaplan 说。“至少对我来说，最有趣的是这是否可以在 3D 中完成。如果有一种形状可以在三个维度上非周期性地重复，那就太好了。这种结构更难可视化，但从计算上来说，证明我们是否足够幸运地找到一个三维形状（一种多形体）并不难，就像 hat 一样，它只能非周期性地平铺。”

“平铺理论作为数学的一个分支是美丽的、有形的，并且有很多有趣的问题需要解决。不乏后续工作要做。”

Hatfest 是对 “the Hat” 发现的庆祝活动，将于 7 月 20 日至 21 日在牛津大学数学研究所举行。该活动的第一天将以针对普通观众的关于平铺的讲座和研讨会为特色，而第二天将以针对广泛的物理学家和数学家观众的演讲为特色。

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本文的作者是滑铁卢大学的 Joe Petrik