nrich problem solving symmetry

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NRICH Transformations

This list supports teaching of transformations in secondary mathematics. It provides investigations, problems and games from NRICH as well as classroom activities on the STEM Learning website that compliment them.

Here are a selection of activities selected by the NRICH team.

• Shady Symmetry   This problem looks at how many symmetrical patterns can be created by shading in either four squares on a 3 by 3 square grid or two triangles on a 3 by 3 isometric grid. Students are then invited to alter the shading and investigate further the symmetrical patterns formed.
• Attractive Tablecloths This problem considers table cloths made using different colours on a 5 by 5 grid. Students consider different rules concerning line symmetry and rotational symmetry. The investigation is extended to symmetry in an n by n tablecloth.
• Growing Rectangles The problem investigates possible lengths and widths for rectangles of a given area. Students consider what happens as various shapes are enlarged. As extension is made to cuboid.

These are just a few of the activities on  transformations  that you can find on the  NRICH curriculum pages .

The activities below, taken from the STEM Learning website, complement the NRICH activities above.

Working with Photos

Quality Assured Category: Mathematics Publisher: cre8ate maths

This activity from Cre8ate maths the enlargements element of the NRICH activity 'Growing Rectangles'. 

The activity consists of four sets of photographs and the task is to identify which have been enlarged from an original. Students must justify their decisions. The final two sets provide lots of room for discussion.

Transforming Shapes SS7

Quality Assured Category: Mathematics Publisher: Department for Education

This DfE Standards Unit covers the majority of topics covered by the NRICH activities on transformations. The activities are an ideal introduction to the NRICH activities.

Topics covered include translation, rotation, reflection and combinations of these transformations. The unit does presume some prior familiarity with transformations and equations of lines of reflection such as y = 3, x = 4, y = x and y = - x

Quality Assured Category: Mathematics Publisher: SMILE

This SMILE resource supports the teaching of the NRICH problems 'Shady Symmetry' and 'Attractive Tablecloths'.

There are two packs of games, investigations and worksheets that cover reflection in one and two lines of symmetry, Rangoli patterns, combined reflections and symmetry of 3-D solids.

Reflection and Translation

Quality Assured Category: Mathematics Publisher: The Virtual Textbook

This 'Virtual Textbook' resource supports the NRICH activities 'Shady Symmetry' and 'Attractive Tablecloths'.

The activities include identifying lines of symmetry in 2D shapes and creating kaleidoscope patterns.

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• The Number System and Place Value
• Calculations and Numerical Methods
• Fractions, Decimals, Percentages, Ratio and Proportion
• Properties of Numbers
• Patterns, Sequences and Structure
• Algebraic expressions, equations and formulae
• Coordinates, Functions and Graphs

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Resources tagged with: Symmetry

There are 35 results.

Attractive Rotations

Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...

Shady Symmetry

How many different symmetrical shapes can you make by shading triangles or squares?

Reflecting Squarely

In how many ways can you fit all three pieces together to make shapes with line symmetry?

Attractive Tablecloths

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes".

Gibraltar Geometry

Take a look at the photos of tiles at a school in Gibraltar. What questions can you ask about them?

Flower Power

Create a symmetrical fabric design based on a flower motif - and realise it in Logo.

Troublesome Dice

When dice land edge-up, we usually roll again. But what if we didn't...?

Paint Rollers for Frieze Patterns.

Proofs that there are only seven frieze patterns involve complicated group theory. The symmetries of a cylinder provide an easier approach.

Topkapi Palace

These images are taken from the Topkapi Palace in Istanbul, Turkey. Can you work out the basic unit that makes up each pattern? Can you continue the pattern? Can you see any similarities and differences in the designs?

Drawing Celtic Knots

Here is a chance to create some Celtic knots and explore the mathematics behind them.

Can all but one square of an 8 by 8 Chessboard be covered by Trominoes?

Notes on a Triangle

Can you describe what happens in this film?

Dancing with Maths

An article for students and teachers on symmetry and square dancing. What do the symmetries of the square have to do with a dos-e-dos or a swing? Find out more?

Rotations Are Not Single Round Here

I noticed this about streamers that have rotation symmetry : if there was one centre of rotation there always seems to be a second centre that also worked. Can you find a design that has only one centre of rotation ? Or if you thought that was impossible, could you say why ?

One Reflection Implies Another

When a strip has vertical symmetry there always seems to be a second place where a mirror line could go. Perhaps you can find a design that has only one mirror line across it. Or, if you thought that was impossible, could you explain why ?

A Roll of Patterned Paper

A design is repeated endlessly along a line - rather like a stream of paper coming off a roll. Make a strip that matches itself after rotation, or after reflection

Symmetric Trace

Points off a rolling wheel make traces. What makes those traces have symmetry?

Emmy Noether

Find out about Emmy Noether, whose ideas linked physics and algebra, and whom Einstein described as a 'creative mathematical genius'.

An irregular tetrahedron has two opposite sides the same length a and the line joining their midpoints is perpendicular to these two edges and is of length b. What is the volume of the tetrahedron?

Pattern Power

Mathematics is the study of patterns. Studying pattern is an opportunity to observe, hypothesise, experiment, discover and create.

Two Triangles in a Square

Given that ABCD is a square, M is the mid point of AD and CP is perpendicular to MB with P on MB, prove DP = DC.

Some local pupils lost a geometric opportunity recently as they surveyed the cars in the car park. Did you know that car tyres, and the wheels that they on, are a rich source of geometry?

Tournament Scheduling

Scheduling games is a little more challenging than one might desire. Here are some tournament formats that sport schedulers use.

Classifying Solids Using Angle Deficiency

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry

The Frieze Tree

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

Frieze Patterns in Cast Iron

A gallery of beautiful photos of cast ironwork friezes in Australia with a mathematical discussion of the classification of frieze patterns.

A red square and a blue square overlap. Is the area of the overlap always the same?

Square Pizza

Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

Prime Magic

Place the numbers 1, 2, 3,..., 9 one on each square of a 3 by 3 grid so that all the rows and columns add up to a prime number. How many different solutions can you find?

Rhombicubocts

Each of these solids is made up with 3 squares and a triangle around each vertex. Each has a total of 18 square faces and 8 faces that are equilateral triangles. How many faces, edges and vertices does each solid have?

An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?

The ten arcs forming the edges of the "holly leaf" are all arcs of circles of radius 1 cm. Find the length of the perimeter of the holly leaf and the area of its surface.

Eight Dominoes

Using the 8 dominoes make a square where each of the columns and rows adds up to 8

A Problem of Time

Consider a watch face which has identical hands and identical marks for the hours. It is opposite to a mirror. When is the time as read direct and in the mirror exactly the same between 6 and 7?

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NRICH - Coordinate Challenge

Subject: Mathematics

Age range: 7-11

Resource type: Other

Last updated

22 February 2018

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Anon1606287245649783

A logic problem from nrich to place 10 different letters on a grid using co-ordinates and knowledge about the reflective and rotational symmetry of the letters. The activity can be completed online using the interactive grid on the website, or the grid and letters can be printed from the PDF document. A high quality resource which could be used in KS2 to teach problem solving, logical thinking, working systematically and methods of trial and improvement.

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Maths: Interactive Symmetry Activities

• symmetry problems
• symmetry objectives
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Transcript of Maths: Interactive Symmetry Activities

Aim:To identify symmetry and investigate with symmetry

Objectives:Identify line and rotational symmetrySolve problems involving symmetry

Nrich: Online interactive symmetry investigationAvailable at: http://nrich.maths.org/7610

Have fun solving the symmetry problems!

Here’s one example of the completed Table Cloth Design…

Maths is Fun: Online interactive rotational symmetry activityAvailable at:

http://www.mathsisfun.com/geometry/symmetry-artist.html

Have fun creating your own designs that have rotational symmetry!

TRANSFORMATION GAME

Maths Week 2010 Symmetry & Art From //mathmuse.sci.ibaraki.ac.jp/urabe

Question Bank - pelangibooks.com Ace Add Maths F4 Online... · dalam bentuk am. 2. ... the equation of the axis of symmetry. persamaan paksi simetri. 3. ... the axis of symmetry.

IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS…web.engr.oregonstate.edu/~zhange/images/symmetry_vis.pdf · Interactive Visualization of Rotational Symmetry Fields on Surfaces

Symmetry and Aesthetics in Contemporary Physicsweb.physics.ucsb.edu/~jatila/sym-lectures/Class_02_2016.pdf · symmetry translational symmetry reflection symmetry screw symmetry =

Targeting Maths · 2019-08-15 · Targeting Maths AUSTRALIAN CURRICULUM EDITION ∑ The Complete Mathematics Program for your school: Student books, Teaching guides, Interactive CD-ROMs,

WALT create designs with rotational symmetry Another of Mrs Robertson’s magical maths lessons!

Mathematics & Music: Symmetry & Symbiosisplynch/Talks/Maths-and-Music.pdf · Music & Maths There are many parallels between music and maths: Structure Symmetry Pattern etc. But music

DEVELOPMENT OF INTERACTIVE MATHS TEACHING TOOL FOR ...

Interactive Voting - 1995 maths paper b

Unit 2- Edexcel Maths (Geometry Revision)glynmathsgcse.wikispaces.com/file/view/Unit2geometry.pdf/...Unit 2- Edexcel Maths (Geometry Revision) Topics: Symmetry of 2d shapes and special

Symmetry Line Symmetry Rotational (Order) Symmetry Translational Symmetry Creating Rotational Symmetry MTH 2-19a MTH 3-19a.

Year 1 School Closure Reading Interactive Learning Linksd6vsczyu1rky0.cloudfront.net/37668_b/wp-content/uploads/...Year 1 School Closure Interactive Learning Links Maths PlanIt Maths

butterfly symmetry - MATHS WEEK IRELAND · Butterfly Symmetry INSTRUCTIONS: 1. Pull out a sheet of paper from your sketchbook, or you can use any kind of thicker paper you have to

Symmetry Line Symmetry Rotational (Order) Symmetry Creating Rotational Symmetry S4.

Vocabulary symmetry line symmetry line of symmetry rotational symmetry center of symmetry order of symmetry magnitude of symmetry.

Easter Egg Symmetry - Number Loving · Maths Helps, Games and Puzzles Visit Easter Egg Symmetry Draw a pattern on one side of the Easter Egg. Place a mirror on the mirror line.

NCERT Solutions for Class 7 Maths Chapter 14icisschool.com/ContentDoc/51201891232.pdf · 2018-01-05 · NCERT Solutions for Class 7 Maths Chapter 14 Symmetry Class 7 Chapter 14 Symmetry

AD Computer Science Maths Building - University of Warwick · Maths Building - Making Interactive Maths (3yrs+) Maths Buskers (5yrs+) WMG - IMC Egg Drop (5yrs+) Snakes and Ladders

Interactive Voting - 1999 Maths Papera

3-1 Symmetry. Symmetry All Around Us Symmetry at the Beach Symmetry at the Beach Line Symmetry & Rotational Symmetry - All you need to Know + Symmetry.

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• Coordinates, Functions and Graphs

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The Problem-Solving Schools' Charter

The NRICH team has developed this Charter to help you reflect on how you currently promote mathematical problem-solving in your school

Values and ethos

We have a shared belief that:

• Mathematical ability is not fixed: everyone can learn and make progress
• Problem-solving often involves taking wrong turns and making mistakes: every learner has the right to struggle and the right to enjoy success
• Everyone should have the opportunity to develop the skills and attitudes necessary to become confident problem-solvers
• Problem-solving can motivate learners to learn new mathematics, apply previous learning and make mathematical connections

Leadership and professional development

In our setting:

• Our staff promote positive attitudes towards problem-solving
• Time is set aside to discuss problem-solving in our meetings
• Our displays, newsletters, website, and social media content celebrate problem-solving for all
• Our monitoring system ensures that priority is given to problem-solving and mathematical thinking
• We engage with printed, online and face-to-face professional development opportunities offered by subject organisations

Curriculum, pedagogy and assessment

We are committed to:

• Regularly embedding non-standard problem-solving opportunities in our maths curriculum for all
• Ensuring that problems, and classroom support, offer opportunities for all to experience both struggle and success
• Allocating time to developing key problem-solving skills and positive attitudes
• Including non-standard problems in our internal/formative assessments
• Liaising with other subjects so that meaningful cross-curricular links can be made

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• Create a safe environment in which learners explore, take risks, and appreciate the value of learning from their mistakes
• Celebrate multiple approaches to solving problems and discuss the merits of the different strategies offered
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• Celebrate the mathematical thinking of every learner

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• Learners to engage with school Maths Club(s) and high quality maths books, ideally stocked by the school library
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• Frequently Asked Questions

Physical Intuition vs. “Math”

A common theme in discussions online of the problems of fundamental theoretical physics is that the subject has gotten “lost in math”, losing touch with “physical intuition”. In such discussions, when people refer to “math” it’s hard to figure out what they mean by this. In the case of Sabine Hossenfelder’s “Lost in Math” you can read her book and get some idea of what specifically she is referring to, but usually the references to “math” don’t come with any way of finding out what the person using the term means by it. Here I’ll mostly leave “math” in quotation marks, since the interesting issue of what this means is not being addressed.

“Physical intuition” is also a term whose meaning is not so clear. Sometimes I see it used in an obviously naive way, referring to our understanding of the physical world that comes from our everyday interaction with it and the feeling this gives us for how classical mechanics, electromagnetism, thermodynamics work. Some people are quite devoted to the idea that this is the way to understand fundamental physics, sometimes taking this as far as skepticism about subjects like quantum mechanics.

Usually though, the term is not being used in this naive sense, but as meaning something more like “the sort of understanding of physical phenomena someone has who has spent a great deal of time working out many examples of how to apply physics theory, so can use this to see patterns and guess how some new example will work out”. This is contrasted to the person lacking such intuition, who will have to fall back on “math”, in this situation meaning writing down the general textbook equations and mathematically manipulating them to produce an answer appropriate for the given example, without any intuitive understanding of the result of the calculation. This is what we expect to see in students who are just learning a new subject, haven’t yet worked out enough examples to have the right intuition.

If the question though is not how to apply well understood fundamental theory to a new example, but how to come up with a better fundamental theory, I’d like to make the provocative claim that “physical intuition” is not going to be that helpful. New breakthroughs in fundamental theory have the characteristic of being unexpectedly different than earlier theory. The best way to come up with such breakthroughs is from new experimental results that conflict with the standard theory and point to a better one. But, what if you don’t have such results? It seems to me that in that case your best hope is “math”.

Here’s a list of the great breakthroughs of fundamental physics in the 20th century, with some comments on the role of “physical intuition” and “math”.

• Special relativity : According to physical intuition, if I’m emitting a light ray and speed up in its direction, so will the the speed of the light ray. The crucial input was from experiment (Michelson-Morley), which showed that light always travels at the same speed. Finding a sensible theory of mechanics with this property was largely “math”.
• General relativity : There’s a long argument about the role of “math” here, but I think the only way to develop “physical intuition” about curved spacetime is to start by learning Riemannian geometry (which Einstein did).
• Quantum mechanics : Here again, a crucial role was played by experimental results, those on atomic spectra. A large part of the development of the subject was applying “math” to the mysterious spectra for which there was zero “physical intuition”. Later on, a better understanding of the theory and better calculational methods involved bringing in a large amount of new “math” to physics, especially the theory of unitary representations of groups.
• Yang-Mills theory : This was pretty much pure “math”: replacing a U(1) gauge theory by an SU(2) gauge theory.
• G ell-Mann’s eight-fold way : Pure “math”.
• The Anderson-Higgs mechanism : The funny thing here is that Anderson did get this out of “physical intuition”, based on what he knew from superconductivity. Particle theorists ignored him (especially when it came time for a Nobel Prize), and their papers about this were often mainly “math”, more specifically argumentation about how the mathematics of gauge symmetry could give a loophole to a theorem (the Goldstone theorem).
• The unified electroweak theory : Looks to me more like “math” than “physical intuition”.
• QCD and asymptotic freedom : David Gross famously had the “physical intuition” that the effective coupling grows in the ultraviolet for all QFTs, based on experience with a wide range of examples. He set a mathematical problem for his student (Frank Wilczek), and when the “mathematics” was finally sorted out, they realized the usual physical intuition for QFTs had to be replaced by something completely different.

Making a list instead of the great disasters of 20th century theoretical physics, there’s

• Supersymmetry : OK, this one is “math”. I suspect though that the problem here is that the “math” is not quite right, but missing some other needed new ideas.
• String theory : As we’re told in countless books and TV programs, this starts with a new “physical intuition”: instead of taking point particles as primitive objects, take the vibrational modes of a vibrating string. Developing the implications of this certainly involves a lot of “math”, but the new fundamental idea is a physical one (and it’s wrong, but that’s a different story…).

20 Responses to Physical Intuition vs. “Math”

Let’s expand some of these a little bit though: Special relativity–>Logic shows that relativity (the Galilean kind) is in conflict with electromagnetism and Galilei’s transformations, intuition shows the third must go and be substituted by something that respects the first two, math was used to derive that.

General Relativity–>Logic shows gravity must be updated to take special relativity into account and the most naive implementation will break the equivalence principle, intuition says the equivalence principle is fundamental and one must find a way to preserve it, logic and some thought experiments (Ehrenfests’s paradox) say that because of time dilation this inevitably involves Riemannian geometry, maths follows

Quantum mechanics—>Experiment gives funny results, intuition shows that to explain them one needs some seemingly counter-intuitive ansatze (Bohr’s momentum quantization, the photoelectric effect), which are then progressively systematized using more and more sophisticated maths.

Yang Mills/Gell-mann/Electroweak theory—>Intuition shows that perhaps it is worth to extent the techniques we used to classify spin to systems where there seem to be approximate degeneracies analogous to spin (isospin/strangeness), group theory follows. Intuition also guided the analogy between strong and weak isospin

Asymptotic freedom—->Before Gross there was the parton model, with a mix of intuition and phenomenological maths by Feynman,Bjorken etc showing that “partons” weakly coupled at high Q² exhibit the sort of scaling seen experimentally.

Higgs—>The necessity of renormalizability follows the deeply intuitive work of Wilson, through there is quite a lot of sophisticated maths to connect this to Gauge symmetry. The result that a theory broken by the Higgs mechanism is still renormalizeable is intuitive (the Higgs is in the IR, gauge symmetry needs to operate in the UV) but needs a lot of sophisticated maths to prove.

In each case intuition has a crucial role in selecting what math to pick. The set of consistent mathematical systems is almost certainly infinitely larger than the set of consistent mathematical systems relevant to physics. So it is an interplay between the two.

The path integral, Feynman diagrams, the Parton model are more physical intuition than math.

Rohrlich, F. The unreasonable effectiveness of physical intuition: Success while ignoring objections. Found Phys 26, 1617–1626 (1996). https://doi.org/10.1007/BF02282125

The process of theory development in physics is a very complex one. The best scientists sometimes proceed on the basis of their physical intuition, ignoring serious conceptual or mathematical objections well known to them at the time.The results soon justify their actions: but the removal of these objections is often not possible for a very long time. Four examples are presented: Newton, Schrödinger, Dirac, Dyson. Some thoughts on this “unreasonableness≓ are offered.

Why are you acting like these are mutually exclusive? It’s clear to me that new ideas always come from a combination of qualitative and quantitative reasoning, in support of each other.

Einstein had the qualitative ideas that led to General Relativity in mind long before he learned Riemannian geometry. He also said that “every true theorist is a kind of tamed metaphysicist.” The taming here is the translation of qualitative insight (“physical intuition”) into quantitative form (“math”).

“Steven Weinberg, who was awarded a Nobel Prize for unifying the electromagnetic and weak interaction, likes to make an analogy with horse breeding: “[The horse breeder] looks at a horse and says “That’s a beautiful horse.” While he or she may be expressing a purely aesthetic emotion, I think there’s more to it than that. The horse breeder has seen lots of horses, and from experience with horses knows that that’s the kind of horse that wins races.”

But like experience with horses doesn’t help when building a racing car, experience with last century’s theories might not be of much help conceiving better ones. ”

That’s a quote from my book “Lost in Math” to say I agree with you.

This is a topic close to my heart. It can be called “physics vs formalism”. In my view, there has never been a successful *unguided* venture into theory space (like string theory) . None of the successful examples that you discussed were unguided. Their advent took place within the context of observed problems revealed by experiments, often amidst several competing theories at the time. Even with general relativity: Einstein spent a long time pondering the problem, considering everyday examples (like a man falling from a window) to build up his “physical intuition,” until he came up with the equivalence principle. Only then did he venture into math, guided by this principle.

“Math” v “physical intuition” is surely about choosing a strategy when confronted with a challenging problem in theoretical physics. The choice obviously depends on the background and experiences of the chooser, and history shows that there is no ‘best’ choice. The real issue is the potential (or lack thereof) for a proper interplay between theory and experiment. All the examples of successful theoretical developments, no matter what strategy was adopted, involve experiment, because experiment in physics is the arbiter of ‘success’. In the cases of supersymmetry and string theory, these are unsuccessful either because these approaches make no contact with experiment, or because the ‘predictions’ they have made have simply not be upheld by experiment. The more substantial issue then arises from theorists justifying their continued commitment to an unsuccessful theory by seeking to change the definition of ‘success’.

Doesn’t the canonical history of general relativity involve a rather “naive” form of physical intuition? “I am falling in an elevator”, “I am moving on a rotating disk”, etc.

Pardon me for being facetious but many years ago I decided that “intuition” is just a word one uses for a body of knowledge learned so deeply and so many years ago that one doesn’t recall not knowing it. E.g., much of at least non-rel. QM seems superficially to be intuitive to me–which is patently ridiculous.

This is a fascinating topic but I come down on the other side. Quantum mechanics, SR and GR were all driven by physical observation and intuition. “Math” was the available tool to be used. If math was dominant, Hilbert would have figured out GR.

Nikita, Yes, it somehow has become “canonical history” that “math” wasn’t important, that Einstein figured out general relativity by using his physical intuition to think through what happens when you fall in an elevator. Enough of this kind of dubious argument about GR. If someone wants to point to a serious discussion of the topic, that fine. Otherwise, enough of that.

After years of talking with physicists, I concluded with some amusement that when they said “physical intuition,” they just meant “guesses of which math to use for physics problems, whenever the guesses turn out to be correct.” 😀

To talk about a somewhat different field, statistical mechanics, Giorgio Parisi won the Nobel Prize in part for inventing the replica method. This is a case where it seems to me that the math is completely broken, but the physical intuition works anyway. But there’s not “no math”. There’s lots of equations, but none of them can be justified rigorously.

It seems to me that papers with few equations, but lots of hand-waving about physical intuition, are likely to be completely wrong.

This is a great idea for a post Peter.

I worry that the term ‘math’ is being used for two very different things: Geometric vs Analytic thinking.

Likely the term ‘Physical intuition” is also being bound to two very different ideas: Constructing Lagrangians vs Unpacking Lagrangians.

And, oddly I think these two subdivisions are mirroring each other. A mathematician trained in Geometry is often much closer to a Field Theorist trying to develop new Lagrangians than (s)he is to most hard core analysts. There are very things that separate mathematical from physical intuition in such cases but there are a few. In particular the concept of what is geometrically natural is slightly different to what is physicalliy natural in a Lagrangian. The “Mexican Hat Potential” quartic potential is obviously geometrically awkward as is the Yukawa Coupling as are the CKM/PMNS matrices. They come from experiment, but they don’t correspond to natural fiber bundle theory that mathematicians would stumble upon. Yet they are natural enough to a field theorist who might not see why the Yang-Mills term is incredibly more geometrically natural because the field theorist may believe that renormalizability is the proxy for naturality.

Conversely, the mathematical analyst might be more focused on keeping control of infinities in mathematical work while the perturbation/regularization/renormalization concerns mirror those in analysis need to keep infinities from spoiling everthing. Yet this is often about unpacking Lagrangians rather than constructing them.

This leads to two related but distinct concepts of elegance and beauty. I have never understood the argument that String Theory is beautiful…but at least I’d like to think I know WHY I don’t understand it. It is because it is largely based on a mathematics problem that doesn’t strike me as the way forward: “How do we treat the metric so that it doesn’t blow up or misbehave when added to the field content to be quantized?” That is really an analysis question to my mind which drags in a lot of geometry kicking and screaming. So you end up with beautiful geometric objects like Calabi-Yau manifolds but for very weird and often ugly reasons.

Ultimately, I think that what we are struggling with here is four quadrants: geometry, model building and phenomenology, QFT, and hardcore analysis on manifolds. We are looking for beautiful geometry to become a compelling natural Lagrangian. That is two quadrants. Then it will need to be unpacked by QFT types to discern what degrees of freedom would be observed at our effective energy scales. And that in turn would need to be made into rigorous analysis to be fully satisfactory. What I don’t understand is why those of us who base oursevels in one of these quadrants or the other too often try to discount those in the other three. It just seems totally self-defeating. One man’s opinion.

Perhaps a slight aside, but as far as I can tell “the math” means anything one does not have a really good intuition about.

And it’s not just physicists who talk this way either. Ask any (well-educated) person on the street what an average is and they’ll explain it in plain English with clarity and precision. Ask them about standard deviation and you’ll get a slightly muddier answer.

Ask them about the third moment of a random variable, and they’ll tell you that for something like that you gotta do the math.

So “math” seems to be the psychological state where one starts manipulating equations with pen and paper and knowledge in the hands begins to take over from knowledge in the mind.

Just my theory.

Robert James Parkinson, I put “math” in quotation marks because I think different people mean different things by it. That it is often getting used by non-mathematicians to mean “stuff I don’t really understand” sounds right.

I like a lot Scott Aaronson’s observation that “physical intuition” often doesn’t mean a non-mathematical understanding of a problem but does mean knowing what the right mathematical understanding of the problem is. To me the big problem of finding a way forward in fundamental theory is exactly that of finding the right mathematics. Mathematics is a set of languages and tools for using these languages, we need to identify which of these is needed to make progress. Eric Weinstein’s comment is along similar lines, with a specific proposal, mine would be different.

Wow this is a wildly out of touch point of view of how physics is done.

To a physicist, “the math” means doing a formal calculation from first principles. There’s often an implication that the calculation is being done with a high degree of generality.

Working physicists often think it’s not a good idea to only use “the math” when doing real world problems. There’s a famous quote by Wheeler: “Never make a calculation before you know the answer.” There are a variety of *heuristic techniques* that physicists use to get a feel for the answer, including: * Dimensional analysis * Knowledge of relevant experimental results * Experience with related calculations * Knowing that the calculation must reproduce some known result as a special case * Toy models In fact, I don’t think this heuristic mode that physicists use is particularly different from what mathematicians use; I think it is quite analogous to what Terry Tao refers to as the “post rigorous” stage of mathematics: https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/

Trying to find “a general solution” to some equation without a *physical question* motivating the endeavor is a classic path to getting lost in equations and not deriving insight from them.

In terms of finding new theories, I think all of your examples can be reframed in terms of a physical motivation. For example, Einstein’s paper on special relativity starts by talking about how it is strange the current produced in a coil of wire moving relative to a magnet has two apparently different explanations in two different reference frames. This motivated him to take equations discovered earlier by Poincaire and others (whether or not he was actually aware that they were previously discovered) and *interpret* them physically in terms of various thought experiments, making them manifestly relevant for physics. Of course math still has a role, especially when it expands our conceptual understanding — like when Minkowski reinterpreted the Poincaire transformations in terms of spacetime geometry. But even then, the benefit of Minkowski’s work was not blindly applying a formalism, but developing new ideas that allowed us to reinterpret existing results in a new framework where different intuition can be applied.

In reality, research is messy, and I think all tools and points of view are needed. But I think the way your article is framed that all discoveries in the 20th century are based on “math” and string theory is based on “physical intuition” is unabashedly political and not a serious attempt to understand the philosophy of science.

Gavin Storm, I think you’re missing the point of my posting. It was specifically an intentionally provocative argument against the idea that, in a situation where current theory doesn’t apply and you don’t have any experimental guidance, the way forward is “physical intuition”, not “math”. I often hear this in discussions of the current lack of progress in fundamental physics (“the problem is that the string theorists are doing math, not using physical intuition”), and I think it’s completely wrong-headed.

You’re arguing about something completely different, about whether one should start trying to solve a problem with heuristic methods or with a rigorous formal calculation. Of course both mathematicians and physicists generally start with the first, later move on to the second.

My argument is about what heuristic method you should use. Should you look to the mathematical tools you have successfully used in the past (your built up “physical intuition”) or should you look for new tools? The best example of the ones I gave in the posting might be the Gell-Mann one. He and everyone else stared a long time at the patterns of strongly interacting states they were seeing, not recognizing any pattern they were used to or anything that fit the mathematical tools they had in hand. He finally got around to talking to a mathematician who told him about the representations of SU(3), giving him a new mathematical tool that explained the patterns.

In the context of string theory, some people argue that it failed because, seduced by Witten, string theorists went out and adopted all sorts of new mathematical tools not previously useful in physics. I think it failed for a different reason: it was a completely wrong heuristic idea based on picking a wrong physical system (the vibrating string) as starting point.

I started my career as a theoretical high energy physicist and am now an experimental biological physicist. I have published in major refereed journals in both fields. I find Jim Baggot’s comment above to be especially relevant and cogent. A simple question is always “What are the experimental data?” that either do or do not validate any “conclusion(s)” coming from math or physical intuition. Experiment is always the final arbiter in science although math, physical intuition as well as experiment can interact along the way to a final determination. Peter your last entry gives some very cogent examples relevant to both Baggot’s comment and what I try to express.

Bob Y, The question I’m interested in is that of research in areas where there is no experimental evidence to interact with. Should theoretical quantum gravity research, for instance, be described as “not physics” because there is no experiment to interact with?

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The Rubik’s Cube Turns 50

Mathematicians and hobbyists have had a half-century of fun exploring the 43 billion billion permutations of Erno Rubik’s creation.

“My method was understanding,” Erno Rubik said of the iconic twisty puzzle that he invented in 1974. Credit... Akos Stiller for The New York Times

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By Siobhan Roberts

• July 1, 2024

Bright and early on the first Saturday in January, Tomas Rokicki and a few hundred fellow enthusiasts gathered in a vast lecture hall at the Moscone Center in downtown San Francisco. A big math conference was underway and Dr. Rokicki, a retired programmer based in Palo Alto, Calif., had helped organize a two-day special session about “serious recreational mathematics” celebrating the 50th anniversary of the Rubik’s Cube. Erno Rubik, the Cube’s inventor, was top of the show at 8 a.m., via videoconference from the south of Spain.

Mr. Rubik, a Hungarian architect, designer, sculptor and retired professor, took part in a question-and-answer session with Dr. Rokicki and his co-organizers, Erik Demaine, a computer scientist at M.I.T., and Robert Hearn, a retired computer scientist, of Portola Valley, Calif.

Dr. Rokicki asked Mr. Rubik about the first time he solved the Cube: “Did you solve corners-first?”

These days, new cubers learn on YouTube, watching tutorials at 1.5x speed. Dr. Rokicki instead recommends the old-fashioned strategy: Set out on a lone path and discover a solving method, even if it takes weeks or months. (It took the computer scientist Donald Knuth less than 12 hours, starting at his dining table in the evening and working straight through to the morning.) Corners-first is a common route, since once the corners are solved, the edges can be slotted in with relative ease. Mr. Rubik said that, yes, he indeed did corners-first. Mr. Rubik, who is known to take a philosophical approach to cubology and to life in general, added: “My method was understanding.”

‘Cubitus magikia’

Mr. Rubik dates the Cube to the spring of 1974. Preparing a course on descriptive geometry and tinkering with the five Platonic solids, he had become especially taken by the cube. But, as he wrote in his 2020 memoir, “Cubed, The Puzzle of Us All,” for quite a while it “never once occurred to me that I was creating a puzzle.”

By about the time of his 30th birthday, in July 1974, he had created the structure, realized its puzzling potential and — after playing with it intermittently for a few months — solved the Cube for the first time. He submitted a patent application in January 1975, and by the end of 1977 the “Magic Cube” had debuted in toy stores in Hungary. Travelers spirited it out “in their luggage, next to other Hungarian delicacies like sausage and Tokaji wine,” he recalled.

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