# Mathematical Mysteries

Analysing the greatest mysteries in mathematics

## Module details

• Offered to 2nd Year students in Autumn term
• Mondays 16:00-18:00
• Planned delivery: On campus (South Kensington)
• 1-term module worth 5 ECTS
• Available to eligible students as part of I-Explore

This course examines mysterious mathematical ideas which do not usually come up in STEM courses. What is infinity, and is there only one kind? What kinds of symmetry are there, and how do they relate to one another? How much do we really know about the familiar counting numbers? Can a shape have a dimension that is not a whole number? Explore these ideas and more in this module.

You will learn to define mathematical concepts such as infinity and fractal dimension, construct rigorous mathematical arguments, and create elegant proofs. The module will also teach you to understand the role of maths in STEM and wider culture and to communicate mathematical ideas in everyday language.

## Accordian

### Learning outcomes

By the end of this module, you will better be able to:

• Use key "mysterious" mathematical concepts such as infinity, incompleteness and fractal dimension to construct mathematical arguments and perform calculations
• Discuss the epistemic value of *rigour*; construct rigorous arguments; critique and refine their own and others' proof-conjectures in the light of this value
• Create elegant proofs; critique and refine their own work and that of others in the light of this value
• Communicate mathematical ideas using everyday language, specialised notation and creative visualisation
• Discuss the role of mathematics in science, engineering and the wider culture, and the history of the world's cultures in shaping it
• Actively value individual peers' disciplinary, cultural and personal perspectives and approaches, within groups and within the cohort at large

### Indicative core content

This module will look at:

• Infinity: cardinality, one-to-one correspondence, countable and uncountable infinities, diagonal arguments, the Continuum Hypothesis
• Symmetry: dihedral groups, wallpaper symmetries and Islamic art, crystals, the Rubik cube, Lagrange's Theorem
• Fractals: self-similarity, geometrical strangeness, fractal dimension, iterated function systems, Mandelbrot and Julia sets
• Mysteries of the primes: Fermat's Little Theorem, pseudoprimes and Carmichael numbers, tests for primality, applications to cryptography
• Logic, formal systems, Russell paradox, incompleteness

### Learning and teaching approach

Contact sessions in the early part of the course will consist of a mixture of activities including instructor-led explanations, class discussions and computer practicals. This will be followed by:

• Online video and audio content
• Literature searches and the preparation of their final project
• Project assessment

Written instructor feedback on the mid-term reflective writing will be provided within one week of submission, and there will be spoken instructor and peer feedback on early drafts of the group assessment. At the start of Term 3, your instructor will also give you feedback (incorporating peer perspectives) on the final version of the assessment and the final piece of reflective writing.

### Assessment

• There will be two short pieces of reflective writing, each worth 5%, submitted in the middle and at the end of term respectively (10%)
• The main assessment will be a group project, culminating in either a presentation, video, podcast, poster, documented visualisation or a documented piece of software (90%)

### Key information

• Requirements: It is compulsory to take an I-Explore module during your degree (you’ll take an I-Explore module in either your 2nd or 3rd year, depending on your department). You are expected to attend all classes and undertake approximately 105 hours of independent study in total during the module. Independent study includes for example reading and preparation for classes, researching and writing coursework assignments, project work and preparing for other assessments
• I-Explore modules are worth 5 ECTS credit towards your degree; to receive these you will have to pass the module. The numerical mark that you obtain will not be included in the calculation of your final degree result, but it will appear on your transcript
• This module is designed as an undergraduate Level 6 course
• This module is offered by the Department of Mathematics