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Master's of Sacred Arts, Individual Courses

Mathematics of Beauty


Class
David Clayton
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Mathematics of Beauty.

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Short summary:

The Music and Art of the Spheres:

The Mathematics of Beauty, Ancient and Modern

Mathematics and geometry are studies of the quantitative aspect of number – answering the question, how much? The traditional approach to mathematics, prior to the Enlightenment, did this too; but it also looked at the qualitative aspect, that is the symbolic character of number that connects it to some other aspect of the truth in the minds of those who are aware of the symbolism. The traditional view is that man is made to see things in the world around him symbolically as much as quantitatively, and the world around him is made to be a symbol. The cosmos naturally symbolizes the glory of God, that is it connects us to the idea of God when we perceive the cosmos through its beauty. And we instinctively see this because God made us that way. This is why this field of study has been referred to as sacred geometry.

 

There is a practical element built into this course in which students will create examples of Islamic tiled patterns, and traditional Christian patterns based upon Gothic floor designs.

 

Rather than focusing deeply on the mathematical calculations themselves, we will study the principles that guided the creative artists, architects and composers of the past so that students will be in a position to draw from it what will be appropriate to their own creative pursuits.

 

Syllabus:

The Music and Art of the Spheres:

The Mathematics of Beauty, Ancient and Modern

 

Mathematics and geometry are studies of the quantitative aspect of number – answering the question, how much? The traditional approach to mathematics, prior to the Enlightenment, did this too; but it also looked at the qualitative aspect, that is the symbolic character of number that connects it to some other aspect of the truth in the minds of those who are aware of the symbolism. The traditional view is that man is made to see things in the world around him symbolically as much as quantitatively, and the world around him is made to be a symbol. The cosmos naturally symbolizes the glory of God, that is it connects us to the idea of God when we perceive the cosmos through its beauty. And we instinctively see this because God made us that way. This is why this field of study has been referred to as sacred geometry.

 

Similarly, the relationships between numbers - numbers and magnitudes in combination - are not all equivalent in value, but some are naturally perceived as more beautiful than others. The recognition of this creates the the pattern of harmony and proportion that is seen in the cosmos, and when it is analyzed can be expressed mathematically. The traditional name for these patterns of beauty was music. Music for ancients was not simply audible music that we think of today (although it did include it) but also any such pattern of beauty created by numbers in relation. It could be in any aspect of the world around us, or in man himself. The assumption is that all point to a universal set of principles, heavenly standard this embodied in the person of Christ and when we apprehend it we have an insight, as Benedict XVI puts it, ‘into the mind of the Creator’.

 

Outcomes: In this course, students will consider how aspects of the culture such as the calendar, art, architecture, and music traditionally reflected traditional ideas of symbolic number, and harmony and proportion. The writings of figures such as Plato and Aristotle, Boethius, St Augustine and St Thomas are examined as sources. There is also an examination of how, far from undermining it, modern science reinforces these traditional ideas.

 

There is a practical element built into this course in which students will create examples of Islamic tiled patterns, and traditional Christian patterns based upon Gothic floor designs.

 

Additionally, students will understand the underlying principles of symmetry and order in traditional symbolic mathematics; traditional ideas of our perception of those patterns and how we can be formed by them and reflected them in our activity and work, so forming the culture. They will be asked to consider how such principles can be applied in different aspects of the culture.

 

Mathematical proficiency needed: rather than focusing deeply on the mathematical calculations themselves, we will study the principles that guided the creative artists, architects and composers of the past so that students will be in a position to draw from it what will be appropriate to their own creative pursuits. Accordingly we will not be doing very much number crunching in this course. However, students will need to be able to understand and use the mathematical concept of ratios and to make any variable the subject of an equation that contains fractions and ratios.

 

Part I

Introduction and the Mathematical Analyses of a Culture of Beauty

 

Reading for Part 1:

The Way of Beauty, pub Angelico Press; by David Clayton - especially section 2 and the Appendices.

 

Additional optional reading.

Modern Physics Ancient Faith by Stephen M Barr.

The True Principles of Christian or Pointed Architecture by A W N Pugin

 

  1. Promotional Video

 

  1. Introduction Where does the mathematics of beauty come from and why is it worth knowing?
    1. Video - Introduction
    2. Video - Introduction
    3. Handout: Text of the introduction

 

  1. Beauty - Our Greatest Weapon in the Culture War Discussion of the nature of beauty - in some ways revisiting things discussed in other courses, but here highlighting that aspect which opens the door to a mathematical Examples proportional relationships in buildings given.
    1. Video - 1 Beauty, Our Greatest Weapon in the Culture Wars

 

  1. Pattern - Introducing the sources of mathematical beauty - intuition, cosmos, man, music, scripture,
    1. Video - 1 Mathematics as the Science of Pattern

 

  1. Analysis and Synthesis - Natural science and the science of beauty are complementary, although at first it might seem
    1. Video - 1 Analysis and Synthesis

 

  1. - relation individual behaviour governed by free will to the pattern of the whole.
    1. Video - 1 Emergence

 

  1. Synthetic, Poetic and Prosaic Knowledge and their connection with Love - Eros and Agape,

the theology of virtuous and loving relationship that leads to human flourishing and is at the root of a culture of beauty.

  1. Handout: Reading 1: Grasping the Whole - Synthetic, Poetic and Loving Knowledge
  2. Handout: Reading 2: A Reflection on Acedia, Eros and Christian Joy
  3. Handout: Reading 3: Why Eros and the Worship of God Are the Keys to Countering Philosophical Error
  4. Video - 1 Grasping the Whole - Synthetic, Poetic and Loving Knowledge

 

  1. The Beauty of Three - Quark, Chord and .Cosmos
    1. Video - 1 The Beauty of Three
    2. Reading: The Way of Beauty, Appendix 2: Trinity and trinity - the Beauty of

 

  1. What influences culture? Mass production? Capitalism? We consider in turn whether the degree to which the following institutions produce ugliness: commerce and trade, capitalism, and the market economy; industrialisation; mass production
    1. Video - 1 What influences culture?

 

  1. How the liturgy influences culture - case study, the Victorian neo-gothic of AWN Pugin
    1. Video - 1 Liturgy influencing culture - case study, gothic revival

 

  1. Faith Freedom and Tradition - the threefold forces that transform the And why love is the greatest ordering principle
    1. Video - 1 Faith Freedom and Tradition
    2. Reading Chapter 5, The Way of Beauty
    3. Further optional reading: The Consolation of Philosophy, Boethius
  2. Mid-term Exam (Part 1) Contributes to 30% of your final grade

 

End of Part 1

 

Part II

Introducing Sacred Number and Sacred Geometry

 

Reading for Part II

 

  • Way of Beauty; Patterns of Thought, the Hidden Meaning of the Great Pavement of Westminster Abbey by Richard Foster
  • Reading: The Way of Beauty, Section II

 

Additional optional reading:

The Gothic Cathedral, Otto von Simson

 

  1. Sacred Number
    1. Video: 1 Symbolism of Zero and One, the point and the circle
    2. Video: 2 Symbolism of Two and Three - why twos need a third. How proportion and harmony spring from this. Means and medians, geometric and arithmetic.
    3. Video: 12.3 Symbolism of Four
    4. Video: 4 Symbolism of Five.
    5. Video: 5 Symbolism of Six and 28 - perfect number (superfluous, diminished)
    6. Video: 6 Symbolism of Seven and Eight at the heart of sacred number.
    7. Video: 7 Symbolism Nine and Ten
    8. Video: 8 Symbolism of Twelve
  2. Video: 13 Rose Windows - the Christian Star Pattern for Today
  3. Video: 14 Mathematical operations - adding, adding 1 - 7+1 or 60+1; subtracting, multiplying, multiplying by 1,000 -
  4. Video: 15 The Rose window of Lausanne Cathedral, Switzerland (1230 AD)
  5. Video: 16 Putting it all together in grand designs eg Ely Cathedral, Westminster pavement. - getting it wrong, follies
  6. Video: 17 The quantitative aspect of number, choosing units of measurement The Development of Standard Units of Measure - what are we quantifying? Choice of unit - what is the measure of all? Earth, Man or God?

 

Geometry

  1. Video 18: Geometry and beauty. What is geometry? Euclid’s 13 books of limited value to us. Plato, Augustine on the value of Saw the world created out of idealised geometric shapes cf Timaeus. Quadrature and doubling the square: another way of creating a mean; Mathes Roriczer. In architecture and geometric patterns. More from the gothic era. Milan cathedral, ars sine scientia nihil est. Regensburg, Bourges, Amiens. Used simple Pythagorean musical harmony as well (continuing practice of Romanesque builders). Euclid’s elements can be found online here. https://mathcs.clarku.edu/~djoyce/elements/elements.html
  2. Video 19: The Westminster pavement and cosmati pavements - through the book by Richard
  3. Video 20: The Byzantine-Arab-Norman churches of 12th century A case study of an exemplary model how to incorporate art from a culture that was hostile to Christianity. In the 13th century William II and Roger II of Sicily built churches in the newly established kingdom created by the Norman conquest of the island

 

which had previously been in Islamic hands. This is important for today because Islamic art may well be the main source of the methods for the creation of Christian geometric art today.

 

  1. Examination and assignments for Part II
    1.          Video 21: explaining the assignments and handouts
    2.          Handout: Written essay - 10% of final grade
    3.   Geometry practical assignments:
      1.          Geometry assignments 1 -4
      2.           Geometry technique handouts 1-4
  •          Examples of past students’ work

 

 

End of Part II

 

Part III Arithmetic and Music

Harmonious Proportion and Consonant Relationships Between Numbers.

 

Reading for Part III:

  • De Institutione Musica by Boethius - pdf file available for download, lesson 26
  • De Institutione Arithmetica by Boethius as presented in the book Boethian Number Theory - A Translation of the De Institutione Arithmetica by Michael Masi, pub Rodopi. The introduction by the translator is very good as well as the original
  • Architectural Principles in the Age of Humanism by Rudolf Wittkower pub Academy Editions

 

  1. Reading the Book of the Word, Reading the Book of Nature (inc. Music, man), the Book of Mathematics - looking for pattern in number and assigning symbolism to it. What is the difference between numerical relationships that are arithmetic and and those considered musical? Examples of proportional relationships in the
    1. 1 Video - Music Scripture Arithmetic
  2. Boethius: De Institutione Arithmetica I - Reading De Institutione Musica, Boethian Number Theory, A Translation of the De Institutione Arithmetica: perfect, superfluous, Odd even male female. Triangular, square numbers.
    1. Handout 1 - Notes to Accompany Boethian Number Theory
    2. Handout 2 - Cicadas and Prime Numbers
    3. Video 1 Introduction - I De Institutione Arithmetica (DIA)
    4. Video 2 DIA Book 1, Book II Ch 39
    5. Video 3 DIA Sidebar - Cicadas and Adam Smith - Prime numbers in nature
  3. Boethius De Arithmetica - II -The Ten Proportions. Definitions of Ratio and proportion described. The symmetry in the operations of creating Harmonic, Geometric, Arithmetic and Fourth of Four proportions why they are used symmetrical Introduce ideas that first three are instrumental and cosmic music. The blurring of the distinction between music and arithmetic.
    1. Video 24.1 - II De Institutione Arithmetica The Proportions Explained
  4. The incorporation of this into the culture - composing visual music; calligraphy; figurative
    1. Video 2 - Examples of proportion in art and architecture
    2. Video 3: The internal proportions of rooms

 

Music - the manifestation of these proportions in sensible form

Theory generated from number alone, source is Wittkower, as used by architects as an application in the wider culture

  1. Boethius: De Institutione Musica
    1. Handout 1 - Reading 1 - the text of De Institutione Musica in translation
    2. Handout 2- Reading 2 - my notes on DIM to guide your reading.
    3. Video 26:1 De Institutione Musica
    4. Handout 3 - Reading 3 - Essay by Dr Tom Larson on the importance of music in education.
    5. Video 2 Dr Tom Larson on Music in Education
  2. Musical proportion in architecture and the culture - a study through the book Architectural Principles in the Age of Humanism by Rudolf Wittkower pub Academy

 

  1. Handout 1 - Reading 1, my notes on Wittkower, to guide and accompany the reading of this book
  2. Video 1 Discussion on Wittkower and music theory
  3. Video 2 Wittkower - Appendix IV discussed
  1. Illustrating Wittkower
    1. Video 1 - Examples of buildings to illustrate Wittkower

Examples of buildings by Roman architects, Giorgi and Alberti Palladio - increased complexity in proportions used goes in harmony with coupling advances in musical theory; Examples of Palladian architecture: 1 By Palladio himself; 2 British, Inigo Jones, Georgian; 3 American colonial

Note, in this video I make reference to an architect called Christopher Alexander - we will talk about him more in the next class.

 

  1. Decline and Hints at Recovery - Groping in the dark - Enlightenment and Romanticism, Jay Hambridge, The Golden Section, Fibonacci series, Fourth of Christopher Alexander, Le Corbusier, Mondrian and abstract art.
    1. Video 1 The rejection of the mathematics of beauty
    2. Reading - Final Chapter of Wittkower - Part IV, 7
  2. Town Layout - what makes community? Not shopping, not government, not cafes Then we must actually worship! Apostolic model of parish community article.
    1. Video 1 Designing for community
    2. Handout 1 - The Apostolic Blueprint for Community
  3. Concluding remarks - Urban Hermit and Suburban Skete how this extends to all human activity
    1. Video 1 Conclusion
  4. Final Exam (Part III exam)

 

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Faculty are permitted to refer to notable past work and achievements (including publications and educational activities not offered by Pontifex University, and even those offered for personal profit) in their published biography on the Pontifex website and course promotions. In the context of educational activities undertaken for Pontifex University, including videos, live or recorded, teachers, can recommend or bring to the attention such work for students (even if for personal profit, for example, books or podcasts) but only with approval by Pontifex University and when it is related to the teaching purpose of the class.  An instructor’s related work will be noted in the syllabus as appropriate.  Instructors may use their own materials as required in their courses and learning events as long as the materials are appropriate for the particular learning event.

 

Failure to comply with this policy will result in a warning or administration modification of course materials. Violations of this policy should be reported to the Provost.

Here is the class outline:

1. Introduction

2. Beauty Our Greatest Weapon in the Culture Wars

3. Mathematics as the Science of Pattern

4. Analysis and Synthesis

5. Emergence

6. Grasping the Whole- Loving Knowledge

7. The Beauty of Three

8. What influences culture

9. Liturgy and culture. Victorian gothic revival

10. Faith Freedom Tradition

11. Midterm Exam

12. Symbolism

13. Rose Windows and the Christian Star Pattern

14. Symbolism of Mathematical Operations

15. The Rose Window of Lausanne Cathedral

16. Putting it all together

17. Quantitative Number and Units of Measurement

18. Geometry and Beauty

19. The Westminster Pavement and Cosmati Pavements

20. Byzantine-Arab-Norman churches of 12th century Sicily

21. Assignments

22. Reading the Book of the Word, Reading the Book of Nature

23. Boethius: De Institutione Arithmetica I

24. Boethius De Arithmetica - II

25. Examples of Proportion in Art and Architecture

26. Boethius: De Institutione Musica

27. Musical proportion in architecture and the culture

28. Illustrating Wittkower

29. Decline and Rise

30. Designing for Community

31. Conclusion

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