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’.

Course Outcome:

By the end of this course the students will be able to:

• Discuss how aspects of the culture such as the calendar, art, architecture, and music traditionally reflected traditional ideas of symbolic number, and harmony and proportion.
• Discuss 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.
• Discuss the underlying principles of symmetry and order in traditional symbolic mathematics.
• Discuss the underlying principles of symmetry and order in 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.
• Apply the underlying principles of symmetry and order in different aspects of the culture.

The writings of figures such as Plato and Aristotle, Boethius, St Augustine and St Thomas are examined as sources. 

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.

System Requirements:

This course is offered asynchronously on our LMS, NEO.  To access the course and complete it successfully, you will need an internet capable device with sound and a current internet browser: either the current or the previous release of Chrome, Firefox and Safari, or Internet Explorer 11+ (for Windows 8 and previous versions), and Microsoft Edge (for Windows 10+) to run the LMS.  In addition, courses require the use of, a PDF viewer, word-processing software compatible with Microsoft Office suite and a current email address. For help with the NEO platform see the Help Center in the LMS, found by clicking on the “?” in the upper right hand corner after you have logged in.  For log in questions, registration questions, or problems with missing content or content malfunctions, contact the teaching assistant, Elizabeth Froula at [email protected].

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

Lesson 1: Introduction

By the end of this lesson the student will be able to discuss the origin of the mathematics of beauty. 

1. • Video I - Introduction
   • Video II - Introduction
   • Handout: Text of the introduction

Lesson 2: Beauty - Our Greatest Weapon in the Culture War

By the end of this lesson the student will be able to discuss the nature of beauty.

1. • Video - 1 Beauty, Our Greatest Weapon in the Culture Wars

 Lesson 3: Pattern - Introducing the sources of mathematical beauty

By the end of this lesson the student will be able to discuss intuition, cosmos, man, music and scripture as sources of mathematical beauty.

1. • Video - 1 Mathematics as the Science of Pattern

 Lesson 4: Analysis and Synthesis

By the end of this lesson the student will be able to compare natural science and the science of beauty..

1. • Video - 1 Analysis and Synthesis

Lesson 5: Emergence

By the end of this lesson the student will be able to relate individual behavior to the pattern of the whole.

1. • Video - 1 Emergence

Lesson 6: Grasping the Whole, loving knowledge - Synthetic, Poetic and Prosaic Knowledge and their connection with Love - Eros and Agape,

By the end of this lesson the student will be able to discuss.the theology of a virtuous and loving relationship that is at the root of a culture of beauty.

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

Lesson 7: The Beauty of Three - Quark, Chord and .Cosmos

By the end of this lesson the student will be able to relate quark, chord, and cosmos to the science of beauty.

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

Lesson 8: What influences culture?  

By the end of this lesson the student will be able to discuss the degree to which the following institutions produce ugliness: commerce and trade, capitalism, and the market economy; industrialization; mass production

1. • Video - 1 What influences culture?

Lesson 9: How the liturgy influences culture.

By the end of this lesson the student will be able to discuss beauty and the work of AWN Pugin.

1. • Video - 1 Liturgy influencing culture - case study, gothic revival

 Lesson 10: Faith Freedom and Tradition

By the end of this lesson the student will be able to discuss the threefold forces that transform and why love is the greatest ordering principle.

• • Video - 1 Faith Freedom and Tradition
  • Reading Chapter 5, The Way of Beauty
  • Further optional reading: The Consolation of Philosophy, Boethius

Lesson 11: 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

Lesson 12: Sacred Number

By the end of this lesson the student will be able to discuss symbolism of numbers.

• • Video: 1 Symbolism of Zero and One, the point and the circle
  • 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.
  • Video: 12.3 Symbolism of Four
  • Video: 4 Symbolism of Five.
  • Video: 5 Symbolism of Six and 28 - perfect number (superfluous, diminished)
  • Video: 6 Symbolism of Seven and Eight at the heart of sacred number.
  • Video: 7 Symbolism Nine and Ten
  • Video: 8 Symbolism of Twelve

Lesson 13:

By the end of this lesson the student will be able to discuss symbolism of the Rose Window.

• • Video: Rose Windows - the Christian Star Pattern for Today

Lesson 14: Mathematical operations

By the end of this lesson the student will be able to discuss the symbolic in mathematical equations.

• •  Video:: Mathematical operations - adding, adding 1 - 7+1 or 60+1; subtracting, multiplying, multiplying by 1,000 -

Lesson 15: The Rose window of Lausanne Cathedral

By the end of this lesson the student will be able to discuss the symbolism of the Rose Window of Lausanne Cathedral.

• • Video: The Rose window of Lausanne Cathedral, Switzerland (1230 AD)

Lesson 16:  Putting it all together

By the end of this lesson the student will be able to discuss beauty in the Westminster pavement.

• • Video: Putting it all together in grand designs eg Ely Cathedral, Westminster pavement. - getting it wrong, follies

Lesson 17: The quantitative aspect of number, choosing units of measurement

By the end of this lesson the student will be able to discuss the development of standards of measure.

• • Video: 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

Lesson 18: Geometry and Beauty

By the end of this lesson the student will be able to discuss the relationship between geometry and beauty.

• • Video: Geometry and beauty. What is geometry? Euclid’s 13 books of limited value to us. Plato, Augustine on the value of geometry saw the world created out of idealized geometric shapes. 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

Lesson 19: The Westminster pavement and cosmati pavements

By the end of this lesson the student will be able to discuss the geometry of the "Cosmati" pavements in Winchester Cathedral.

• • Video: The Westminster pavement and cosmati pavements - through the book by Richard

Lesson 20: The Byzantine-Arab-Norman churches of 12th century

By the end of this lesson the student will be able to discuss incorporating elements from a non-Christian culture into Christian sacred art..

• • Video: 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.

Lesson 21: Examination and assignments for Part II

Examination and assignments for Part II

• • Video: explaining the assignments and handouts
    •  Handout: Written essay - 10% of final grade
    • Geometry practical assignments
      • Geometry assignments 1 -4
      • 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

Lesson 22: 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 those considered musical? 

By the end of this lesson the student will be able to discuss patterns present in the natural world.

• • Video - Music Scripture Arithmetic

Lesson 23: Boethius: De Institutione Arithmetica I - Reading De Institutione Musica, Boethian Number Theory, A Translation ofthe De Institutione Arithmetica: perfect, superfluous, Odd even male female. Triangular, square numbers.

By the end of this lesson the student will be able to discuss numbers as defined in "De Institutione Musica," by Boethius.

• • Handout 1 - Notes to Accompany Boethian Number Theory
  • Handout 2 - Cicadas and Prime Numbers
  • Video 1 Introduction - I De Institutione Arithmetica (DIA)
  • Video 2 DIA Book 1, Book II Ch 39
  • Video 3 DIA Sidebar - Cicadas and Adam Smith - Prime numbers in nature

Lesson 24: 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. The incorporation of this into the culture - composing visual music; calligraphy; figurative

By the end of this lesson the student will be able to discuss proportion and symmetry as defined in "De Institutione Musica," by Boethius.

• • Video 1 - II De Institutione Arithmetica The Proportions Explained

Lesson 25: Examples of proportion in art and architecture 

By the end of this lesson the student will be able to recognize proportion and symmetry in art and architecture..

• • Video 1 - Examples of proportion in art and architecture
  • Video 2: 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

Lesson 26: Boethius: De Institutione Musica

By the end of this lesson the student will be able to discuss Boethius' work on music, "De Institutione Musica."

• • Handout 1 - Reading 1 - the text of De Institutione Musica in translation
  • Handout 2- Reading 2 - my notes on DIM to guide your reading.
  • Video: 1 De Institutione Musica
  • Handout 3 - Reading 3 - Essay by Dr Tom Larson on the importance of music in education.
  • Video 2 Dr Tom Larson on Music in Education

Lesson 27: Musical proportion in architecture and the culture - a study through the book Architectural Principles in the Age of Humanism by Rudolf Wittkower pub Academy

By the end of this lesson the student will be able to discuss musical proportion as it relates to architecture.

• •   Handout 1 - Reading 1, my notes on Wittkower, to guide and accompany the reading of this book
  • Video 1 Discussion on Wittkower and music theory
  • Video 2 Wittkower - Appendix IV discussed

Lesson 28: Illustrating Wittkower

By the end of this lesson the student will be able to discuss specific examples that demonstrate the principles of Rudolf Wittkower.

• • 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.

Lesson 29: 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.

By the end of this lesson the student will be able to discuss the decline of proportion, symmetry and harmony in art and architecture.

• • Video 1 The rejection of the mathematics of beauty
  • Reading - Final Chapter of Wittkower - Part IV, 7

Lesson 30: Town Layout - what makes community? Not shopping, not government, not cafes Then we must actually worship! Apostolic model of parish community article.

By the end of this lesson the student will be able to discuss the application of proportion, symmetry and harmony in town planning.

• • Video 1 Designing for community
  • Handout 1 - The Apostolic Blueprint for Community

Lesson 31: Concluding remarks - Urban Hermit and Suburban Skete how this extends to all human activity

By the end of this lesson the student will be able to discuss the wider application of the "hermit" and "skate"models.

• • Video 1 Conclusion
  • Final Exam (Part III exam)

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