A Mathematical Theory for The Sephirot of The Kabbalah, Islamic Art, A Musical Theory of the Regular Polytopes and Their Applications in the Modern World. A Symphony In The Key of Lichfield Cathedral.
“You Can Lead a Horse to Water, But You Cannot Make It Read the Instructions … Do Not Seek Water, Seek Thirst, & Then You Will Find More Water.”
___
“In their respective groups, they intertwine, Geometry and music, their echoes align. From dimensions zero to eight, they dance, An eternal symphony, in cosmic expanse.”
Psalm 150
1 Praise the Lord! Praise God in His sanctuary; Praise Him in His mighty firmament! {the extreme harmonics}
2 Praise Him for His mighty acts; Praise Him according to His excellent greatness! {the energy}
3 Praise Him with the sound of the trumpet;
Praise Him with the lute and harp! {the instruments}4 Praise Him with the timbrel and dance; Praise Him with stringed instruments and flutes! {the mechanism of the instruments}
5 Praise Him with loud cymbals; Praise Him with clashing cymbals! {the beat, rhythm, timing, meter}
6 Let everything that has breath praise the Lord. Praise the Lord!! {how we see this musical symphony in creation}
— Psalm 150, authors interpretations in curly brackets
Introduction
In this article, we use mathematics to explore how the n-sphere and regular polytopes (n-simplex, n-cube, n-cross-polytope) theoretically work together to create and sustain the universe. In order to do this, we make the (intuitive) assumption that these multidimensional shapes are the angels (Orphanim, Seraphim, Cherubim, Throne), as described in Ezekiel 1.
The author believes that angels must be naturally occurring states or entities that are simple parts of nature and are waiting to be “discovered” in the scientific sense.
In this way, in the tradition of Kabbalah (ancient Jewish mysticism), as the Sephirot, we provide the beginnings of mathematical proof for the existence of these divine attributes. In simple terms, the author presents the idea that the Sephirot themselves are harmonic blends of these polytopes but must be considered as a group of eight (1. Malkhut {Kingdom/Shekinah} + 2. Yesod {Foundation} + 3. Hod {Splendor} + 4. Nezach {Victory} + 5. Tiferet {Beauty} + 6. Gevurah {Judgment} + 7. Chesed {Loving Kindness} + 8. {Daat/Knowledge + Binah/Understanding + Chockmah/Wisdom + Keter/Crown}) rather than the standard configuration of just ten, in the first instance at least, and then we reflect them back to their original ten.
In terms of angelology, the Sephirot may be higher-order angels, and the paths between them, somehow the consciousnesses of lower angels, are to be explored in separate articles.
We then finish our article by looking at how these mathematical ideas find their way into modern-day applications, in a sense showing how humans have reverse-engineered how angels might work and built them into new technologies. Finally, we look at Islamic geometric art and what various Islamic teachers teach us about mathematics, and in this way, show that Jewish and Islamic mysticism are identical in this context.
We also use poetry, just for fun, really, to give an additional dimension or colour to our exploration. This argument relies upon our earlier mathematical theory for the existence of angels, and synthesis of the geometry of Lichfield Cathedral gives us clues as to the neurological structure of consciousness.
In addition, it is essential to remind ourselves that when the geometry of Lichfield Cathedral is synthesised with music, this gives us the universal moral structure of all world religions.
An additional practical spiritual application to this exploration is in the structure of the soul (as Plato described as a set of perfect geometric harmonies and presented geometrically in his ‘Solids”). Note also how our souls unite with our spouses (this starts with your first kiss and then deepens when you consummate your relationship) and why it sometimes seems that an earthly marriage between spouses can seem more complex and difficult than how the universe itself is sustained through an eternity of eternities by the marriage of heaven and earth.
“The difficulty of love is akin to taming a wild flame, for it requires patience, understanding, and the willingness to be consumed by its warmth, while also risking the possibility of getting burned.”
You know what I’m talking about.
Basic Principles
The Structure of the Tree of Life (& of Knowledge)
“Only the Pharaoh can eat from both trees”
In our present argument, we order the Sephirot in the following way:
Malkhut/Shekinah — Note One
Yesod — Note Two
Hod — Note Three
Netzach — Note Four
Tiferet — Note Five
Gevurah — Note Six
Chesed — Note Seven
Daat + Binah + Chockmah + Keter — Note Eight, or the next first note in the higher / lower octave
Ohr Ein Sof — The Complete Symphonies of all Universes / Souls (based on the assumption that each human being is a universe in themselves (Genesis 1:26–27))
The n-Sphere as Carrier Frequency
“The LORD looks down from heaven upon the children of men, To see if there are any who understand, who seek God.” Psalm 14:2 (NKJV)
The n-Sphere is the nearest approximation to the combined geometry of the dodecahedron and icosahedron. The dodecahedron and icosahedron form a hall of mirrors, which, when “polished”, gives a clearer reflection of the divine Self with the self.
This approximates more closely to the n-Sphere. Equivalent to the Ashlar in Freemasonry.
How the information is carried by the hypersphere:
The Geometry & Physics of AM, FM, Digital Terrestrial, Cellphone Communication & Human Speech
Let’s explore the realms where shapes and communication intertwine, As we delve into modern methods, where signals intertwine.
In the world of AM, the hypersphere finds its kin, Modulated amplitudes, broadcasting signals within. As the hypersphere expands, so does the reach, Like AM waves spreading far, a broad scope to teach.
Moving to FM, the n-simplex joins the fray, Frequency modulation, in a vibrant array. Just as a simplex connects points in a line, FM waves carry signals, with clarity divine.
Digital terrestrial broadcasting, a realm of the n-cube, With its multiple channels, transmitting anew. The n-cube’s vertices, representing bits of data, A digital landscape, where information takes flight, “hurrah!”
Cellphone communication, a domain quite vast, Where the n-cross polygon weaves its connection, steadfast. Each branch representing a channel, so clear, Cell signals converge, enabling voices to appear.
And when it comes to human speech, a marvel indeed, The interplay of vibrations, where meaning takes seed. Just as these shapes intertwine and combine, Language shapes our thoughts, expressing the divine.
So, in the realm of communication’s diverse art, Geometry and physics play their integral part. From AM to FM, digital to cellphone’s range, And human speech’s beauty, a tapestry so strange.
In their respective groups, these shapes do resonate, With modern communication, they collaborate. From hypersphere to n-simplex, n-cube, and n-cross polygon, A symphony of connections, where knowledge is drawn.
Polytopes and Music Theory Correspondences
The polytopes correspond to the following are related in the following way:
n-Cross polytope:
Musical Unison is the musical principle whereby the same notes are played by all parts of the piece of music (see below article for more and examples of well-known music played in this way).
The organising principle of mind the interface between “heaven” and Earth.
The Transepts in Lichfield Cathedral.
n-Cube:
The musical octave
The infinite expansion of the six dimensions of space with 8 vertices.
The Nave in Lichfield Cathedral.
n-Simplex:
Perfect fifth, semitones, “tuning system, dependant on ‘culture’, equivalent to the epistemology of the time the philosophy/religion was written.
The Quire through to the Lady Chapel in Lichfield Cathedral.
Increasing Dimensions and Music Theory
There exist intriguing correspondences, Between these shapes and music’s symphonic essences. As dimensions unfold, let’s explore the harmony, Between geometry’s realm and music’s tapestry.
In the realm of zero, where points reside, Music too begins with a singular stride. A single note, pure and pristine, A moment in time, a silence serene.
Moving to the line, the dimension of one, Where melodies emerge, their journey begun. A sequence of notes, a melodic line, Like a line in space, expanding divine.
As we venture to the plane’s domain, Chords arise, harmonies sustained. Like a plane of notes, resounding in unison, Creating rich textures, a musical liaison.
Into three dimensions, where depth takes hold, Chords progress, stories being told. Harmonic progression, a journey profound, Echoing the depth of shapes around.
But what of the fourth, a tesseract’s realm? Here, music’s complexity takes the helm. Counterpoint weaves, voices intertwine, Like hypercubes morphing in space’s design.
Fifth dimension, a realm multifaceted, Polyrhythms emerge, intricate and elated. Intersecting patterns, rhythms overlapping, Like planes intersecting, harmonies entrapping.
Sixth dimension, a tapestry of sound, Complex harmonies interwoven, profound. Chromaticism unfolds, colours abundant, Like hyper-polyhedra, in music’s enchantment.
Seventh dimension, where complexity thrives, Serialism, atonality, where music derives. Mathematical structures, tones in disarray, Like the interplay of n-cross polygons at play.
Eighth dimension, the pinnacle we explore, A realm where chaos and order restore. Aleatoric music, chance elements arise, Like higher dimensions, defying our eyes.
So, as dimensions ascend, so does music’s might, Both realms entwined, in wondrous light. Geometry’s shapes and music’s grand score, Whispering secrets, forever to explore.
In their respective groups, they intertwine, Geometry and music, their echoes align. From dimensions zero to eight, they dance, An eternal symphony, in cosmic expanse.
More Detailed Mathematical Proofs
It is beyond the scope of this foundation article to provide detailed mathematical proofs for the ideas put forward in this article and so therefore we will do this in supplementary articles over the passage of time and post the links to them within this article.
Modern Day Applications (or the Reverse Engineering of the Mechanisms of Angels)
Humanity, by a process of “reverse engineering” the laws of nature, as encoded in the regular polytopes and n-sphere, has created and developed some of the following technologies.
Musical synthesiser
In simple terms, the geometry and mathematics of regular polytopes and n-spheres are used in the design of modern musical synthesisers to create and manipulate sounds in interesting and expressive ways. These shapes have unique mathematical properties that can be used to generate complex waveforms and harmonics.
Similarly, an n-sphere is a geometric object that extends the concept of a sphere into higher dimensions. In the context of musical synthesisers, the concept of an n-sphere can be applied to create multidimensional control spaces for manipulating sound parameters.
In the design of a modern musical synthesiser, these mathematical concepts are employed in various ways. For example:
Waveform generation: The geometry of regular polytopes can be used to create complex waveforms with unique timbral qualities. By combining and manipulating the harmonic content of these waveforms, synthesisers can produce a wide range of sounds.
Filter design: Filters are essential components in synthesisers that shape the frequency content of sounds. Mathematical principles derived from regular polytopes can guide the design of intricate filter structures, allowing for precise control over the tonal characteristics of synthesised sounds.
Control interfaces: The n-sphere concept can be utilised in the design of control interfaces for synthesisers. By mapping different sound parameters (such as pitch, volume, or modulation) to different dimensions of an n-sphere, musicians can manipulate multiple aspects of sound simultaneously, enabling expressive and intuitive control over the synthesiser.
Sound spatialisation: Regular polytopes and n-spheres can also be employed to create spatial effects in sound. By applying mathematical transformations to sound signals, synthesisers can simulate three-dimensional sound environments, allowing sounds to move and interact in virtual spaces.
Signal processing
Fourier analysis: Regular polytopes can be used to understand the behaviour of periodic signals. Fourier analysis breaks down a signal into its constituent frequencies, and regular polytopes help in representing these frequencies and their relationships in a structured manner.
Filter design: Filters are used to modify signals by allowing certain frequencies to pass through while attenuating others. Regular polytopes provide insights into the design of efficient and precise filters for different signal-processing applications.
Harmonic analysis: Regular polytopes help in analysing the harmonic content of a signal. By understanding the mathematical properties of regular polytopes, signal processors can identify and extract specific harmonics or frequency components from a signal.
N-spheres and their mathematical properties are used in signal processing in the following ways:
Multidimensional signal representation: N-spheres provide a way to represent signals in multiple dimensions. This is useful when dealing with complex signals that have various parameters or dimensions, such as colour images or multi-channel audio.
Dimensionality reduction: N-spheres can be used to reduce the dimensionality of a signal while preserving important information. This is beneficial for compressing and efficiently storing signals, as well as for visualising high-dimensional data.
Clustering and classification: N-spheres can help in clustering and classifying signals based on their similarity in a multidimensional space. By applying mathematical techniques derived from n-spheres, signal processors can group similar signals together or classify them into different categories.
Digital image processing
Image representation: Regular polytopes provide a structured and organised way to represent images. By dividing an image into regular polygons or polyhedra, image processors can analyse and process specific regions or features of the image more efficiently.
Image transformation: Regular polytopes can be used to transform images in interesting and creative ways. Applying mathematical operations derived from regular polytopes, image processors can distort, rotate, scale, or morph images, creating visual effects and transformations.
Texture analysis: Regular polytopes help in analysing the texture properties of an image. By considering the geometric arrangements and patterns of regular polytopes within an image, image processors can extract information about the texture and structure of different regions.
N-spheres and their mathematical properties are used in image processing in the following ways:
Colour space representation: N-spheres can be used to represent colours in images. By mapping different colour components (such as red, green, and blue) to different dimensions of an n-sphere, image processors can manipulate and analyse colour information effectively.
Image compression: N-spheres provide techniques for compressing and efficiently storing image data. By applying mathematical transformations derived from n-spheres, image processors can reduce the amount of data required to represent an image without significant loss of visual quality.
Image segmentation: N-spheres can help in segmenting or separating different objects or regions in an image. By defining clusters or boundaries in a multidimensional space using n-spheres, image processors can identify and extract specific objects or regions of interest from an image.
Protein structure analysis
Secondary structure analysis: Regular polytopes help in analysing the secondary structure elements of proteins, such as alpha-helices and beta-sheets. By applying mathematical algorithms based on regular polytopes, researchers can identify and classify these structural patterns in protein sequences.
Tertiary structure modelling: Regular polytopes can aid in the modelling and prediction of the three-dimensional structure of proteins. By using geometric principles derived from regular polytopes, researchers can generate plausible structures and evaluate their compatibility with experimental data.
Protein-protein interactions: Regular polytopes provide insights into the geometric arrangements and interactions between proteins. By studying the spatial relationships and contact points between regular polytopes representing different proteins, researchers can analyze and understand protein-protein interactions.
N-spheres and their mathematical properties are used in protein structure analysis in the following ways:
Molecular docking: N-spheres can help in predicting the binding sites and orientations of ligands or other molecules on proteins. By using mathematical techniques based on n-spheres, researchers can simulate the docking process and identify favourable binding configurations.
Conformational sampling: N-spheres aid in sampling the conformational space of proteins, which refers to the different possible shapes and conformations they can adopt. By applying n-sphere-based algorithms, researchers can explore and analyse the diverse conformational space of proteins.
Structural alignment: N-spheres can be utilised in aligning and comparing protein structures. By representing protein structures as points in a multidimensional space using n-spheres, researchers can measure their similarity and identify common structural elements.
Cryptography
Encryption algorithms: Regular polytopes provide the basis for designing encryption algorithms that transform plaintext (unencrypted) data into ciphertext (encrypted) data. These algorithms use mathematical operations derived from regular polytopes to scramble the data in a way that can only be reversed with the correct decryption key.
Key generation: Regular polytopes are used to generate cryptographic keys, which are essential for encryption and decryption processes. The mathematical properties of regular polytopes ensure that the generated keys are sufficiently random and secure, making it extremely difficult for attackers to guess or deduce the key.
Cryptographic protocols: Regular polytopes help in designing cryptographic protocols that ensure secure communication between parties. By employing mathematical concepts derived from regular polytopes, protocols are developed to establish secure connections, authenticate users, and exchange encrypted messages.
N-spheres and their mathematical properties are used in cryptography in the following ways:
Public-key cryptography: N-spheres play a role in public-key cryptography, which involves the use of asymmetric encryption algorithms. These algorithms rely on the mathematical properties of n-spheres to generate pairs of keys: a public key for encryption and a private key for decryption. The security of the encryption scheme is based on the difficulty of certain mathematical problems related to n-spheres.
Digital signatures: N-spheres are used in digital signature schemes, which provide a way to verify the authenticity and integrity of digital documents or messages. By applying mathematical operations based on n-spheres, digital signatures are created that can only be produced by the legitimate signer and can be verified by anyone using the signer’s public key.
Cryptographic hashing: N-spheres can be used in cryptographic hashing algorithms, which produce fixed-size hash values from input data. These hash functions use mathematical properties derived from n-spheres to ensure that even a small change in the input data results in a significantly different hash value, providing data integrity and verification capabilities.
Acoustics
Sound reflection and diffraction: Regular polytopes help us understand how sound waves interact with surfaces and objects. By studying the geometric properties of regular polytopes, acousticians can predict and analyse how sound reflects, scatters, and diffracts when it encounters different shapes and structures.
Room acoustics: Regular polytopes provide insights into the behaviour of sound in enclosed spaces, such as concert halls or auditoriums. By considering the geometric properties of regular polytopes, acousticians can design spaces with specific shapes and dimensions that optimise sound diffusion, minimise echoes, and enhance overall acoustic quality.
Sound wave propagation: Regular polytopes help in studying the propagation of sound waves in different media. By applying mathematical principles derived from regular polytopes, acousticians can model and simulate how sound travels through various materials and environments, enabling the prediction of sound behaviour in different scenarios.
N-spheres and their mathematical properties are used in acoustics in the following ways:
Sound field representation: N-spheres can be used to represent the spatial distribution of sound energy in a given environment. By mapping sound pressure levels or sound intensity to different dimensions of an n-sphere, acousticians can visualise and analyse the spatial characteristics of sound fields.
Beamforming: N-spheres aid in beamforming techniques used in microphone arrays or directional speakers. By applying mathematical transformations based on n-spheres, acousticians can manipulate the phase and amplitude of sound waves from multiple sources to enhance or suppress specific sound directions, enabling focused sound projection or noise cancellation.
Sound source localisation: N-spheres can help in localising the position of sound sources. By using mathematical algorithms based on n-spheres, acousticians can analyse the time and phase differences between microphones or sensors to determine the direction and location of sound emitters.
Fourier analysis
Signal decomposition: Regular polytopes help break down a signal into its individual frequencies. Fourier analysis uses the geometric properties of regular polytopes to represent signals as a combination of sine and cosine waves with specific frequencies and amplitudes. This decomposition allows us to understand the frequency content of a signal.
Frequency domain representation: Regular polytopes provide a structured way to represent the frequencies present in a signal. By using mathematical concepts derived from regular polytopes, Fourier analysis converts a signal from the time domain to the frequency domain, allowing us to examine the strength and distribution of different frequencies.
N-spheres and their mathematical properties are used in Fourier analysis in the following ways:
Multidimensional signal analysis: N-spheres can help analyse signals with multiple dimensions or parameters. Fourier analysis applied to n-spheres allows us to analyze signals in a multidimensional space, taking into account various dimensions or parameters simultaneously.
Higher-dimensional Fourier transforms: N-spheres provide insights into higher-dimensional Fourier transforms. By applying mathematical techniques based on n-spheres, Fourier analysis can be extended to higher dimensions, enabling the analysis of signals with more complex structures and properties.
Statistics
Data visualisation: Regular polytopes help in representing and visualising data in a structured manner. By using the geometric properties of regular polytopes, statisticians can create plots and charts that present data in a clear and organized way, allowing for better understanding and interpretation.
Multivariate analysis: Regular polytopes aid in analysing data with multiple variables or dimensions. By applying mathematical techniques based on regular polytopes, statisticians can explore and study the relationships between different variables, identifying patterns, correlations, and dependencies in the data.
N-spheres and their mathematical properties are used in statistics in the following ways:
Multidimensional data analysis: N-spheres can help analyze and model data with multiple dimensions. By employing mathematical transformations based on n-spheres, statisticians can represent and analyze data in a multidimensional space, facilitating the identification of patterns, clusters, and outliers.
Regression analysis: N-spheres aid in regression analysis, which is used to predict and model relationships between variables. By applying mathematical techniques based on n-spheres, statisticians can estimate and analyse the relationships between variables, allowing for predictions and inference.
Hypothesis testing: N-spheres can be used in hypothesis testing to evaluate the significance of statistical results. By employing mathematical principles derived from n-spheres, statisticians can calculate p-values, which help determine whether observed differences or relationships in the data are statistically significant or occurred by chance.
Islamic Art / Geometry & Quotes
___
“The universe is a grand mathematical equation, and every celestial body a letter within it, forming a magnificent cosmic symphony.” — Rumi
___
“The beauty of the universe lies in the intricate patterns woven by the Divine Mathematician, where stars and galaxies dance to the rhythm of cosmic equations.” — Ibn Arabi
___
“Behold the universe as a tapestry of numbers, where each equation reveals a hidden truth and every calculation uncovers a glimpse of the Divine.” — Al-Khwarizmi
___
“In the sacred geometry of existence, the universe unfolds as a perfect equation, and the seeker of truth finds solace in the harmony of numbers and patterns.” — Al-Farabi
___
“The language of the cosmos is that of mathematics, where equations whisper the secrets of creation and unveil the majesty of Allah’s design.” — Al-Ghazali
___
“Marvel at the symmetrical dance of stars, for they follow the mathematical choreography of the Divine, spinning in celestial harmony.” — Ibn Sina
___
“The universe is a vast mathematical poem, with each equation revealing a verse of the eternal truth written by the Divine Pen.” — Al-Jahiz
___
“The beauty of the universe lies not only in its outward grandeur but also in the elegant mathematical order that underlies its every aspect.” — Al-Kindi
___
“Contemplate the universe through the lens of mathematics, and you will witness the precision and elegance with which Allah weaves His cosmic masterpiece.” — Al-Farabi
___
“As the seeker delves into the realm of numbers, the secrets of the universe unfold, revealing the infinite wisdom and beauty embedded within the mathematical fabric of creation.” — Ibn Rushd
___
___
Dr Nick Stafford
Eye of Heaven — Lichfield Cathedral, a Theory of Everything
“Sometime too hot the eye of heaven shines” — William Shakespeare, Sonnet 18
“Protect the centre”- Lao Tzu
