Duerer's Solid In Key Of C In Space and Time For You And Me

Foreword - This is one of the pieces I've been working on (the table is another artwork I recently completed), there are currently several thousand measured, cut, painted and glued pieces with several hundred to go to compete this piece. The boarders will be finished in clear acrylic rod, allowing the colours underneath to be visible as they're integral to the art as a whole. My low resolution camera could make anything look drab and often does, sorry about the image quality, this is just a peek for my artistic friends and anyone else that may be interested, enjoy. 

I know! You will have to bear with me! I'm also a poet/comedian/waffler and just could not resist a rhythmical title for such a musical sculpture. A slight deviation from the Title of the work where this solid was originally expressed – A woodcut by Albrecht Duerer titled, “Meloncholia I”, but to those of you who understand music and the work of this great Renaissance artist, the sentiment and humour expressed by Duerer is also expressed in this modern title, but I rhymed mine so....

I will start by explaining some of the properties of the sculpture, such as the colours and shapes in the area of physics, geometry, time, space etc. then discuss the music portion of the sculpture. It will be difficult to keep the explanation simple, but lets give it a go and see what happens.

We could talk for hours on what, ‘Albrecht Duerer’ meant by producing his solid (Truncated Rhomboid), no doubt it would be a fantastic experience, but so many others have covered this topic in detail. Of particular interest is the essay by David R. Finkelstein, School of Physics at Georgia Institute of Technology, which can be found here:  http://www.arendlandman.nl/wp-content/uploads/2012/02/Albrecht-Du%CC%88rer-Melencolia_code_finkelstein.pdf and http://mathworld.wolfram.com/DuerersSolid.html . If you are not familiar with Duerer’s Woodcut, “Meloncolia I,” then I would suggest reading these other websites as a precursor to this document. I am building my own version of Duerer’s solid, but I have taken it further as you will see in my explanation. I will not include every single aspect of what I have done, in order to save time and leave it to your own imagination. Some secrets are too precious to divulge and others, too sensitive for publication. One does not want to force a cube to assume the shape of a cylinder in the hope that the cylinder will not be the end result.

Truncated Polygons

The truncated polygon faces that make up the truncated rhomboid, have different colour willow branches cut and laid across the plane of geometric non-Euclidean 3d space, the so-called egocentric coordiantes: up and down, left and right, forward and backward. The directions (manifolds) are represented by the 72-degree point of each truncated polygon when unified within the Euclidean space-time geometry of the truncated rhomboid. There is a relationship between the six directions, i.e. up and down is a pair so we will give the boarders a colour paring relationship, which will be reflected throughout the whole of the sculpture. We have four colours and eight sides, we will make pairs out of these colours giving four possible pair combinations, we will then use three of the pairing to represent the six directions in non-Euclidean 3d space, and the final pair of colours will be used on the two small equilateral triangles, which I will discuss later. The six faces divided into groups of two, which give three pairs of directions. The specific colours are placed in the boarders around the outer edge of each of the truncated polygon faces.

If we look at the direction arrows of the truncated polygons, they can be thought of as individual particles in an isolated system with no correlation to one and other on the microscopic process level. The six directions unified by time and space in the truncated rhomboid structure. The boarders surrounding each of these truncated polygons are representing entropy within each of the isolated systems. Time and space are the correlation of all of the particles into a unified macro process, laying down the second law of thermodynamics to shine light (excuse the pun), on the symmetrical nature of time and the irreversibility phenomena found in nature, thus marking the relative direction of time from past to future within the non-Euclidean geometric space. Also, the boarder (Entropy constraints) colour groupings equate to 432 = 4 colours, 3 sets of two boarders and 2 sets of directions in each of the three sets – a very significant number.

If we look at the colour pairing in detail, we will see that there are four sets of two colours. We divided the shapes into rational groups of 6 x truncated polygons and 2 x equilateral triangles. If we look at the truncated polygons first; these have three functions; first, we will discuss two of them and then the other one later. The Truncated polygons represent non-Euclidean geometric 3d space, as such, we only used three pairs of colours: Green/ Yellow, Blue/Orange, Blue/Yellow, to represent 3d space. The second function of the truncated polygons is the boarder constraints that we mentioned earlier and the third function is the musical composition instruction, which we will discuss in detail later on. Therefore, to recap, we assigned the truncated polygon faces, which represent non-Euclidean 3 dimensional space without time (fourth dimension), with three colour pairings and we have one colour pairing left which is "Green/ Orange.” Green = G = time and Orange = O = space. The space portion of the sculpture we leave empty, which frees us to discuss the time element. G or green represents the time element (Triangle) of the sculpture. G represents the fourth dimension time Gamma = C = Light. In order for us to comprehend the fourth dimension in its entirety, though, it is necessary to have access to the fourth or higher dimensions, this access is represented by using the fourth colour pairing, so that we can cross from a 3d non Euclidean geometric space (Truncated polygons), to a four dimensional Euclidean space-time geometry. We must use the access key in order for us to enter fully into the fourth dimension but we don't have it! We can only see its effects from a 3d perspective - the missing key is represented by the missing Orange colour in the space triangle; it is an allusion to the following metaphor: that we can never enter higher dimensions from within our 3 dimensional universe. We can only comprehend fourth dimensional time in the three dimensional perspective - a mere glimpse behind a warped mirror - but that we can only see equal and all lower dimensions from our own 3d perspective. We can only imagine or plot a partial shadow of higher dimensions. For a better understanding of the phenomena of which I speak, I recommend a book on this subject of dimensions called, ‘Flat Landers’. There are also plenty of ‘YouTube’ videos explaining the concept of "Flat land.”

Truncated Rhomboid.

The Truncated rhomboid solid contains the 137-quantum fine-structure constant, a stretched cube within an inscribed circle divided by fifths. Before we speak about the time portion of the sculpture, let us note that the estimated time of the beginning of the nascent Universe is 13.7 billion years ago. As the universe expanded, both the plasma and the radiation filling it grew cooler. When the universe cooled enough, protons and electrons could form neutral atoms. These atoms could no longer absorb the thermal radiation, and the universe became transparent instead of being an opaque fog. Cosmologists refer to the period when neutral atoms first formed as the recombination epoch and the event shortly after, when photons starting to travel freely through space rather than constantly scattering with electrons and protons in plasma, referred to as photon decoupling. The photons that existed at the time of photon decoupling have been propagating since then, albeit, fainter and less energetic with the onslaught of time. The reason for this is because the expansion of space causes their wavelength to increase over time (and wavelength is inversely proportional to energy according to Planck's relation). This is the Melancholy in this Sculpture and possibly in Durer's too and the joke is still on you, him, and yours truly... what do you think existentialism was all about?

The melancholy in this sculpture most definitely belongs to the Big Bang model for the formation of the universe. A story of sorrow that is seen as a reflection in all processes ever since. To explain the recombination epoch mentioned above in greater depth, we look to the Inflationary Cosmologists who predict that after about 10⁻³⁷ seconds the nascent universe underwent exponential growth that smoothed out nearly all inhomogeneities. The remaining inhomogeneities caused by the quantum fluctuations in the inflaton field caused the inflation event. After 10⁻⁶ seconds, the early universe was made up of a hot, interacting plasma of photons, electrons, and baryons. As the universe, expanded, adiabatic cooling caused the plasma to lose energy until it became favourable for electrons to combine with protons, forming hydrogen atoms. This recombination event happened when the temperature was around 3000 K or when the universe was approximately 379,000 years old. At this point, the photons no longer interacted with the now electrically neutral atoms and began to travel freely through space. This event resulted in the decoupling of matter and radiation. As mentioned previously, the colour temperature of the decoupled photons has continued to diminish ever since; now down to 2.72548 ± 0.00057 K… see the trend.

Now we move onto the two triangles, time and space.

The triangles divide the 5 sided Truncated polygons by 3 sides = 1.6666666 an approximation of Phi.

Triangle 1: This equilateral triangle represents time as the fourth dimension. The willow projecting out from adjacent to the mid point of the triangle's base lines, actually projects out to the length of an invisible inscribed circle, the point between the base line of the triangle and the edge of the circle is precisely phi thus representing time. We know that time equates to phi constant because of G-factor equations for Electrons and Protons, which are as follows: = -2/ sin (Ø) and = 2 Ø/ sin (1Ø), respectively. 

The National Institute of Standards and Technology (NIST) states these gfactor constants as per the table below.

	gfactor electron	Gfactor proton
Phi-based	-2.0022334732293	5.5848781529840
Per NIST	-2.0023193043718	5.5856947010000
Variance	-0.004287%	-0.014619%

While the phi-based approach is not exactly equal to the NIST constant, with quantum level constants, there will always be a difference between theoretical work and empirical work.  This can be due to the irregularities in the metal of the test equipment, stray background radiations, or a host of other minute causes. The results of any empirical measurement always show some degree of statistical spread consequently. Such constants are adjusted as new measuring techniques and better materials become available. Therefore, any theoretical value that is within one part in a thousand has scientific value.  

The Time Triangle has three points connecting the three sets of two truncated rhomboids, thus the time is free from entropy and has potentiality, following the direction of confined space within the non-Euclidean geometric shape or truncated polygons. The colour pairings of these two equilateral triangles are important and I will discuss them later as a whole.

Triangle 2: This triangle represents space; it is blank for this reason. Nevertheless, it still has a value of Orange on the colour combination pairings.

Okay, so now we take the two equilateral triangles as a whole and assign the last colour pairing out of the combinations we produced earlier. These colours are green/time and orange/space. You will notice that the artist has encrypted his initials into the artwork by using that colour pairing and using the first initials of those colours Green and Orange we get GO, which stands for Gwylym Owen. Gwylym has put himself directly in space-time. The encryption goes a little further too - If we take the cross section line in the middle of the time triangle where the willow intersects up to the invisible inscribed circle then we continue that line across we make the shape of an "A.” A = Alpha in the Greek Alphabet, it represents the beginning, "A" also has a value in the music portion of the sculpture but I will deal with that as a whole later on. The colour of the triangle is Green, if we take the first letter and apply it to the Greek alphabet, then phonetically we have gamma as the nearest sound, Gamma is the third letter of the Greek alphabet, if we convert three into our alphabet then we get "C" and C = the speed of light. If we then look at the space triangle the colour would be orange as it is the only colour not yet used and is paired with green, the first letter "O" is phonetically the nearest sound to Omega , Omega = end. If we convert Alpha and Omega to binary 01(off/on), then insert gamma (light) into the middle we get 0.01.1 or put in another way, the expression of the point between off and on as it progresses in sine wave fashion within space-time.

I'll repeat myself here in the hope that you won't be confused by the overlapping of artistic concepts and physical elements of the sculpture. The time triangle is Green and the space triangle would be orange if it wasn't empty (space). The colour pairing, Orange/ Green has been carefully selected, not only to encrypt the artist’s initials into the sculpture, but also to signify light, the beginning and the end. If we use the invisible "A" section in the Time triangle, we can call this Alpha (beginning). There is no symbol in empty space so we use the numerical symbol for zero, therefore; we will use the initial “O” from the colour orange. O = Omega (End). Omega is angular frequency 360 degrees or one radion, if we then take a full turn or one radion around the circumference of the 3d Truncated rhomboid we end back up to the Time triangle. As we have already used the “A” in the the time triangle to denote Alpha, we will instead use the colour Green to finish our one radion rotation. If we use the first initial of the colour green then we get G = Gamma which is the third letter of the English alphabet, which equates to C and the speed of light = C. Therefore, if we convert Alpha to Zero and Omega to one, binary values (off/on), then we get 01. Let us insert Gamma into the middle as it is in the Alpha through space (Omega) to time (Gamma) rotation previously described and we get 0.01.0, written as a sine wave, this represents all life, the beginning, which is potential energy until the wave function collapses; Alpha, Gamma, Omega. Gamma is the life source; the sine wave shows us the gaps between beats of pulsating bundles of energy releasing from its alpha states as the quantum wave function collapses and entropy plays its part in marking the timeline of that energy/ light/ proton/ life - A linearly polarised light wave frozen in time.

Music

Music be the sound of life, of love and of our universe as a whole.

Outer Portion Of The Truncated Rhomboid.

The following is a description of my thought processes in creating the music part of this sculpture. The Duerer solid is inscribed within a circle divided by the five points of the truncated polygons, making six perfect fifths. I want the music to represent the same relationships so I have decided to use the C-major scale, because C = light. The required perfect fifth notes from the Ionian mode, in C-major scale have been transcribed into this sculpture: the four colours we started with (Blue, Yellow, Green, Orange), and assigned each of these with four notes of the scale. I then made three pairs out of those colours, I then rearranged the order so that I wouldn't get confusion/ overlap of notes because I am not using gaps to denote the start and finish of a note. I also further rearranged the colour combinations so that “G” was once again matched with green, Emphasising the “G” for “Gwylym,” but more importantly, the “G” for gamma, the “G” for light. These colours and colour combinations are F= MAJOR = Blue/ Yellow, C = minor = Orange, G = minor = Green, D = MAJOR = Yellow/ Blue, A = MAJOR = Green/ Orange, E = minor = Yellow B = minor = Blue (notice the colour combination ratio explained below).

 I wanted to tune the notes to the frequency A = 432Hz which is the old international standard concert pitch, which has now changed to 440hz - there was considerable concern raised by thousands of musicians at the time. Many of the old bells, Buddhist gongs and many other string and percussion instruments were at one time, tuned to a frequency of “432Hz.” The “440Hz” frequency, causes waves to crash into one another and in some cases can cancel each other out. By tuning the concert pitch to “432hz” we ensure the tones are pure and full. The transcription was achieved by the ratio in the chosen colour pairings, let me explain: Green, Orange, Blue, Yellow  is a set of 4,  Blue/Yellow, Green/ Orange, Yellow/Blue is a set of 3 and there are 2 sets in total, so we get 432, representing the Hertz of concert pitch. 432 squared equals the speed of light = C. The “A” portion of “A = 432Hz”, is obtained by looking at the time triangle, where the golden ratio intersection which marks the invisible inscribed circle, which again creates the 6 x fifths or Phi. This intersection can be imagined to cross through the triangle and out of the other side, thus creating the "A", the visible part of the crossbar of this  A-section is painted in black and white representing the Intervals T-T-S-T-T-T-S or otherwise called the Ionian mode of the C major scale, Where T = tone and S = semitone. Another special value of the colour combinations is that once again we can find the Artists initials encrypted into the sculpture and again, in the time and space elements of the sculpture - If we take a look at the C on the scale you will notice its colour is Orange = O = Space = Omega = Owen. Now look at the other important note on the scale which is integral to this whole artwork, of course, it is the “G” note and you will also find the colour Green = Time = Gamma = light = Gwylym. Now just for good measure so that you know it wasn't a fluke the artist in the style of Duerer's great sense of humour has chosen the colour combination Green and Orange to represent the A note, thus making Gwylym Owen = A = 1 or number 1, as Duerer had done hundreds of years previously.

The transcription of the seven pure notes of the C-major scale, starting from key of, "Bb Major,” was achieved by moving down the perfect fifths on the scale chart, starting from “F”, yielding the required seven notes or seven pure tones. 6 x perfect fifths is also encrypted within the sculpture: 6 x truncated polygons and the truncated polygons have five sides, which yields 6 x perfect fifths and seven natural pitches or harmonic notes contained in the Ionian mode which are F-C-G-D-A-E-B. The perfect fifths are very powerful notes that feed back into themselves just like the polygon ideograph hidden in Duerer's solid, reinforcing the strength and grandeur of this sculpture just as the perfect fifths reinforce the strength and grandeur within nature, reflected in the composition of music, space, time, geometry, physics etc. The sculpture then, is art-imitating nature in its purest form. The form is one of self-replication on the micro and macro, internal/ external levels, which give a hint about four-dimensional objects - i.e. Perfect 4D cubes and how we interpret their repeating nature, in the 3D perspective. So, to recap, the outside Faces of the Truncated polygons have these notes: “F-C-G-D-A-E-B”, laid out in their original scale order; one scale length after the other, running horizontally across the truncated polygon faces and descending down the scale of time. The transcribed colours create a pattern like this: “Blue Yellow Orange Green Yellow Blue Green Orange Yellow blue...”, this pattern is a full length scale starting from “F” and finishing at “B” and then repeating, it has not got any overlap of colours repeating within the scale as long as the scale is kept in tact. 

As mentioned earlier in this text, the Duerer solid is inscribed within a circle, thus making six perfect fifths. I want the music to represent the same relationships so I have decided to use the C-major scale because C = light. I have used bits of cut up willow branches painted in the same four colours as the boarder colours around the shapes of the truncated polygons. I have cut these willow pieces to different lengths that equate to 72 beats per minutes. I have measured the space across the face of the truncated polygons in lines; each line has gaps that equate to 0.5 cm increments, for each drop down to the next line. If we measure line lengths, they are between 10cm and 15cm long (up or down depending on the portion of the shape). There are 26 lines in each truncated polygon. If we add up all the seconds of each line then we get 364cm or seconds in total. 364 is the twelfth Tetrahedral number = 12 scales of music. Of interest too is the number 364 is also the leap year. If we multiple 364 seconds by the 6 sides of the truncated rhomboid, we get a total playing time of 2,184 seconds converted to minutes (/60), we get 36.4 minutes. If we do some time adjustments for the Gregorian calendar, equinoxes etc, a very special relationship between the origin of time, space, music and Humanity becomes startlingly apparent, we could also find many other relationships between these parameters if we want to do the calculus.

We squared the circle of time using a square triangular number - i.e. the time triangle inscribed by an invisible circle (of time 360 degrees) around the distorted, stretched cube, which makes up Duerer's solid. 36 is both the square of 6 and a triangular number, making it a square triangular number. It is the smallest square triangular number other than 1, and it is also the only triangular number other than 1 whose square root is also a triangular number, now you see why time is represented by a triangle in this sculpture. Time is linked to the human element within this sculpture, represented by setting the tempo of the seven pure notes, at the average heartbeat of Humans, which is 72bpm. Each pure note played at 1.2 cm's or beat per second (bps), which equates to 72 bpm. The colour combinations will have to adhere to this rule too, so the single colours that denote the Major notes will be cut at 1.2cm, and the combination colours that represent Minor notes will be cut at half that @ 0.6cm; keeping the Minor notes at the same tempo of 1.2cm, thus encrypting the average heart rate beat of human beings. Another wink in the direction of Duerer's “Meloncholia I” and what I believe him to be telling us in that work. I will be as cagey as Duerer on this point, which definitely isn't moot, in fact, I will go to my grave with this delicious thought, proving that a sense of humour rarely changes down the ages; I tip my hat to you Duerer.

Okay, so we now have a timeline and an ordered scale in C-major that is repeating the seven pure notes at a tempo of 72 bpm for 36 minutes total on the Outside of the 3d structure. If we divide 364 by the five sides of the truncated polygon we get 72.8, if we then add 36 to this number we get 108.8, if we subtract the number of total sides, which is 8 from the end of this figure, then we get 108. Eight is the number of planets in our solar system. 36 minutes X 5 sides of the Truncated polygons x 2 of the triangles time and space = 360 or one radion or full circle. 108 degrees is the interior angles of a regular pentagon, which again is to be found in the skeleton of the shape. Moreover, 108 is the hyperfactorial of 3 since it is of the form . 108 is a number that is divisible by the value of its φ function, which is 36. 108 is also divisible by the total number of its divisors (12), hence it is a refactorable number. The number 108 is considered sacred in many Eastern religions and traditions, such as Hinduism, Buddhism, Jainism, Sikhism and connected yoga and dharma-based practices. The individual numbers 1, 0, and 8 represent one thing, nothing, and everything (infinity). 108 represents the ultimate reality of the universe as being (seemingly paradoxically) simultaneously One, emptiness, and infinite, which is ultimately expressed in the rest of the sculpture too.

Inside portion of the shape.

On the Inside of the shape I have encrypted the Om pronounced Aum. The boarders of the truncated polygon faces have 55.5 millimetre lengths, starting at the 72 degree point and travelling clockwise under the influence of time; the colour of these boarders are green (the final colour left from the set of four colours), We know that G = Gamma = C = Light, from the outer encryption explained earlier on. The reason why I chose green light to surround the Aum chants is because green light at a wavelength of 555 nanometres has an energy level of 216 KJ/ mol, a typical energy of everyday life. The numerical value of the Planck constant depends entirely on the system of units used to measure it. When expressed in SI units, it is one of the smallest constants used in physics. This reflects the fact that, on a scale adapted to humans, where energies are typically of the order of kilojoules and times are typically of the order of seconds or minutes, Planck's constant, the quantum of action (the U Syllable in the Aum, the hidden sound), is very small. The syllable consists of three phonemes, a (Vaishvanara), u(Hiranyagarbha), and m (Ishvara), which symbolises the beginning, duration, and dissolution of the universe and the associated gods Brahma, Vishnu, and Shiva; or Alpha, Gamma, Omega; or 123, respectively. Green is how the time travels with light and green = Gamma = Light. If we look for Orange, we see that it is the colour representing the, "A", from the Aum symbol or Alpha, the beginning. “A” is represented externally on the outer portion of the sculpture and once again, internally; drawing attention to this one true source, light. If we look at the “A” note, this one true source is reflected internally, by virtue of being the only set of one, which has the same set of initials made up of space and time, i.e. The Orange and Green notes (C,G). The “G” note represents the “U” from the Aum sound and “G” also represents C = Light. The O = Orange = Space, this colour is the "C" note on the scale we created, representing C for light and Orange for space, the artist is drawing attention yet again to the source of our being and placing himself squarely in the middle. The internal sound of the sculpture will be laid down with the willow using the same 72 beat mechanism and time line as the external level, adjusting for the difference between the 7 notes on the outer side and the 3 syllable sounds of the Om sound, on the inside. The Internal sound plays initially, then shortly followed by the major pure tones on the external part of the sculpture. Both external and internal elements are at 72bmp but staggered to create a two-wave interference pattern.

As a finishing touch and reaching somewhat, the date of the artwork can be found by looking at the space triangle which has 3 sides, multiply this figure by the time Triangle = 9. Take the AUM symbol and assign the number one, two, three respectively and add the 3 to the 9 = 12. Then add the 8 sides of the truncated rhomboid = 1220, which happens to be the day that Trial by ordeal was abolished, a great date for the World to celebrate but not the one we're after, besides, we still had confessions by torture until the 16th Century, so… Therefore, we will use the 6 Polygons x 2 Triangles + (6+2) gives us the 20 and voila we have the date.

The Artwork is drawing the viewer’s attention to the connectedness of our universe. That there is a relationship between all things and that relationship is light. Light is the source, when we arrange the harmony for others to see, we are allowing them to open up to the idea that we are all connected beings who share in infinite proportions everything which we experience as individuals.

I have plans for five designs in different materials, with variations on this theme, such as semi-precious lapidary set in black onyx or marble, for instance. If you are interested in commissioning me to produce a one-off piece of art which is current, relevant, interesting and likely to increase in value as my career progresses, I look forward to hearing from you. I'm a painter, sculptor, poet with a keen interest in bringing the arts and sciences into a more collaborative relationship for the benefit of everybody. We share a space in creativity and as long as the boundaries are respected it is possible for a great deal of good to come out of such collaborations.