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.
