please note that this website is under construction
Images / Vorgestellte Gegenstände
The Shape of the Whole
Unverborgenheit / Unconcealment
'das (Nicht)-Dingliche der Sprache '
Prima Materia / Squaring the Circle
The pages in the top left-hand corner of the menu outline the horizon of my thoughts. For an essential outline of what I am trying to do please go to: ORIGIN / Emergence of the Whole.)
The Language and Form of Causality
"The ultimate paradox of thought: to want to discover something that thought itself cannot think." S.K.
(Each form / idea / thing / Vorstellung / Gegenstand is carved from a single piece of lime wood in the dimension of either 280 or 240cm x 9cm x 9cm - the given and fixed paradigm through and within which each idea must transubstantiate and materialise. The 40cm distance between form and ground allows for the rise of the visual field. Gluing is not permitted. )
(In a line features such as surface, volume and edges are indistinguishable/inseparable; the uniform surface of the line offers no detail or reference points for the eyes or the mind to hold on to - thus the seemingly bodiless line is literally overseen. Even in the act of carving, volume or depth are reduced to a continuous extension of surface - depth itself stays out of reach. A circle has neither beginning nor end; it can neither be constructed from nor be decomposed into parts - nothing can be taken away without thus destroying it entirely. It is not possible to split it up further to find more fundamental units or more elements that would constitute its being and history. There is no deeper ground to reach. A circle, as expressed in the mathematical symbol of pi, is immeasurable, infinite and irrational. The wooden circular loop is itself of circular cross-section - detail and whole are identical.
The thought of a hole piercing a single line turns the line into a circle. This thought changes the status of the line in wood from being a rigid material and singular thing into a flexible and infinite loop.
In the mind and in wood this loop/circle can be 'folded' into the volume of wood in ways that change the essential property of a circle: outside and inside are no longer opposed along the circle line, but instead, they envelop each other and merge. The folded loop not only changes the fundamental properties of the circle but also the status of the piece of wood itself, as the circle can be 'folded' into the volume of wood in ways that the overall length of this loop far exceeds the size of the actual wooden beam. The wood itself has become flexible, fluid and potentially infinitely stretchable. In the shape of the loop, wood takes on properties/qualities of the mind; thought and wood are indistinguishable.
The loop in wood is a monad-like and literal Zero in the system of my thought - an irreducible, powerful, fundamental yet infinite starting 'point', essentially defined by the hole in its centre. Something fundamental is necessarily a limit.)
homeomorph / featureless circles / unknots = topologically 'the same'
emergent / resultant - ?
What is the link between psychic and physical events ?
The Original Shape - The Detail And The Whole / V-E+F=
Information / Reliquiae
(That which hangs stands - and vice versa. As the spectator circumambulates the form, every time s/he reaches 90, 180, 270 and 360 degrees of this circle (the centre of the 'sculpture' being the centre of the circle), the shape appears as the up-side-down reversal of the previous position, creating a wave-like motion. The contour -of where the straight line segments meet the curved lines- is identical with the Aleph shape.)
The Detail And The Whole 1
That which hangs stands - and vice versa. As the spectator circumambulates the form, every time s/he reaches 90, 180, 270 and 360 degrees of this circle (the centre of the 'sculpture' being the centre of the circle), the shape appears as the up-side-down reversal of the previous position, creating a wave-like motion. The contour -of where the straight line segments meet the curved lines- is identical with the Aleph shape.)
Xylem / "Certain Lines Split"
The thing / Gegenstand / the object of my studies never appears in its totality / it is essentially unknown and unknowable in its entirety. The wooden things are like conic sections of this unknowable whole; they examine the same contours in the light of different questions coming from different fields. The same contours lead to a range of different questions which are linked by the forms in wood.
no name, detail (destroyed) continuous line yet reversal of form
9+4= 13+5= 18+6=24+7=31+8=39+9=48+10=58 = 240
Information / Horizon (Coincidence )
I discovered this number sequence used in the Horizon series during an extended stay on Eilean Shona, a small island off the West Coast of Scotland - an extremely remote environment, with no electricity, no telephone reception or any means of contact to others, and accessible only by kayak - a place of unspeakable wilderness and beauty. There the elements play out their grand rhythmic movement and pattern without disturbance, each element and detail existing tightly woven into its precise place and role within a complete wholeness which incessantly changes and remains the same all along. There I saw Nature, I was able to see Nature for the first time, as it is without me, without me looking. I was not part of this wholeness.
This outsider view on everything outside of me, this absence of my own 'subject-hood', was the most profound experience. It has fundamentally shaped my view on how-I-am-in-this-world - and on Art.
To focus on elements in Nature which are constantly moving whilst their essence remains unchanged: the shape of a waterfall, the relentless movement and rhythm of the tide and waves, the shape of a spiral shell seemingly turning upwards: movement without moving. In an attempt to understand the structure and 'movement' of a spiral shell - seemingly growing from one point as if out of nothing and widening -potentially infinitely- outwards, I started to relate numbers to this shape. Intuitively I began with 9, a value large enough to manifest within the paradigm of my work in wood, and stopped randomly at some point to add up the values: the result was 240 - which happens to be exactly the length of the wooden sculptures in centimeters. Surprised and alerted by this coincidence I applied this number sequence to my basic tool / the infinite loop as a coordinate system structuring the circle, as well the lines in drawings and balsa models - which resulted in chessboard-like patterns, as if drawn in perspective, outlining a horizon and vanishing point.
The subject does not belong to the world but it is a limit of the world. L.W.
Is information natural?
On which side does information belong / where does it originate - Nature or human nature?
Is information the link between the two?
'I know it' I say to someone else; and here there is a justification. LW OC 175
Spirit Level / Sprachwerkzeug / Sprachhandlung
Four square lines along the edges of the beam and four square diagonals across the four surfaces have been removed. The base of the wooden beam is shaped at a right angle in relation to two of the diagonals on two adjacent surfaces - so when placed on the floor these diagonals are now vertical seen from the front of each surface they sit on. Gravity / a lead weight connected to the beam from underneath the floor tile, holds the leaning beam - and the two diagonal lines in a vertical position. The two other diagonals and the four edges of the beam now appear to be toppling over.
Invertible Cube by Paul Schatz
The short edges act as hinges which allow this form to undergo a surprising and fluid sequence of different combinations of triangular, hexagonal and square/cubic forms on the inside and outside of the shape. (for more info see: 'Invertible Cube' in menu on the left)
The Same Thing / Slow Motion (Zeitlupe)
The outline of the two 'sculptures' follows the outline of one segment of the Invertible Cube in motion; the wooden forms are two stills of the motion of this one form which itself is only present as an image in the mind. The stills are taken fractions of a millisecond apart / with a maximum degree of proximity in time and space between them, thus rendering the forms in wood near indistinguishable.
The rounded lines in the two forms 'touch' in slightly different places. This difference is only measurable in relation to the cubic base of the wooden beam and cannot be determined with certainty when looking at the relationship of their circular surfaces only (so when not taking the square reference frame into account). Without the cubic reference frame the points of the lines 'touching' look and are the same.
As the spectator moves around the two forms they behave like one of the forms and its own shadow.
Movement of one segment of the Invertible Cube
from: Paul Schatz - Die Welt ist umstülpbar –
Rhythmusforschung und Technik. 1975
"Anhalten zum Herauslösen aus der Prägeform"
The edges of one segment of the Invertible Cube repeated (with the number sequence of Information/Horizon applied) outline a twisted surface (sample carved in lime wood).
Parallel Diagonals / Objective Physical Reality Rohling
Connecting the diagonals on four sides of the beam as if by minimal surfaces results in a shape of surprising complexity. Diagonals on four surfaces of the wooden beam, when exposed as the edges of a volume, reveal that they are -paradoxically- straight, curved and parallel all at once. A line that is not curved is parallel to a line that is curved (this is a reality in wood; not an abstract mathematical reality.) Lines parallel to diagonals on parallel surfaces touch at the centre, yet are further away from each other at the ends. The distance between the points at the end of two parallel lines is greater than the distance between their midpoints.
Opposing surfaces or edges in a cubic volume are parallel / at the same distance to each other. Yet isolated points or squares along those parallel surfaces can be at a distance to each other which is greater than the distance between these parallel surfaces. 'Parallel' in a cubic volume in an absolute sense only applies to either the totality of the parallel surfaces (or edges), or to single points which sit exactly opposite each other at a 90 degree angle. The relationships of all other points which are not part of this definition are of greater 'freedom' and difficult to define. I chase excatly those elements. The paradox relationship between elements which are parallel and not parallel at once is exposed in diagonals which run along the parallel surfaces of the cubic volume, yet also connect corners which are at a greater distance from each other than the distance between those parallel surfaces.
In Parallel Diagonals these diagonals are laid bare as edges of the volume and reveal that they are curved and straight, as well as parallel and diagonal all at once.
Parallel Diagonals 1 unfinished
("...to think of the thing as being the forces that act on matter, rather than matter on which forces act, to just show enough matter to expose the forces, or rather: to expose the incomplete notion of matter - as if it was possible to exist independently of forces.")
"No, no, you are not thinking, you are just being logical." Niels Bohr
to look for situations where geometry behaves differently from classical geometry -
"das (Nicht)-Dingliche der Sprache zum Ausdruck bringen"
(Should it be possible to work from Nature, to develop 'the rules of the game' as if from 'the other' side, not from language and logic, but from within matter? To find that which resists the 'language game' / to find that which resists within the language game?)
"Ontic Residua - in principle unknowable once the knowable structures of reality have been factored out: [...] models can be increasingly informative only about the relations that obtain between the objects, but not about the first-order, one-place predicates (the intrinsic nature) qualifying the objects in themselves. " Luciano Floridi
In the mind, a line is single, bodiless and identical with itself. The single line is an idea(l). When trying to carve one line / a circle in wood -in the process of transubstantiation from mind into matter- the 'outline' of a single line must be drawn onto the surface of the wood as two lines which is necessary in order for it to become an object that can be handled. Only when the surface has become circular (when the edges have been smoothed and eradicated) the line will again be singular.
In wood a single line can only be directly shown as a single edge of a cubic volume.
The edges and vertices in a cubic volume of wood are immaterial features and boundaries - they are bodiless information - and it is these singular 'bodiless' wooden lines which unfold the most irrational and paradox qualities.
The shape of a circular loop with the line that is round in cross-section is fixed and decided in an instant before I even begin to carve, and there are no surprises or changes of mind in the process of production. But the work with the cubic volume is completely unpredictable- it appears to emerge from the volume itself; potential new shapes and relationships 'mushroom' along all the edges, and I constantly have to make choices between many possible forms and leads. The circular loops spring - in an unplanned and most spontaneous manner- from my mind; the square / cubic lines spring in this same manner from matter itself.
It is strangely contradictory that the 'irrational' circular loops should be the far more definite and fixed things in the process of 'production', and that the seemingly rational and measurable cubic volume should emerge in this boundless and irrational/incomprehensible way.
As a result, the work with the cubic volume has made it much more difficult to say when a work is finished. Examining the cubic volume itself is like crossing a border behind which objects - as complete and finite things- do not seem possible. (For more on the relationship between circle and square go here: Prima Materia / Givens.)
Where is the limit where the 'language game' (L.W.) ends, at which matter reveals only itself ?
Parallel Diagonals 2, destroyed
"For, being some 'thing' does not preclude something from belonging to grammar."
Parallel Diagonals 3, destroyed
(Two opposing surfaces in a cubic piece of wood are necessarily parallel to each other. “Parallel" applies to each and every part of these surfaces - there are no areas or points which could be excluded from this 'parallel-ness'. A diagonal drawn on one side of this piece of wood is necessarily parallel to the surface on the opposite side. If another diagonal is drawn at a different angle on the opposite surface, then these two diagonals will necessarily not be parallel to each other - despite both diagonals sitting on two parallel surfaces. How can this be understood - two lines which are not parallel, but which clearly and necessarily are parallel to each other at once? As they each sit on parallel surfaces, each point of which is part of the attribute ‘parallel’ -?
This situation necessarily reveals the need for attributes to be applied: the line thus must have a 'top', 'bottom', 'left' and 'right', defining the seemingly bodiless line as a multi-dimensional body. Geometry here is produced by the Logic the situation requires.
The 'top' and the 'bottom' of the lines must accept the way the surface provides space, but 'sideways' the line has the 'autonomy' to 'negate' the attributes that apply to this surface. Lines on parallel surfaces can be described by attributes which are not totally determined by this surface. Can one say that these lines have a degree of freedom-? How can this divergence of the line from the whole be best described and understood? 'Freedom': the possibility of the detail not being fully determined by the whole, whilst being fundamentally determined by the whole ... ?
The totality of a line is congruent with the surface it sits on. The totality of surface is not part of the totality of the line. The surface can only determine the line in the area the line covers. The surface being parallel to another implies that its attributes apply to the lines it carries. The diagonals on parallel surfaces being not parallel to each other implies that the lines are entities with attributes either relating to the relationship between lines (on surfaces), or to the relationship between surfaces as whole entities. The 'freedom' of one set from another despite being contained in it? The question of the line and surface is a question of the relationship between the part/the detail and the whole. Surfaces and lines are physical and logical sets which partly contain each other. Attributes can satisfy the logic for sub-sets or parts, yet on the whole a paradox is produced.
Matter reflecting (ordinary) language as relative/particular/only able to reflect one aspect at a time out of an infinitude of aspects?
Everything we do -and think- is embedded in a ‘geometrical’ physical situation / it does not seem possible to think of something that is not embedded in a physical situation. Even spontaneous thought, images that can 'pop' up in the mind, seemingly without history, seem 'embedded' in this way.
The diagonals which run along the four (parallel) sides of the beam have been exposed as edges. This diagonal edge is a single borderline between two surfaces either side. Lines parallel to these diagonal edges are the two outer borders of one surface between them.
To Come to the Surface 1 / Inside = Outside
To Come to the Surface 2 - surface = the borderline between representation and 'thing'
To Come to the Surface 3
To Come to the Surface / Segment / Limit (Either / Or) , destroyed
The Shape of (Its Own) Information 1 / Self-Evidence
to make a 'sculpture' that is its own 'raison d'être'
A line/loop, square in cross-section, when placed on one side of the beam, has got 4 surfaces and 4 edges. Yet when folded around one corner of the beam as in the shape above, the properties of the loop change: whereas locally this square line still has got 4 surfaces and 4 edges, globally it now only has 1 edge and 2 surfaces. Taking the loop across a 90 degree edge has 'swallowed' 3 edges and 2 surfaces.
This single remaining edge is a Trefoil Knot (the edge of a Moebius Band). It can be removed without erasing the shape (see photo below); instead removing the single edge produces new edges as it splits the line in half and at the same time doubles its length. This process can potentially be repeated ad infinitum - making the line infinitely thin and long without ever erasing itself.
The Properties of Strings/Encircling a Spatial Tear
Trefoil Knot 1 / The Shape Of Its Own Information 2
The edge of 'The Shape Of (Its Own) Information' (a Moebius Band) is put through the paradigm of my work for observation, revealing again the shape of its own information.)
The knot is shaped like two offset halves of the Aleph form (see the very first photo at the top of this page).
To Come To The Surface 4
"If a curved surface is developed upon any other surface whatever, the measure of curvature in each point remains unchanged." Carl Friedrich Gauss, Theorema Egregium, 1827
"Physical knowledge is the meeting place of inner images and outer facts." Wolfgang Pauli
Zero Intrinsic Curvature 1 began with the blend of several vague yet distinct notions / flavours / desires: the desire to carve a window similar to those in Islamic architecture, and wanting to work with circles as well as with the diagonal shift in the pattern of light and shadow on the surface of the moon as it rotates around itself and revolves around the earth and the sun through time (synchronous rotation).
I knew that, in order to get close to the window notion, I needed an uninterrupted surface (which the square beam does not offer) and thus knew intuitively that I had to work with a rounded beam of wood. But the curved, continuous and uniform cylindrical/circular surface, unlike the cubic volume, offers no point of entry to begin to start measuring and applying lines. I thus 'detached' the curved surface of this cylinder, traced it on a flat and rectangular sheet of paper onto which I was then able to draw the pattern - which was then 're'-applied to the surface of the wood. But this produced a seam/cut all along the length of the beam and the pattern which exposed the cross-section of the lines, the nature of which puzzled me for a long time. Only much later did I understand that intuitively I had grasped -and made visible- an intrinsic property of the rounded beam:
The curved surface of the cylinder is intrinsically flat. The intrinsic curvature of the surface of the cylinder is zero - equivalent to that of a sheet of paper.
In 1827 Carl Friedrich Gauss had formulated the Theorema Egregium - the remarkable theorem- which states that the curvature of a surface can be determined entirely by measuring angles and distances on the surface itself, without any reference to the way in which the surface is embedded in the ambient space (the fact that the cylinder surface clearly is curved in the way it sits in space is not relevant for the intrinsic curvature of this surface - which is flat).
The Gaussian curvature is an intrinsic INVARIANT OF SURFACE - a property which does not change when the surface is being bent or crumpled (but does change when stretched). It is closely related to the Euler Characteristic and the Polyhedron Formula defined by Leonard Euler (1707–1783): V-E+F= - a topological invariant and intrinsic property of surface which does not change under deformation.
The pursuit of this Invariant has played a central role in subsequent works (The Detail and The Whole).
What is remarkable to me is that by intuition and through wood carving I was able to discover for myself and able to 'understand' a mathematical truth and property of a surface. It has re-affirmed my confidence in my tools. (How far can my intuition take me?)
The ratio between the diameter of a circle and its circumference is expressed in the mathematical symbol and value of π (pi).
π thus determines that the circle at the base of the rounded beam of a diameter of 9 cm must have a circumference of 28.27 cm - which in this 'sculpture' is consequently also the width of the flat surface onto which a circle pattern is applied. Consequently the circles at the top and bottom 'end' in this pattern with a diameter of 28.27 cm wrap around the cylinder perfectly without cutting or overlapping. The diameter of the circle at the base of the wooden beam provides its circumference; this circumference provides the surface, the width of which determines the size of the circle inscribed.
The vertical 'cut' or 'seam' which runs lengthways along the cylinder surface is not only an expression of the intrinsic flatness of the cylinder surface but also the limit expressed by pi - the fixed ratio between circle and diameter / between cylinder base and cylinder surface.
Squaring the Circle
π [pi] / Calendar / Phases Of The Moon - Angular Momentum
Each circle is one day; each column of circles one month; the twelve columns together represent one year.
Rolled across / wrapped around the surface of the beam this entire calendar is projected onto the surface without overlapping, with the edge of the diagonal pattern spiralling around the beam. Even when more 'months' are added, as long as the parallel diagonals cut across the entire pattern from the beginning to its end, then it will still wrap around the surface without overlapping. With increasing 'time' (with more 'years' added to this calendar) the spiral becomes increasingly dense and horizontal.
The loop is here shaped in a way that the idea of two parallel lines is evoked. This loop pierces the centres of the two circles and touches on their circumference at once. When rotating this loop anti-clockwise in the mind (around the two imaginary rotation points at both ends of the loop), then the two circles move away from each other. Once the circles reach the 'ends' of the loop, the loop is suddenly outside of the two circles/it no longer pierces their centre - without any need to fracture the actual circles.
Sprachwerkzeug: Standard For The Use Of The Word 'Symbol'
Invariant: The Original Shape / Self-Emerging / Swayambuh (Rohling)
Is it possible to think of something that is not embedded in a physical situation?
Spontaneous self-emerging thought, images/forms that 'pop' up in the mind, seemingly without history, examined in matter.
"Nature as self-repetition is the core of the symbolic: an entity or a process which is conceived as eternal because it is reenacted again and again in the guise of the symbol. Inexhaustibility, endless renewal, and the permanence of what they signify are not only attributes of all symbols but their true content."
Dialectics of Enlightenment
"Changing the fundamental categories of causation from that of local forces to a higher-level concept such as information." P.Davies
Geometry of Logic / The Original Shape: Information+Matter
Each carved thing of a spontaneously conceived idea of a form which does not have a (conscious) history of thought turns out -in matter- to have a logic of form which is implied: the new physical reality of the carved thing emerges as the 'explication' / unfolding of a history and of information which is implied in the original thought (originally conceived as new).
Decisions must be made about this form in the process of carving, like the shape and treatment of edges, which must be derived from the logic of form as it emerges in wood.
In the process of carving, idea and wooden thing are both cause and effect, exchange perpetually and become indistinguishable in wood. The original idea transubstantiated into the cubic volume of wood must materialise according to rules and forces that govern matter.
Information implied in The Original Shape (photo above) is carved as separate pieces of (its own) information: its edges, surfaces and vertices. The Euler Characteristic V-E+F= which is closely related to the the Polyhedron Formula defined by Leonard Euler (1707–1783), states that the number of vertices (corners), minus the number of edges, plus the number of (sur-)faces provides a topological invariant and characteristic of surface and form which does not change under deformation.
The Original Shape: The Detail and The Whole
to relinquish, re·lin·quished, re·lin·quish·ing, re·lin·quish·es:
1. To retire from; give up or abandon.
2. To put aside or desist from (something practiced, professed, or intended).
3. To let go; surrender.
4. To cease holding physically; release:
Relic, Latin reliquiae, meaning "remains" or "something left behind".
The Original Shape: The Detail and The Whole
SurFACE 1 (after having been cut)
A body in which gravity plays a dominant role cannot be rigid.
The Original Shape: The Detail and The Whole
The Idea of an Idea of an Idea (Downward Causation?)
Paper models of 'The Original Shape Surface 1' embody their own inbuilt 'logic' or 'grammar'. Moved by the force of their own shape, matter and gravity, and only initiated by light manual manipulation, they 'jump' and then relax into different shapes - pushing themselves towards equilibrium. The sum of all possible formations is the grammar of this form - the physical structural rules governing what is possible. There are n-number of ways in which this physical situation (shape) can be realised. The more ways in which something can happen the higher the entropy.
In 'The Detail and the Whole' that which hangs stands - and vice versa. As the spectator circumambulates the form, every time s/he faces one of the four sides of what used to be the beam of wood / when s/he reaches 90, 180, 270 and 360 degrees of this circle (the centre of the 'sculpture' being the centre of the circle), the shape appears as the upside-down reversal of the previous position, creating a wave-like motion. Seen from what used to be the corner of the beam both aspects merge.
The detail / the contour - where the three straight line segments meet with the curved line sections- is identical with the Aleph (saddle) shape, and at the same time replicates the interlocking of the two half-circles at the centre of this 'sculpture'.
Circumambulating the Square
Significance Of Number 9
Each 'sculpture' is based on a square. As the spectator circumambulates the vertical form s/he is offered a distinct number of views at each corner and surface of the wooden beam. When counting these distinct different aspects of each work starting from the corner of the beam a repeating sequence develops:
Numbers written in pencil count the distinct views from corners and faces continuously spiraling outwards around the square: Starting at one corner with position 1, followed by the frontal view of a surface (2), followed by a corner view (3), and so on. When adding the positions of each side of the beam, the cross-sums of each of these sides result in a number sequence (marked in red in the diagram above), which repeats in each new round:
1+2+3 = 6
3+4+5 = 3
5+6+7 = 9
7+8+9 = 6
9+10+11 = 3
11+12+13 = 9
6+3+9 itself = 9
This sequence - 6 3 9 - only repeats if at the point of reaching the end of the first round, counting does not close the circle to return to position one, but instead extends out in a spiral-like manner into a second round - and continues with position no 9 instead.
For the 639 sequence to repeat infinitely, counting must be continued in this spiral-like manner around the square and consequently away from or 'up' the length of the beam.
The cross-sums of each of the counted positions (marked in green) reveal that the number sequence 1, 2, 3, 4, 5, 6, 7, 8, 9 repeats infinitely as the spiral continues to grow around the square (yet this sequence itself heads inwards - 1 being further on the periphery and 9 nearing the centre of the square):
10= 1+0 = 1
11 = 1+1 = 2
12= 1+2= 3
13= 1+3 = 4
14= 1+4= 5
15= 1+5 = 6
16=1+6 = 7
Thus by circling the square 9 times one sequence is complete.
It seems of relevance to note that only at one point / corner - at the original point of entry- the connection to the next round can be made. Thus one of the four corners of the square is different from the other three.
Once these sequences had been identified on the outside of the square, they could then be extended towards the inside of the square (diagram below).
Inside the square:
Following the sequence of the counted positions around the outside of the square, consequently on the inside position 1 is followed by 2, position 2 followed by 3 and so on. When adding each of these pairs of the first outer and the first inner round together, the cross-sums (marked in pencil) are: 3, 5, 7, 9, 11, 13, 15 - and then not followed by 17, as the sequence would suggest, but 9 (!) instead.
The cross-sums for the two opposing sides of the diagram on
the left (1+8) = 9
the right (5+4) = 9
at the top 6+7 = 4
at the bottom 2+3 = 5
but added : 5+4 = 9
A trinity of 9's at the centre of the square.
And again the diagonal relationship as expressed in the values at opposing corners:
3+11 = 5
7+15 = 4
5+4 = 9
And again the cross-sums of the values along the sides is 9 :
11 + 13 + 15 = 9
3 + 5 + 7 = 9
3 + 9 + 15 = 9
7 + 9 + 11 = 9
Four sides of the square
4 x 9 = 36 = 9
9 comes after 8 and before 10.
10+8 = 9
8+9+10 = 9
Complete numerical symmetry inside the square.
The diagram below magnifies the inside of the square with focus on the repeating number sequence from 1 to 9 at each corner and each side of the square. It is magnified to contain 9 numbers of these sequences on the inside of the square.
At the centre there is now: 1, 2, 3, 4, 5, 7, but then not followed and concluded by 8 but 9 again instead.
But the cross-sum of this sequence 1+2+3+4+5+6+7+9 = 37 = 10 = 1.
Whereas if the last digit was 8 (and thus again not closing the circle but stepping outside to continue in a spiral-like movement, the cross-sum would again be 9: 1+2+3+4+5+6+7+8 = 36 = 9
A circle has got 360 degrees. Continuous division by half result in cross-sums the result of which is again 9:
360 degrees =9
180 degrees =9
90 degrees =9
45 degrees =9
22.5 degrees =9
11.25 degrees =9
But how does 9 relate to π ?
Early calculations / approximations for pi:
When taking the cross sums of three digits in this series the result is again the repeating 369 sequence:
Leibniz / Gregory Formula:
Leibniz' proof of this formula was found by the quadrature of the circle.
Again the cross-sums add up to 9:
1+3+5 = 9
9 links circle and square ... it seems to 'represent' growth and movement whilst ensuring complete symmetry in both, square and circle. Whereas approximations to pi -and from there patterns of 9 and 369- were found by division or the quadrature of the circle, for the square these patterns were found by circling the square in an outwards spiralling additive manner.
ORIGIN/ EMERGENCE OF THE WHOLE
To begin to consider any-thing is retrospection. My work proceeds exclusively by reaching backwards at each instant: focussed on its own emergence and origin, on the origin of each idea and spontaneous emergence of a form in my sub-/conscious mind, on the origin of forms in and from the cubic volume of wood itself, and on how these forms may be related to fundamental elements in Nature and to the ultimate Origin of Space and Time.
To contemplate the Origin / the Fundamental from all angles and (new) knowledge available to me in present-day culture is to reach for an absolute limit - of form, matter and thought. To consider that which emerges as new and as close as possible to both sides of non-existence: between emergence and annihilation, and thus to think of the emerging space or form in-between as absolutely fundamental and necessary. To ask what is absolutely fundamental, non-arbitrary and necessary in the realm of 'sculpture' or 'art'; to always examine what else can be taken away without destroying it / the form / 'the sculpture' or 'art'. Or: to find this limit / to define what is fundamental as if from the other side, by pushing the work through a thought process which the object as 'a sculpture' / as 'art' may no longer be able to handle. To define an absolute limit is a paradox. Each paradox encountered is a limit. Numerous paradoxes in wood have surfaced over the course of my work and appear absolutely essential, fundamental and perhaps even original-? The forms which emerge and manage to exist in this way are the most pared down and bare forms possible in wood. Stripped down to an absolute limit of what form and thought in matter can be, they are the 'purest' / simplest -and most fragile- tools of thought possible. To examine forms at the moment of their emergence limits my field of activity to the minutest realm, which upon close inspection opens up and reveals - if not 'histories'- then evidence not of a 'pre'- but a permanent / all-pervasive (?) existence of logic and of 'laws' in matter determining and shaping everything that is new according to their own 'rules', thus allowing me to contemplate the question and the concept of the new and of free will (information itself calling into question the concept of the 'new' and 'free will').
So originally my work in wood was triggered by music / the Cello Sonatas by Benjamin Britten (a fact that to this day is crucial to my system of thought). At first, I had tried to translate the nature of this music into drawings (sets of straight lines drawn with a ruler). Puzzling qualities observed in these drawings led to my attempt to try and translate these into three-dimensional matter/wood - which led me to discover my most fundamental tool: the idea of a circle 'folded' into and carved from wood.
The introduction of the idea of a circle/loop produced radical paradigm shifts in the status of the wooden matter, my relationship to it, as well as the fundamental properties of the circle. When 'folded' into wood, inside and outside of the circle are no longer opposed along the circle line, but envelop each other and merge. Matter and mind become equally indistinguishable as the wood 'takes on' qualities and properties of the mind, and becomes flexible, fluid and potentially infinitely stretchable, and the limit of where the mind ends and the wood itself begins becomes a matter of deep uncertainty. In the process of carving (carving thoughts / carving wood), causality -a clear sequence of events / the arrow of time- cannot be defined with certainty as cause and effect have no fixed and definitive 'location' and appear to be located on 'both' sides. The observation of the material thing in wood, external to me, is introspection at once: of thoughts, language and images inside my own mind. The quest for the Origin through carving is my quest for an understanding of the relationship between cause and effect, between language and matter, and of Time.
Over the course of my work, it became clear that my findings right at its beginning had touched on something that seemed far 'bigger' than the individual forms that I had made thus far. I realised that the work in the very first and few individual manifestations had emerged as a complete Whole, and my work ever since has been the attempt to try to analyse and understand this wholeness. It is this emergence of the Whole at the very beginning of my work which is responsible for its retrospective character. Every single detail and aspect in this process is of significance; absolutely nothing is accidental or unimportant. The work oscillates between being this emergence and being about its emergence. My work is: trying to understand what it is that I do when I work. It is best understood when told as a story which is its own history; a circular narrative where that which develops and is understood much later turns out to have already been present in its very first seemingly all-encompassing beginning.
It seems that because of the focus on Origin and on that which is fundamental the work makes horizontal and vertical connections which reach far beyond itself. Carving wood has not only led me to explore but has -strangely- enabled me to understand concepts and questions from a wide range of fields. It has come as a complete surprise to me to find essential concerns in my own work echoed in central themes in science and philosophy, in particular in the philosophical implications of quantum physics, topology and the information revolution. This surprise has over time given way to the understanding that my work has evolved as a kind of 'cosmological model', where the immediate findings about what is fundamental and/or original in the process of the emergence of 'a sculpture' can serve as scaffolding for thought in a much more fundamental and therefore wider and bigger context.
The physical act of carving acts as a motor, producing a literal and extra-ordinary insight by taking me deep into the conscious and subconscious mind-matter-language-body situation that I myself am, and which cannot be accessed and understood through logic and language alone. Carving has not only given me an insight into the extraordinary complexity of introspection / the relationship between myself and 'the world', but has also enabled me to see that the impossibility of a clear separation between subject (myself) and object (wood) in the process of introspection has a formal similarity with atomic/quantum physics, the discovery of which had forced scientists to abandon the ideal of causality and certainty, and with it everyday language as well as any visual models which were unable to describe, or account for, the quantum event. The loss in quantum physics of causality -of a clear subject/object separation- was the condition for a total loss of visualisability. The examination of this formal similarity -between introspection and processes in quantum physics-, and of the loss - the breakdown or limit of language / visual form of causality, as well as the limit and role of measurement itself- and the role symbols may play in visualising joint processes in mind and matter - are absolutely central to my work. I believe these to be some of the most central questions which should be addressed in art today (yet seem to be a frontier which today is largely ignored and neglected - which may be one fundamental explanation for the stillstand in contemporary art over the last few decades?).
To explore the philosophical implications of quantum physics and the information turn in the age-old and simple medium of wood / in the realm of art, to me is Mount Everest - the greatest challenge possible - which is from the outset: utopian.
The Copenhagen Interpretation of quantum mechanics, the writings of Niels Bohr and Werner Heisenberg, the work of Wolfgang Pauli und C.G.Jung (whose close friendship was crucial to each of their most fundamental discoveries), Merleau-Ponty's 'The Visible and The Invisible" and Ludwig Wittgenstein's 'On Certainty' are some of my most constant and seemingly infinite sources for my meditations in wood.
The exploration of each of the above-mentioned fields and works has in turn fundamentally shaken and changed my understanding of 'sculpture'. Scientific revolutions and paradigm shifts have radically transformed our understanding of the nature of 'reality' over the last three centuries, developments to which contemporary art generally seems to turn its back. Deeply rooted in the tradition and history of art, carving is my tool not only to explore various concepts but to observe and measure how these conceptual revolutions and paradigm shifts, in turn, affect 'sculpture' and 'art'.
Carving is an in itself abstract and invisible window-like 'machine' through which thoughts coming from this activity itself, as well as from other fields can be observed. Thoughts, words and ideas are run through this machine-like tokens, enabling me to measure, analyse and crystallise their essence, and to contemplate, test, measure and attempt to verify language and knowledge. The given and fixed paradigm of the same volume of wood used in each work of 9cm x 9cm x 240cm is the in-itself seemingly invisible frame which actively shapes the emerging worldview. Thus I root myself in this world.
The process of carving is a tool which is produced through the very use of itself. The wooden things (if they survive - and most don't) are left-overs / a by-product of this process. In this context, it is very interesting to me to see that, for example, Baruch Spinoza manufactured highly sophisticated optical lenses whilst developing his philosophy, or that Isaac Newton was deeply involved in alchemy whilst developing a scientific 'rationalist' world view at the same time. I have no doubt that their immediate involvement in and experiments with matter were absolutely instrumental for the development of their most progressive and essential philosophical and scientific ideas. A pity Kant did not carve! Every philosopher should have her stone. Woodcarving to me is alchemy.
Yet - Spinoza's lenses went on to be sold and used as optical instruments, and thus fell back from their status and role of being philosophical tools to being mere practical tools like hammers and nails. Yet in his writings, I believe one can sense and perhaps trace the lenses in their original function. Should it be possible to make lenses, or in my case 'sculptures', that are nothing but this in-between - held right at the point of production -of being a machine for thought- and before being sold on to do daily things in other realms - like being telescopes - or 'art'? This kind of 'sculpture' would require a response from the audience that is very different from the one that is currently given to contemporary art.
'Sculpture' was an entry point at the beginning of my work, at which I took the field of art for granted. But over time the idea of 'sculpture' and 'art' as such, as presupposed 'givens', and for which one may strive, through my work have completely disintegrated - and with it the possibility of my work automatically having a 'given' audience that 'belongs' to the field of art.
The work has a dynamic of its own, creating a reality of its own, a realm of its own - and this reality falls between the given categories of art and philosophy as if in a blind spot. This falling-in-between-the-categories of my work is a direct expression of the nature and status of these wooden 'things'. They withhold something from each of these realms.
What exactly is withheld?
Is perhaps that which is withheld the most fundamental part of what I do?
As an injury has brought my carving activity to a (preliminary?) halt, the 'machine for thought' that this work is to me continues to work, albeit deprived of its heart, and thus of the acuity that can only come from a complete immersion in matter.
"Further, from what has just been said—namely, that an idea must, in all respects, correspond to its correlate in the world of reality,—it is evident that, in order to reproduce in every respect the faithful image of nature, our mind must deduce all its ideas from the idea which represents the origin and source of the whole of nature, so that it may itself become the source of other ideas."
Tractatus Intellectus Emendatione - On The Improvement Of Understanding, Baruch Spinoza, 1661
The Circle is the Infinitive.
It exists before I draw it, or even think of it. It is the Mother Image - hovering above. From it forms emerge.
Where is the origin of the circle / where is the origin of my knowledge of the circle? Are these two or in fact one and the same 'thing' or 'event'? Does the knowledge of a circle require the knowledge of any other thing / does its existence depend on things? Is it in itself entirely free from sensual perception / what is the relationship between the idea of the circle and my senses? In how far is it founded in the material world? Is a circle the origin or the result of movement? Has the idea of the circle got a physical foundation / is its existence linked to gravity? Is there a borderline between the idea of a thing in the context of my thought and an object in the context of reality? Did I learn of the circle's existence? Did I discover it by experience? Perhaps by seeing the moon as a child? When as children for the first time we draw a human being - we draw a line which closes in on itself / a circle from which four lines emerge - but how do we find this circle? Do we 'simply' intuitively know? "An a priori intuitive idea the origin of which cannot be established through reason."- ? Can the thought of a circle itself be called an experience? Does the idea of the circle emerge individually / does it form part of our 'collective consciousness' / does the centre of the circle represent the 'Self' (C.G. Jung)? Both, Jung and the quantum physicist Wolfgang Pauli, sought to understand the link between the physical world and the psyche; Pauli likened the nucleus of an atom with the Self ("the totality of the psychic, as the conscious plus the unconscious").
Carving is to offer the circle from within and without to all of my senses, to show it to and to involve my whole being; to run it through my body, my thoughts, language and knowledge, my emotions. When I carve and pulverised wood fills the air that I breathe, gritting my teeth, I am inside the wood, I am inside the Circle.
the Dialectic of Enlightenment
At the beginning of my work in wood I was exclusively interested in and focussed on considering and carving forms that had appeared spontaneously as instant images in my mind (variations of a circle folded into the wood), yet over time the cubic volume of wood itself came into view: a given paradigm with its own information, and thus an active receptacle for my thought. Thus two strands of thoughts or two 'classes of objects' with different kinds of origin have developed, which have come to serve as a mandala-like framework which produces and structures my thoughts.
Squaring the Circle / Wholeness:
circle + square =
psyche + matter =
introspection + extrospection
innate + empirical
spontaneous + causal
Atalanta Fugiens, 1617, by Michael Maier
Atalanta Fugiens, 1617, by Michael Maier
"Trotzdem ist der Blick auf das Entlegene, der Hass gegen Banalitaet, die Suche nach dem Unabgegriffenen, vom allgemeinen Begriffsschema noch nicht Erfassten die letzte Chance fuer den Gedanken [...]
Die Zirkulationssphaere, deren Male die intellektuellen Aussenseiter tragen, eroeffnet dem Geist, den sie verschachert, die letzten Refugien in dem Augenblick, in dem sie es eigentlich schon gar nicht mehr gibt. Wer ein Unikum anbietet, das niemand mehr kaufen will, vertritt, selbst gegen seinen Willen, die Freiheit vom Tausch."
-T.W. Adorno, Minima Moralia