# 03 – Topological transformations

When I put scalars together I get a lattice, like the image below (figure 4). Thus when I put deformed scalars together, I get the same lattice when I don’t draw the deformed part of every unit. However, the latter is the part of the volume of the unit that alters in a continuous way, synchronous with the other units around.

figure 4
Every unit is a reduced scalar thus the deformed part of the unit will rearrange its shape in a way that nears a symmetrical shape (the sphere). However, increasing the reduced scalar is impossible because of all the other units around. The post “Composed basic field” in the topic “Beyond spacetime and quantum fields” describes the details of the combined scalar and vector field more extensive.

When I change the imaginary cylinders into membranes I can draw the schematic quantum unit in a more “realistic” way. So figure 5 shows a schematic representation of a unit that is a reduced scalar.

figure 5
The schematic shows the invariant volume but it is clear that the unit cannot decrease the strokes of the membranes when the other units around don’t allow this. In other words: the direction of the future alterations of the unit is restricted by the alterations of all the other units around.

Figure 5 shows the registers (and sign bit) of every membrane and it is clear that the internal transfer of volume from 1 membrane to another membrane are equivalent to topological transformations. So I can add digital comparators to the registers but that is not enough to control the alterations of the binary quantum units.

figure 6
A binary quantum computer cannot have an enormous amount of quantum units. Therefore, it is necessary to simulate the lack of units in a controlled way. The schematic above (6) shows a solution when the alterations are controlled from the outside of the quantum computer. For example with the help of a control unit (computer).

However, there is something missing. The control unit must detect the magnitudes of all the registers of quantum units thus the schematic in figure 5 is too simple.

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# 02 – Invariant

The main law in (quantum) physics is the law of conservation of energy. It is like a cylinder with a piston inside. I can move the piston to and fro but the sum of the lengths at both sides of the cylinder is a constant (invariant). So when I move the piston to the left – position 1, see the image below – I get 2 values: 15 and 17 (sum 32). However, I can express every position of the piston with the help of a sign bit – a negative or a positive position (see the scale division). In this way I can describe every position of the piston at both sides of the cylinder with only one number and a sign bit.

figure 1
Quantum fields are not composed of cylinders with a piston inside. The typical feature of the basic quantum fields – fields that are existent everywhere in the universe – is the spatial and the homogeneous structure of the volume. So when I want to design a quantum computer I have to simulate the structure of the basic quantum fields with the help of scalar and vector properties.

The unit that composes a spatial structure that have scalar and vector properties is a deformed scalar with an invariant volume. Figure 2 shows the cross section of the unit. The volume of the scalar is grey and the deformed (vectorized) part is blue. To keep it simple the deformed part of the unit is drawn like a cube but keep in mind that the symmetric volume shows any vectors (blue volume).

figure 2
The only mathematical true scalar is a sphere. So when the cause behind the deformation of the unit is removed, the shape of the unit will become a sphere again. Thus the scalar expands (increases) at the positions of the red arrows (figure 2). However, the deformation of the unit is caused by the existence of all the other units around so theorizing about the alterations of the unit is senseless without considering the environment of the unit. A surrounding that is filled with units which have exactly the same properties as every other unit in the universe. That is why the vectorised part of the unit and the size of the scalar (dotted circles) can be altered by other units.

figure 3
The image above (3) shows one unit (M) in the centre of its surroundings. Now it is clear that every alteration of the shape of one unit will affect the other units around in a direct way. So all the alterations of the shape of the distinct volumes of the units are synchronized because the volumes of the units are identical and invariant (all the units tessellate space).

Suppose I substitute the red arrows in figure 3 with the cylinder in figure 1. Every alteration of a unit is an internal transfer of volume from one face (or faces) to another face (or faces). Just like the altering volumes inside the cylinder when the piston is moved to and fro.

The result is an alteration of all the binary numbers that represent the joined faces between the units at the same moment. Conclusion: the principle simulates the quantum computer in a perfect way.

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# 01 – On quantum computing

Quantum computing is a new way to calculate quantum algorithms. The idea behind quantum computing is to use the quantum superposition or quantum entanglement to get more possibilities in relation to a traditional digital computer. Quantum computing will speed up computing too because of the use of qubits (quantum states of electrons).

Quantum computing is a promising development in computer science. However, is it possible to get equivalent results when we re-examine the way we use the traditional Arithmetic Logic Unit? Because modern IC’s are not expensive or complicated in relation to quantum qubits.

The next posts are about the theoretical possibility to design a quantum computer with the help of digital components (IC’s).

On Quantum Computing” is the original comprehensive website.

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# 01 – The end of math

The title – the end of mathematics – is exaggerated of course. It doesn’t mean that mathematics will vanish as a branch of science. The purpose of the title is to express a change in understanding the origin of mathematics. A bit like a paradigm shift.

It isn’t a new subject because the ancient Greek philosophers had already the opinion that mathematics is – in some way – the foundation of the universe (Pythagoreans). In 2007 Max Tegmark brought this idea back to live with his publication “The Mathematical Universe” (MUH).

The project “Beyond spacetime and quantum fields” describes a hypothesis about the mathematical structure of the universe. The hypothesis showed to be correct and this questions the theoretical structure of mathematics in relation to the “mathematical universe”. In other words, the posts are about the foundations of mathematics.

The end of mathematics” is the original comprehensive website.

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# 04 – Composed basic field

A solitaire element that has an internal – spherical shape forming – mechanism is a true scalar. That means that the scalar is determined by only 1 value – its radius – irrespectively its position in space and time.

There are no solitaire elements in our universe thus every element is a deformed scalar. That is why the invariant volume of every element is divided in:

• a deformed part of the volume (vectorized volume),
• a not deformed part of the volume (decreased scalar).

The result of the existence of the deformed elements is the creation of a composed vector and scalar field. This composed basic field is everywhere in the universe and tessellates space.figure 4
The image above shows the scalar part of the elements when all the decreased scalars have the same radius (so it is a flat scalar field). The different colours are used for visibility.

When every element has an identical decreased scalar and an identical vectorized volume the hypothetical symmetrical element shows to be a rhombic dodecahedron. So all the rhombic dodecahedrons together tessellate space. The image below – figure 5 – shows one symmetrical element with the decreased scalar inside.

figure 5
The internal spherical shape forming mechanism of the elements is never in equilibrium because every element is a deformed true scalar. The consequence is the continuous alteration of the shape of the elements because the volume of every element is identical and invariant.

So how do the elements deform?

The deformation of an invariant volume means that some volume of the element becomes surface area and vice versa. The mechanism behind the deformation of the invariant volume – actually a topological homeomorphism – is showed in figure 6 (cross section of the joint face between 2 rectangular bodies).

figure 6
The only way to keep the volume of both bodies constant is to move 2 or 3 blue arrows or green arrows. Thus the sum of the transfer of volume is null.

An element is a deformed true scalar. That is why the alterations of an element are similar. The radius of the inscribed sphere – the undistorted decreased scalar – can only increase or decrease at the points of contact between the scalars of the adjacent elements.figure 7
The image above shows the schematic cross section of 2 adjacent elements. The internal spherical shape-forming mechanism is visualised by concentric circles.

It is easy to conclude that alterations by the decreased scalar – actually an internal transfer of volume – can only happen at the contact point between both elements.

figure 8
The image above (8) shows the combined basic field. The white spheres are the decreased scalars (scalar field) and the arrows show the alterations at a certain moment between the vectorized parts of the volumes of the elements (only the resulting vectors are drawn).

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# 03 – The all-inclusive set

Figure 3 – set C – represents the existence of a non-hierarchical underlying reality. There exists no fundamental difference between the objects – like set A and set B – and the surroundings of these objects. In other words, when I imagine that I move my arm through space it is the underlying reality that forms my moving arm from point to point.

figure 3
There is another essential aspect: the reality of figure 3 envelopes an all-inclusive volume. In addition, we have to conclude that this volume can not be totally homogeneous. Because in daily reality we observe different properties between phenomena. So we have to conclude that the all-inclusive volume must have an internal structure. This implies that the underlying reality – that forms all the phenomena in the universe – is an all-inclusive mathematical set, consisting of elements.

Conclusion: there exists no volume in the universe that does not consist of one or more elements.

An element has two kinds of properties: invariant properties and variable properties. When every element only has variable properties, the law of conservation of energy and physical constants like Planck’s constant cannot exist.

Conclusion: each element must have at least one invariant property.

At the level of the micro cosmos, we cannot detect differences between the proportions of the same phenomena under equal physical conditions. For this reason, it is not logical to conclude that the volume of each element ought to be a variable property. Thus, the quantity of the volume of each element has to be identical.

An identical quantity of the volume of each element does not indicate that each element has an identical size of the surface area. Because daily reality shows that everything in the universe is altering continuously. This observation rules out the possibility that every element is static. Therefore, the surface area of each element is variable and has the ability to deform.

Summarised:

• Each element has an invariant volume.
• The quantity of volume of every element is identical.
• The surface area of an element is deformable.

Unfortunately, the named properties of each element do not explain why all the phenomena in the universe alter continuously. Thus every element must have an unknown property that is responsible for all the observed alterations.

All the phenomena do have volume and all the phenomena alter with respect to each other. These properties – mentioned above – are observed everywhere in the universe. However, there is still another widely observed phenomenal property in the universe: the domination of the spherical shape everywhere in the cosmos.

In view of this, we can draw the following conclusions:

• The universe is an all-inclusive set of elements.
• Every element has identical properties.
• The volume and surface area of each element within the all-inclusive set is the result of an internal – spherical shape forming – mechanism.

Despite of the logic behind the concept of figure D there remains a dubiety, which is hard to ignore. The description of the underlying reality as a mathematical set, which is composed of elements with identical properties, seems far too easy in relation to the multiplicity of all the observed phenomena in the universe. Therefore, in the next chapters some explanation about the functioning of this mathematical set.

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# 02 – Empiricism

Modern physics is empiric science – empiricism – and it is beyond any doubt that it is extremely successful. Physicists have discovered the constituents of matter and the relations between the distinct phenomena. The results are described in the Standard model of particle physics.

In spite of the research of physicists all over the world during nearly a century, there is still no accepted Theory of everything in physics. That’s a bit worrying so it raises a question about the suitability of the empiric method to search for the Theory of everything.

What are physicists doing when they observe the properties and mutual interactions of the phenomena in the universe?

Phenomenological reality is like a mathematical set, so we can represent phenomena with the help of a Venn-diagram (figure 1). Set A and set B have constituents (properties) in common and this is represented by the intersection (AB). So when we want to relate phenomena to each other with the help of experiments, the measurement is the intersection between set A and set B.

figure 1
Unfortunately, the search for the theory of everything is not a search for the relations between phenomena at the lowest scale (elementairy particles and force fields). It is the search for the “underlying” reality. A reality that forms all the distinct phenomena in the universe. Thus it is the search for the not composed properties that create the composed set A and composed set B. In other words, set A and set B are part of an all-inclusive set that envelopes everything in the universe: set C.

figure 2

The Venn-diagram in figure 2 looks nice but it is a wrong representation of reality. It merges the phenomenological view – set A and set B – with an all-inclusive view (set C). That is why we cannot represent the composed phenomena A and B in figure b with the help of 2 sets because set A and set B cannot have an intersection. It cannot differ from the properties of the structure of set C because phenomenon A and B emerge from set C (see figure 3). The absence of the intersection AB in relation to the present of set C proofs the uselessness of the results of measurements when physicists try to find the Theory of everything.

This simple model shows that empiricism is not the right scientific method to explore the Theory of everything. Because it is impossible to deduce the properties of set C with the help of the results of experiments that are related to the properties of physical phenomena like particles and force fields (composed phenomena).

figure 3
The solution to by-pass these problems is simple. Research in the field of the Theory of everything must not focus upon the observed properties of the phenomena but on the general properties that are observable everywhere in the universe (the properties of set C). Moreover, the description must be done with the help of correlated mathematics. The next post is the start of an explanation how this can be done.

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