A geometric approach to scientific constants
For many the complexity of scientific equations reads like a foreign language. Specially designed mathematical symbols can convey complex ideas quickly to the trained eye, but this requires dedication and years of training. But what if there was a way to decode the complexity, in simple enough terms for anyone with an eye for art and knowledge of geometry to understand.
Dimensionless Science aims to do just that. At its foundation is the idea that all scientific constants and formula can be perfectly described using simple ratios, and geometric constants. Through a process of dimensional analysis, we can examine the nature of scientific constants and there relationship to each other. Through this we are able to establish exact values for important constants such as:
|c||Speed of light||3|
|e0||The electromagnetic constant||1/36π|
|U0||The magnetic constant||4π|
|Z0π||Characteristic Impedance of the Vacuum||12π|
|α||Fine Structure Constant||4π/√3|
|h||Planck Constant||(√3³/√4³) /π²|
The Von-Klitzing constant predicts the exact quantisation of electrical phenomena, at the quantum level of reality. Each state a either a whole integer or whole fraction of Hall conductance. The value is so precise that it can even be detected from a single hydrogen atom.
The Von-Klitzing can be defined by two completely different equations. The first is derived from the Plank Constant (h) and the electric Constant (e). The second is found from The Fine structure Constant, the Magnetic constant and the Speed of light. Both are satisfied with the exact values provided by Dimensionless Science. This demonstrates the integrity of the proposed values.
This animation shows how electrons are moved by magnetic fields – The Hall Effect.
After James Clark Maxwell had established the laws for electromagnetism, (1861-1862), Investigation began into the detailed nature of the interactions between electric currents and magnetism. These fields appeared set to be set at a 90° axis to each other, (the left-hand rule). In 1979, Edwin Hall found that an asymmetric charge distribution would build up across certain materials. When the electron was discovered 18 years later, is was shown that the Hall effect was due to the movement of electrons, which formed the basis of our modern theory of electricity.
As scientific measurements have become more precise,? the Hall Effect was identified at the level of a single electron. It was discovered that electrons exhibit quantised states of charge that are a whole number of whole fraction of the Von-Klitzing Constant. It has the striking property that is fraction integers relate to electron-electron interactions.
This animation shows the quantised states of the Quantum Hall Effect.
Exact ratio = Exact science
Dimensionless Science correlates with ALL scientific formula
Dimensionless Science has define exact values for over 40 scientific constants. These include further correlations for the Compton wavelength and Hartree energy that in a similar way to the von Klitzing constant can be defined through different equations the involve different constants. It also correlates with every equation that formulates the Fine Structure Constant.
Constructed from exact ratios
Given this overwhelming level of mathematical precision we are led to conclude the the Universe can be explained through simple ratios that find a perfectly described by laws of geometry.
Space is Geometric
The idea that geometry defines the universe is a logic conclusion,when we consider that scientific experimentation is conducted within 3D space. Mathematically the standard model relies of vectors, and tensors, that are similarly placed within a co-ordinate system of space.
A Scalable system
Dimensionless Science avoids the need for a fix coordinate system, by defining constants as ratios, which are found through simple compass constructions. As such the system is scalable from the macro to the micro.
Due to this nature Dimensionless Science can more easily explore the domain of the 4th+ dimensions, which also exhibit the geometric maths of ratio.
Science Without Maths
As ratio can be ascribed to geometry, dimensionless Science offers a unique window into the working of the universe without the need for complicated equations.
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Speed of light
|Constant||Symbol||Traditional Value||Units||Dimensionless Ratio||Dimensional Value|
|Speed of light||c||299792458||m/s||3||3 * 108|
|Magnetic Constant||μ0||1.25663706212 * 10-6||10−6||4π||1.259937061 * 107|
|Electric Constant||ε0||8.8541878128 * 10−12||C2/N m2||1 / 36π||8.41941283 * 10-12|
|characteristic impedance of the vacuum|
|Z0||376.730 313 461||Ω||12π *10||376.69911184…|
k = 1 / μ0 * ε0
|k||8.987 742 438 × 109||N·m2C-2||1 / 4π * (1 / 36π)||9 * 109|
|Mass of electron||9.109 3826(16) * 10−31||(((φ * √3 ) / (4/3)) / π2 ) / 32||9.038409121 * 10-31|
|Planck constant||h||6.62607015 * 10-34||J s||√(33/43) / π2||0.06581003923 * 10-34|
|Planck constant||h||4.135667696 * 10-15||eV s||(√(33/43) / π2) / (5 / π)||0.04134966716 * 10-13|
|Reduced Planck constant||ђ||1.05457168(18) *10−34||J s||(√3/4) / (4/3) / π3||0.01047399305 * 10-34|
|Reduced Planck constant||ђ||6.582 119 15(56) * 10-16||eV s||((√3/4) / (4/3) / π3 ) / (5/π)||0.006581003923 *10-13|
( ̄ђ * c / G)1/2
|MP||2.176 45(16)×10-8||Kg||((((√3/4) / (4/3) / π3) * 3 ) / (2/3) )1/2||0.2171012868 * 10-7|
ђ / mP * c
|Lp||1.616 24(12) * 10-35||m||(√3/4) / (4/3) / π3) / (mp*3)||0.0160815768 * 10-33|
LP / c
|Tp||5.391 21(40) * 10-44||s||(√3/4) / (4/3) / π3) / (mp*3) / 3||5.360525599 * 10-3|
|elementary charge||e||1.602176634 *10−19||C||5 / π||1.5915494 * 10-19|
(charge quantum ratio)
|e/h||2.417 989 40(21) * 1014||A J-1||(5 / π) / (√(33/43) / π2)||2.418399152 * 1013|
|magnetic flux quantum|
|Φ0||2.067 833 72(18) * 10-15||Wb||(√(33/43) / π2) / (2 * (5 / π) )||2.067483358 * 10-15|
|G0||7.748 091 733(26) * 10-5||S||(2 * (5 / π)2) / (√(33/43) / π2)||76.98003589 * 10-6|
|inverse of conductance quantum|
1/ (2e2/h) = 2 / RK
|G0-1||12 906.403 725(43)||Ω||((2*(5 / π)2) / (√(33/43) / π2))-1||0.01299038106 * 106|
|Kj||483 597.879(41) * 109||Hz V-1||(2 * (5 / π)) / (√(33/43) / π2)||48.36798305 * 10-6|
|von Klitzing constant|
|Rk||25 812.807 449(86)||Ω||(4π*3) / (2 *(4π/√3))||2.598076211 * 103|
|von Klitzing constant|
|Rk||25 812.807 449(86)||Ω||(√(33/43) / π2) / ((5 / π)2 )||0.02598076211 * 10-7|
|von Klitzing constant||Rk||25 812.807 449(86)||Ω||3√3 / 2||2.598076211 * 103|
|Fine structure Constant||α||7.297 352 568*10−3||none||4π / √3||7.255197457 * 10-3|
|Inverse Fine structure Constant||1/α||137.035999 11||none||√3 / 4π||0.1378322239 * 103|
|first radiation constant 2π * hc2||c1||3.74177138(64) * 10-16||W m2||2π *(√(33/43) / π2) * 32)||3.721470044 * 10-16|
|first radiation constant for spectral radiance|
2 * h * c2
|c1L||1.19104282(20) * 10-16||W m2 sr-1||(2x (√(33/43) / π2) * 32||1.184580706 * 10-16|
|electron mass||me||9.1093826(16) * 10-31||Kg||9 / π2 * 10||0.9118906528 * 10-30|
|Electron rest energy|
|8.18710506-14||J||(9 / π2 * 10) * 32||8.207015875 * 10-30|
|Electron rest energy|
|0.510998928-14||M eV||((9 / π2 * 10) * 32) / (5 / π)||5.156620156 * 10-30|
√(2e / me)
|593096.9818||m s-1||√((2 *(5 /π) / (9/π2 * 10))||0.5908179503 * 106|
|5.4857990945 * 10-4||u||√(9 / π2 * 10)||0.5908179503 * 10-3|
|electron mass energy equivalent in MeV||Me * c2||8.1871047(14) * 10-14||J||√(9 / π2 ) * 9||8.207015875 * 1014|
|electron charge to mass quotient||−e/me||−1.758820 12(15) * 1011||C kg−1||(−5 / 9π)||-1.745329252 * 1011|
|de Broglie wavelength|
for an electron
λ=h / mv
|λde||1.226425907||(√(33/43) / π2) /|
(√((2 *(5 /π) / (9/π2 * 10)) * (9/π2 * 10))
(√3/8) / (π/√π)
|1.22150628 * 10-6|
α2 *me * c / 2h
|R∞||10973731.568 525(73)||m−1||64√3||110.8512517 * 104|
|R∞ c||3.289841960360 * 1015||Hz||(64√3) * 3||332.5537551 * 1013|
|R∞hc||2.179 872 09(37) * 10-18||J||(64√3) * 3 * (√(33/43) / π2)||21.88537567 * 10-19|
|R∞hc in eV||13.605 6923||eV||(64√3) * 3 * (√(33/43) / π2) / (5 / π)||13.75098708|
α / 4π * R∞
|a0||0.529 177 2108 * 10-10||m||1 / 192||5.208333… * 10-7|
|Bohr magneton||μB||9.274 009 68 *10-24||m2 A||((5/π *((√3/4) / (4/3) / π3) / 2 * (9/π2)|
(5√3) / (96π2 )
|9.140283227 * 10-24|
in eV T−1
|5.788 381 804 * 10-5||eV T−1||(5√3) / (96π2 ) / (5/π)||5.743009327 * 10-2|
|13.996 2458 * 109||Hz T−1||5/36||0.1388888… * 1011|
|46.686 4507||m−1 T−1||5/108||0.0462962963 *103|
e2 / 4π * e * a0
|Eh||4.359 744 17 * 10-18||J||(5/π)2 / (4π * (5 / π) * 1/192)||4377.075133 * 10-21|
2 * R∞ * h *c
|Eh||4.359 744 17 * 10-18||J||2 * 64√3 * (√(33/43) / π2) * 3||4.377075133 * 10-19|
|quantum of circulation||h/2me||3.636 947 550 * 10-4||m2 s−1||(√(33/43) / π2) / 2*(9 / π2 )||0.0003608439182|
|quantum of circulation||h/me||7.273895101||m2 s−1||(√(33/43) / π2) / (9 / π2 )||0.07216878365 * 10-2|
h/me * c
|λC||2.426310238(16) * 10-12||(√(33/43) / π2) / ((9 / π2 ) * 3)||0.2405626122 * 10-10|
|λC||386.1592678(26) * 10-15||m||(√(33/43)/ π2)/((9 / π2 )*3)/2π||3.828672885 * 10-12|
λC / 2/π=
α * a 0=
α2 / 4π *R∞
|λC||NOTE: Equivilance of these equations is actually λC / 2/π and not|
|m||(√(33/43)/π2)/((9/π2 )*3) *2/π|
(4π/√3) * 1/192
(4π/√3) / (64√3)
|0.03778748675 * 10-13|
|classical electron radius|
α2 * a0
|re||2.817940325(28) * 10-15||m||(4π/√3)2 * (1 / 192)||0.2741556778 * 10-14|
|Thomson cross section (8π/3) * re2||σe||0.665245873(13) * 10-28||m2||(8π/3) * ((4π / √3)2 * 1/192)2||0.6296701333 * 10-27|
|Fermi coupling constant|
NOTE GF= phi2 +1?
|GF/(hc)3||1.16639 *10-5||GeV−2||(Φ2 +1) / ((√3/4)/(4/3)/π3)*3)||116619.7307 * 10-10|
What is Dimensionless Science?
What is Dimensional analysis?
What are fundamental constants?
Traditional Measurement Systems
Dimensionless Science (Process)
Powers and Dimensions
Dimensionless Science is the examination of mathematical formulas and scientific constants as ratio that is expressed through the lens of geometry. From the perspective of quantised reality, it assumes that this arises out of the geometric nature of space itself. Energy that fills this space does so in conformity to this structure. However, as we are embedded in space, we cannot directly measure this nature. 4D space need to be conceptualised, as it is beyond our realm of experience. Dimensionless Science provides a framework that does not require any direct measurement. Rather it extrapolates the ratios found in existing scientific formulas, that can then be ascribed to certain geometric functions. Dimensionless Science recolonises the relativistic nature of time and space, which exhibit time dilation and length contraction when travelling at high velocity. Therefore, rather than using units of space and time as its foundation, Dimensionless Science drives its base units from the energy in the vacuum, whereby the speed of light is set to 3 and the constant u0 to 4π. From this basis, exact values for Z0 and e0 are ascertained, which provides the foundation for the exact values for a vast number of Scientific Constants to be ascertained.
When we think about measuring a quantity of something in reality we generally choose the correct kind of units to represent the phenomena we are examining. Water is a fluid and is defined in terms of litres, or pints. Solids we weigh in KG, or ounces. Time we measure in seconds and minutes and space unit of length, such as the meter or centimetre. It does not make any sense to ask how fast is a loaf of bread. These different types of measurement are called dimensions. Dimensional analysis asserts that only quantities that exhibit the same dimension may be compared add, or subtracted. This is called dimensional homogeneity. However, phenomena that hold different qualities may be multiplied or divided.
To be clear, we can take the weight of 2 people and add or subtract them as the quantities exhibit the same dimension, and so the result is meaningful. However, the same cannot be said if we take the weight of one person and the height of another. In this case addition and subtraction of the quantities hold no meaning.
However, we can take height and weight and divide or multiply them them to produce a relationship between the two. This implies that mathematical functions that use multiplication and division are in fact dimensionless operations, that can accept units of different qualities. This is interesting as most scientific equations are also predicated on ratio, the relationships of quantities found through division or multiplication of various values.
For example, the equations for Newton’s Laws of motion have values for mass and distance, that are make sense, regards of which units are chosen to define weight and distance. When changing between different systems of measurement we simply employ a conversion factor, which is also a dimensionless quantity, as it mealy express one unit of measure to in units of another. Reality is not affected by out choice of measurement unit and so units are arbitrary, compared to the relationship of different phenomena, that is an express of simple ratio.
The concepts of dimensional analysis are actually already found strewn throughout science, which has adopted a variety of different measurement systems adapted for examining different types of phenomena. Such systems adjust the scale of measurement so that key values so they can be more easily expressed. Often certain base units are set to 1, which makes them dimensionless, at when squared that value does not change. Against these base values such as the amp, or the Kg, different systems of measurement have arisen such as SI or CSG, that we will describe in detail shortly.
In the 1920?s Arthur Eddington set out on an extensive investigation of the scientific constants in an attempt to create a overarching mathematical system. However, inaccurate predictions for the value of the fine structure constant created serious problem for his hypothesis. Despite this failure, he enquiry did bring about a recognition for the importance of dimensionless constants, which has since been integrated into critical aspect of the ?Standard Model?, which contains 19 fundamental dimensionless constants describing the masses of the particles and the strengths of the electroweak and strong force, etc.
British theoretical physicist, Michael Duff, who is a pioneer of supergravity, has even suggested that, ?varying of dimensional constants is operationally meaningless; variation of dimensionless fundamental ?constants? of nature is operationally well-defined and a legitimate subject of physical enquiry.?
Whilst such statements add weight to the importance of a dimensionless approach to science, to date all attempts to produce a convincing model have failed, and currently there are no legitimate contenders who are actively pursuing the idea. Yet if such a system were developed it would undoubtedly have a huge impact on science, and even our very perception of the universe.
In 2015, after 2 years of extensive practice the art of compass construction, and an investigation into the regular 3D polyhedra, Colin Power began to recognise that these fundamental forms could be used to express scientific constants and formula. From the seed of an idea, he began to investigate the results of science from the macro tot the micro, and began to discover the same pattern presents at each scale. Through his investigation of the Atom he proposed a new geometric model that defined the spherical harmonics of the electron in simple 3D polyhedra. After constructing a working model, he began to formulate his ideas, the he terms Atomic Geometry. When he turned his attention to the nature of 4D+ geometry, he found an expression of space that satisfied the criteria for a 4D Aether Theory. He realised that 4D Polyhedra include a component of time, and could also act like an incompressible fluid. From this he was able to conceptualise a 4D Aether theory, however, what was needed was a new kind of mathematics that could describe the 4th dimension, without being tied to the notions of space / time (the meter of the speed of light). Whilst searching for a solution to this problem he came across Dimensional Analysis, and realised that it represent the perfect tool to explore scientific constants from the perspective of ratio rather than quantity. Dimensionless Science is the consequence of this work. This mathematical system can be applied to reality without the need for any kind of man-made axis as the various ratios express themselves in terms of pure geometry. Spacial orientation is thus defined by the geometry itself, instead of the standard notion the geometry is defined by space. This idea suggest that the reason for the quantisation of reality is due to the nature of 3D polyhedra, that nest inside one another, form different shells. It also provided solutions for the ultraviolet catastrophe, and the problem of black body radiation in classical theory. As it is based in the mathematics of ratio, it becomes possible for the student to remember the exact values for large numbers of scientific constants. Additionally as complex scientific concepts are presented as tactile geometry that can be constructed, the application of Dimensional Science is suitable for almost all ages. The field of Dimensionless Science is still in inception, however, new discoveries are being made everyday that validate it findings. This in turn opens the door for a whole new approach to science, that also reveals some fascinating facts about out universe, and how it is constructed.
When we hear the term Scientific Constant, we might like to think of it as a part of the universe that is fixed in nature. However, constants like The speed of light (c), or the Planck constant (h), are not strictly of this nature as they have associated units. The speed of light is express in meters / seconds. As the relationship of time and space can change due to relativistic effect such as length contraction and time dilation, such units are subject to change. The notion of fundamental physical constants is often applied to certain universal, but dimensional constants, such as the Planck Constant (h), The speed of light (c), and the Gravitational Constant (G), which are all defined by specified units. In truth we should reserve the word fundamental physical constants for those that are not associated with any units. Now, for some the idea of having a scientific constant without units seems rather odd, after all, science is about measuring things ,right?
In many cases this might be true, but there are clear exceptions.
The most obvious place to find dimensionless constants is mathematics, specifically in the field of geometry. In electromagnetism a Gaussian object is an imaginary shape used to calculate the flux, (amount of energy that moves in and out of a closed system), inside a theoretical closed system. Such techniques employ simple geometric formula to form circles, spheres and other surfaces, which can then be used to help calculate charge. Every geometric form has its own formula from which it can be created, normally as a ratio of its side-length or radius. For example, we define circles and sphere, etc., using the mathematical ratio π. In mathematics the notion of π is dimensionless as it is the expression of a ratio between a circle circumference and its diameter. Another dimensionless geometric constant comes out of the concept of powers, such as 22, which transforms a line (length of 2) into a square divided into 4. These ratios and mathematical functions are more than a set of linear numbers. They are transformation of space from one type to another, i.e. from a straight line to a curve, or line to a square. In addition to those mentioned previously, other types of dimensionless values include percentages, and angles. In fact all pure numbers, such as π, e, φ, ½, √2, etc., and units of a set number, such as the dozen, can be considered dimensionless.
Outside of the realm of mathematics there are scientific constants that are dimensionless and universal in nature. Probably the most recognised is the fine structure constant (?). This is not expressed in units of any kind, and yet it is found throughout the universe, from the structure of the atom to the redshift of distant quasars. Due to its wide spread appearance, our identification of its exact value in Dimensionless Science sheds new light about its nature.
Aside for the fine structure constant, another important constant is the magnetic constant u0. This has traditionally been given the dimensionless value of 4?. This value is derived in the SI unit system through the definition of the Ampere, which considers two infinite parallel wires that carry the same charge. The Amp is defined as the force per unit meter length between the two at a set at distance of 1 meter.
Whilst the SI unit system has been widely adopted, the fields of magnetics is a notable exception, where the general preference towards the centimetre-gram-second (CGS) system of electromagnetic units (EMU). The key difference is the CGS sets the value of Coulomb’s constant (k) to one, defined by a unit value called electrostatic units (ESU). In this way the Coulomb’s constant becomes a dimensionless. Often when converting between Coulombs and ESU the numerical difference is absorbed by the factor c, the speed of light.
1 Coulomb (k) x Speed of Light = 1 Electrostatic Unit (ESU)
The Third type of Universal scientific constants appear as the experimental mass values of fundamental particles, such as electrons, protons, and neutrons. Presently, there is no concise theory to explain why these masses should be the set at their specific values. This standard model of science has defined values for a plethora of fundamental particles including Quarks, Lepton’s, and Bosons. These mass values have been ascertained in the main through experiments with particle colliders. However, it is worth pointing out that the majority of the mass of these particles come from their energetic interactions. For example Protons and Neutrons are made up of a group of three that appear in two orientations, up and down, in a 1 : 2 ratio.
[image of quarks]
Traditional Measurement Systems
In a drastic redesign of the SI system on 20 May 2019, the International Committee for Weights and Measures (CIPM) altered the nature u0 changing it from a dimensionless constant to a value that now needs to be determined experimentally. This move came about through a decision to try and focus measurement standards upon the newly fixed Planck constant.
In the CGS system, the magnetic and electric fields in a vacuum are equal in value and dimension, despite their different units. Whereas, the SI system, these fields are converted through the fixed constant μ0 with a value of 4π, as they have different values and dimensions. The consequence of this recent shift is a departure from such a simple convention, and away from the idea of dimensionless science.
Dimensionless Science (the process)
Whilst the current trend in science is to seek more and more accurate definitions of constants through experimental results, Dimensionless Science goes in the opposite direction. Instead it seeks to examine the relationship of various equations, and find a system of coherence in geometric terms. The ratios are developed without consideration of scale, which is analysed once the geometric concepts have been established. Therefore we can say that the process of investigation follow three simple steps.
1. Access the ratios found in scientific formula
2. Using the estimated ratios, derive a geometric expression from the formula
3. Examine the relationship in terms of scale. For this we have developed a special set of scaling principles that we call Universal Scaling.
Based on the above 3 principles, we can begin to see that specific methodologies must be employed, as often we consider objects in a higher dimension than 3D. As higher dimensional objects cannot be measured in a 3D reality, they are best explored in a dimensionless context first, and then appropriated to our 3D reality subsequently.
especially useful for considering higher dimensional objects
outside of our 3D world.
There are certain numbers that do not translate well onto a standard calculator, without fully understanding the limitations of linear sums. For example, if we take the number one and square it (1²), the answer on a calculator will be one. Cube it (1³) and the result is still one. In fact the number one can take to any power without changing its apparent value. It is dimensionless. However, I can draw a line (1), and make a square from it (1²), and then I can draw a cube (1³).
Therefore we can see that numbers can be deceiving, and so when we endeavourer to make sense out of a dimensionless system of scientific constants, the geometry of the concept must also be deeply considered.
Another clear example how numbers can be deceiving is found in the nature of ratios. In fact there are many different kinds of ratio, the simplest being a fraction made of two whole numbers. Geometrically a fraction can be seen as a line length of a certain number of units whose total length is divided. Fore example the fraction 3/2 is a line of unit length 3, divided in half.
When we add another term to the fraction we create a three term ratio. These are different in nature as they have certain are predictive qualities. They can grow or shrink in size at a defined rate. For example, the fraction 3 : 9 can also be written as 1 : 3. However, when we add another term to the start we define a different perspective, dependant on the value added. For example:
1 : 3 : 9 or √3 : 3 : 9
Here we see the two expression 3 : 9 with different values added to each. The first (left) begins with 1 and moves through the multiplication of 3. Therefore the next number after nine is 27. The ratio on the right has √3 in its first place, and moves through a series of square powers. Therefore the next number after nine will be 81. Only when we added the third term did the ratio reveal its true nature.
One of the reason many people believe that the human are unable to model the universe, is the fact that all scientific measurement has to have some kind of axis in order to define reality. From the simple number line to flattened Minkowski space-time, one thing all scientific investigation employs is some kind of mathematical space that is governed by co-ordinate sets of numbers. However, it is recognised that if the axis is moved or rotated then the outcome values will also change. This video from numberphile explains the problem in more detail.
Numbers and Free Will ? Numberphile
In effect the universe does not care about up or down, neither does it take measurements each time it performs and action. It is humans who construct the units framework of space through which our investigations are metered. So how can we get away from this nature.
Dimensionless Science employs the art of compass construction that requires no measurement in order to define specific geometric relationships. Just as everything is relative in the universe, so the same can be said of compass construction.
In this example we can move the two dots at the centre of each circle into any orientation and any distance, and the overall images remains in tact.
The only factor the defines the scale of any particular form is the initial opening of the drawing compass. This we make dimensionless by setting its value to ONE. In addition to not requiring any kind of axis or scale, the results are embedded into simple images that we believe make understanding and remembering scientific concepts a far easier task. Additionally, we can also move from the 2nd dimension and into the 3rd by building various polyhedra and nesting them with specific ratios. We have found that from this perspective people find it far easier to comprehend the 4th and higher dimensions. Based on our experience we have developed our 4D and 5D thinking programs that are aimed at improving our mental ability to visual geometrically. In this way, the universe becomes easier to comprehend.
One important point to note here is that Geometric thinking is very different to the idea of imaginary numbers. The development of the complex number plain arose out of a mathematical need to locate the position of √1 on the number line. This was accommodated through the addition of a 2nd axis placed at 90° to the first, through the zero point. Consequently all values on this new axis have their foundation in the concept of √1. However, as we have demonstrated, in the mathematics of geometry, the nature of powers is to create forms of a higher dimensional order. When we consider the nature of root values, we find that they are found dividing various forms. For example, a square with a side-length of ONE will have a diagonal that is √2. In 3D, a cube with the same side length is divided through opposite corners by √3.
Therefore geometric thinking is a very specific approach to mathematical space, that directly employs the nature of geometry in its constructs. In many respects we feel that this is appropriate medium for exploring the nature of the universe, as space-time can be express in terms of 4D constructs. Therefore by adjusting our mind to towards geometric thinking will enable us to grasp the foundations of Universal knowledge more easily and deeply.