Three dimensional science

Science has three dimensions and theories have four. There are consequently no theories in science and only corrupted science in theories. Theories are not science. They are 4-d stories about science. Science is about pairs of objects and pair diagrams and nothing else. Theories are about single objects and single object reference frames. Theories are not even about the same things as science.

The 3-d language of science

The language of science is the language of surveying. It is the language of triangulation, trigonometry and geometry. It is a language of angles and ratios – sine, cosine, and tangent – triangles, tetrahedrons and combinations of triangles and tetrahedrons. It is a language of numbers and relative speeds. In this language, clocks and measuring tapes both measure metres, but they measure metres along two different directions. To distinguish the directions I shall call metres measured by a measuring tape, ‘metres’ and metres measured by a clock, ‘mεtres’. The angle between the directions of metres and mεtres is 90-α where sin(α) is the relative speed of a pair of objects. When α is small the relative speed is small and when α is 90 degrees, the relative speed is 1 and the angle between metres and mεtres is zero.

When the speed of a car is measured by a speed camera, both tape and clock are used, the tape to draw a scale on the road in metres and the clock to measure in mεtres the interval between two photographs of the car crossing the scale. The camera measures (a) the number of metres, x, between the two images of the car displayed on the photographs and (b) the number of mεtres, y, between the two photographs. A suitably equipped car could also take photographs and measure the same relative speed. The relative speed measured by the camera is x/y=sin(α). Since x and y are measured in equivalent units, metres and mεtres, speed is a number between 0 and 1.

Pair diagrams in 3-d science

Once obtained, the number x/y=sin(α) can be used to draw fig 1.

Fig 1: A pair diagram consisting of similar triangles. The ratio x/y=sin(α) defines the diagram and the relative speed. They are the same whichever of the pair takes the photographs from which x and y are obtained.

(Fig 1 is to be contrasted with the one-object reference frames of theories. These are based on perpendicular time and distance axes fixed in one object of a pair and an angle β whose tangent, as opposed to sine, gives the speed of the second member of the pair relative to the first member. )

In fig 2 a second pair diagram is added in which α is close to 90 degrees. It shows that as α approaches 90 degrees, the relative speed of a pair approaches 1, like sin(α). Relative speed is therefore a number between 0 and 1.

Fig 2: A second triangle shows how S approaches 1. As α approaches 90 degrees, (12) and (13) close like a pair of scissors, becoming collinear and identical at S=1.

The person making the measurements of the relative speed of a car and a road knows whether the road is aligned north-south or east-west but that is not relevant to the measurements. It is important to avoid inserting information not obtained by the measurements. The measurements obtain x/y and nothing else. Separately x and y contain no useful information.

The Doppler shift

A 90 degree zig-zag line drawn inside the triangles of figs 1 and 2 diminishes by a factor D at each turn. D is the Doppler parameter for a relative speed of sin(a). D is the relative rate of aging. When two observers approach each other, each sees the other’s image to be aging D times faster than he is aging and when they retreat from each other, each sees the rate of aging of the other’s image to be D times slower than his rate.  Since figs 1, 2 also define the relative speed, S, figs 1-3 relate D to S and C.

Fig 3: The Doppler parameter is a number between 1 and infinity

When observers converge, each sees the other’s image to be blue-shifted by the factor D, and when they diverge, each sees the other’s image to be red-shifted by D. From figs 1-4 and trigonometry the following equations are obtained.

D=(1+S­)/C                          (1)

D-1=(1-S­)/C                        (2)

D2=(1+S)/(1-S)                  (3)

S=(D2-1)/(D2+1)                (4)

In these equations, S=sin(α), C=cos(α) and D, are all numbers.

In 3-d science, wavelength and frequency are reciprocals. If wavelength is λ metres, frequency is 1/λ reciprocal mεtres. The product of wavelength and frequency is 1, the maximum speed.

The relative speed of two vehicles travelling in opposite directions

If two vehicles travelling along the road in opposite directions have speeds S1=sin(α1) and S2=sin(α2) relative to the road, they have Doppler parameters D1 and D2 relative to the road. Doppler factors are multiplicative so their Doppler parameter relative to each other is given by (5).

D3=D1D2                       (5)

From (3) one can substitute each of the three D’s in (5) with an expression in the corresponding S. Then after some reorganisation one obtains equation (6) for the speed of the vehicles relative to each other.

S3= (S1+S2)/(1+S1S2)                 (6)


If the sides of the triangle 123 in fig 2 are all multiplied by a number m=1, they become m, mS and mC. Then the number, mS, can be given the new symbol p. Applying Pythagoras’ theorem to the triangle yields equation (7).

m2=p2+(mC)2                 (7)

m2 can be identified with total energy, p2 with kinetic energy and (mC)2 with potential energy. A pebble falling from a height into a black hole has initial values, p=0 and C=1 and all the energy is potential energy. As the pebble falls, potential energy is converted into kinetic energy. The angle α, governing the relative speed of pebble and fixed surroundings, is initially zero but it grows towards 90 degrees as S approaches 1 and C declines towards zero.

Though initially, α is zero, after falling y metres the pebble’s speed, S=sin(α) is S=gy. The number, x, of metres fallen is x=gy2/2. Differentiating this equation yields dx/dy=S=gy. The angle α grows with S, at first linearly, but then slowing as S approaches 1, α approaches 90 degrees and C approaches zero.

That completes what I have to say about 3-d science for the moment. In the remaining two sections of this article I shall first give some comparisons with theories to enable the reader to see how science and theories are related by ideas rather than observations or logic. Then I shall give a detailed study of Einstein’s 1905 theory of relativity, focussing particularly on a rather simple mistake that Einstein made because the correction of the mistake makes his theory easier to understand.

Part II: Ideas in 3-d science and 4-d theories

The unit of a second

In theories a second is a unit of the fourth dimension, time. In 3-d science it is 300 million mεtres. Like mεtres, seconds are measured by clocks.

The speed of light

In 4-d theories 300 million is the speed of light through 3-d space and 1-d time in metres of space per second of time. In 3-d science 300 million is the number of mεtres in a second of mεtres.

The speed of Time

In theories, time advances at the speed of light. In 3-d science, images are swept into the past at speed 1 as part of their continual updating.

Fig 4: A photograph taken between mirrors, 1.6 metres apart. An image 1.6n  metres from the camera had already receded 1.6n metres into the past at speed 1, when the camera shutter opened.

Fig 4 consists of a succession of images of the volume between the mirrors. Earlier images are deeper in the past and smaller, than later images.

The location of scientific data

In theories, data are assumed to be located in 4-d space and time. In 3-d science data are located in the 3rd dimension of pictures on our retinas. The 3rd dimension is clearly illustrated in fig 4.


In theories metres measure the space between two objects and seconds measure the time taken for light to travel between the objects. In 3-d science metres and seconds measure two sides of a triangle in a pair diagram encoded in pictures on retinas.

The angle between Space and Time

In theories the angle between space and time is always 90 degrees. In 3-d science the angle between metres and mεtres depends on the relative speed.

Wavelengths and frequencies

In theories wavelengths are measured in metres of space and frequencies in reciprocal seconds of time. In 3-d science wavelengths and frequencies are measured in metres and reciprocal mεtres.

Part III: Einstein’s 1905 theory

In 1905 Einstein set out to discover why theories were failing to describe the Doppler Effect in a way that agreed with observations. By a combination of intuition, logical inferences and trigonometry he found the equations of 3-d science without finding the pair diagrams from which they come. He consequently tried to fit the equations of 3-d science to the reference frames of 4-d theories. That, in a nutshell, is what Einstein’s 1905 theory of relativity is about. That simple fact is concealed though by the multiple consequences of a simple mistake.

There are two ways in which one might replace the ‘sin(α)’ of 3-d science by the tangent of 4-d reference frames. One could replace sin(α) by tan(β) or one could divide sin(α) by cos(α). What isn’t logical is to do both these things in the same theory. It has to be one or the other. Einstein did both and that reduced his theory to chaos. That needs clearing up first before a valid comparison can be made between science and theories.

Einstein’s mistake

In the equations he acquired from 3-d science, Einstein replaced S=sin(α) by (v/c) where v=tan(β) and c=tan(45), or 1. His equations are as follows.

D2=(1+v/c)/(1-v/c)                                    (8)

v/c=(D-D-1)/(D+D-1)                                  (9)

D3=D1D2                                (10)

(v/c)3= ((v/c)1+(v/c)2)/(1+(v/c)1(v/c)2)                  (11)

Einstein’s second set of equations can be obtained by renaming the sides of a triangle. In these equations, sin(α) is converted to tan(α), rather than tan(β), by multiplying sin(α) by γ=1/C. The inconsistency between the two sets of equations is the error.

Fig 5: Einstein’s second set of equations

In fig 5 one begins with the upper triangle whose sides are 1, S and C. Then one renames S as v/c  and C as 1/γ. Pythagoras’ theorem applied to the renamed triangle yields equation (a). In a second step, one multiplies all three sides by γmc2 while at the same time defining E and p by equations (b) and (c). Pythagoras’ theorem applied to this version of the triangle gives equation (d).

Now γ is a number, 1/C, greater than 1, so the lower triangle is γ times bigger than the upper triangle and equations (b-d) describe a different triangle than equations (8-11). That inconsistency is Einstein’s mistake. It need not detain us because it is easily corrected by removing γ from the theory and consistently replacing sin(α) by (v/c) where v=tan(β) and c=tan(45)=1. Fig 5 is then converted to fig 6.

Fig 6: Equations (b-d) made compatible with (8-11)

When Einstein’s equations are made consistent with each other his theory loses all its startling features – ‘time dilation’, ‘length contraction’, ‘proper’ and ‘contracted’ lengths, ‘proper’ and ‘dilated’ time, ‘time-like’ and ‘space-like vectors, the ‘twin paradox’ and the ‘Lorentz transformation’. Einstein’s ‘constant speed of light’ is based on the 45 degree zig-zag line in fig 3.

When all these distracting features have been removed, Einstein’s theory is reduced to the replacement of sin(α) by tan(β) in order to fit 4-d reference frames rather than pair diagrams. The only special features of his theory that remain are the 45 degree light-cone and the rule that relative speed can not exceed the speed of light, a device which restricts β to a range of 0 to 45 degrees.

The twin paradox

As a result of his mistake, Einstein thought that when one twin goes on a journey and returns to his brother, the travelling twin experiences the proper time, t, of the double journey because he is present at all three of the events which define the journey, and that he also experiences the contracted distance, d/γ, that goes with the proper time, while his brother experiences the dilated time, γt, and the proper distance, d. Both experience the same relative speed d/γt. Einstein consequently thought that when they were reunited, the travelling twin had aged by t and his brother by γt. The paradox is that it should be possible to treat either twin as the traveller.

If d/t=tan(α), d/γt=sin(α), so Einstein’s theory has the correct relative speed but, as one should expect, it falls apart when trying to describe that speed in terms of time and distance.

In 3-d science, there are no concepts of time or distance. When Einstein’s error is eliminated by suppressing γ and retaining 1/C, the twin paradox vanishes, along with time dilation and all the other products of the error.

When corrected, Einstein’s theory is the way it is because he failed to discover pair diagrams when he discovered their equations. If he had discovered pair diagrams he would surely have used them instead of 4-d reference frames and then the history of science, and probably of philosophy, would have been very different over the last 114 years.


The discovery of pair diagrams makes 3-d science completely independent of theories. It also makes Einstein’s theories of relativity redundant. When you know the form of pair diagrams you don’t have to use reference frames.

Thank you for joining me.

Stan Clough        8 Dec, 2019.


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