Correction on my previous blog and some interesting implications …

Andrey Mokhov spotted that to satisfy the actual inverse Pythagorean we need to have alpha=1/2 rather than 2. That’s right. Indeed, what happens is that if we have alpha = 1/2 we would have (1/a)^2=(1/a1)^2+(1/a2)^2. This is what the inverse Pythagorean requires. In that case, for instance if a1=a2=2, then a must be sqrt(2). So the ratio between the individual decay a1=a2 and the collective decay is sqrt(2). For our stack decay under alpha = 2, we would have for a1=a2=2, a=1/2, so the ratio between individual decay and collective decay is 4.

It’s actually quite interesting to look at these relations a bit deeper, and see how the “Pythagorean” (geometric) relationship evolves as we change alpha from something like alpha<=1/2 to alpha>=2.

If we take alpha going to 2 and above, we have the effect of much slower collective decay than 4x compared to the individual decay. Physically this corresponds to the situation when the delay of an inverter in the ring becomes strongly inversely proportional to voltage. Geometrically, this is like contracting the height of the triangle in which sides go further apart than 90 degrees – say the triangle is isosceles for simplicity, and say its angle is say 100 degrees.

The case of alpha = 1/2 corresponds to the case where delay is proportional to the square root of Voltage, and here the stack makes the decay rate to follow the inverse Pythagorean! So this is the case of a triangle with sides being at 90 degrees.

But if alpha goes below 1/2, we have the  effect of the collective decay being closer to individual decays, and geometrically the height of the triangle where sides close up to less than 90 degrees!

Incidentally, Andrey Mokhov suggested we may consider a different physical interpretation for inverse Pythagorean. Instead of looking at lengths a, b and h, one can consider volumes Va, Vb and Vh of 4-D cubes with such side lengths. Then these volumes would relate exactly as in our case of alpha=2, i.e. 1/sqrt (Vh)=1/sqrt(Va)+1/sqrt(Vb).


Charge decay in a stack of two digital circuits follows inverse Pythagorean Law!

My last blog about my talk at HDT 2019 on Stacking Asynchronous Circuits contained a link to my slides. I recommend you having a particular look at slide #21. It talks about an interesting fact that a series (stack) discharge rate follows the law of the inverse Pythagorean!

It looks like mother nature caters for a geometric law of the most economic common between two individual sides.

My Talk on Stacked Asynchronous Circuits at HDT 2019

I just attended a Second Workshop on Hardware Design Theory, held in Budapest, collocated with 33rd International Symposium on Distributed Computing

The HDT’19 workshop was organised by Moti Medina and Andrey Mokhov. It had a number of invited talks, and here is the programme:

I gave a talk on Stacked Asynchronous circuits.

Here is the abstract: In this talk we will look at digital circuits from the viewpoint of electrical circuit theory, i.e. as loads to power sources. Such circuits, especially when they are asynchronous can be seen as voltage controlled oscillators. Their switching behaviour, including their operating frequency is modulated by the supply voltage. Interestingly, in the reverse direction if they are driven by external event sources, their switching frequency determines their inherent impedance which itself makes them ideal potentiometers or voltage dividers. Such circuits can be stacked like non-linear resistors in series and parallel, and lend themselves to interesting theoretical and practical results, such as RC circuits with hyperbolic capacitor discharges and designs of dynamic frequency mirrors.

Here is the PDF of my slides:

questions about energy current (ExH slab)

Mac Rynkiewicz from Australia, who has been exploring the problem of the existence of aether, and recommended to me to read papers by Vladimir Demjanov, a Russian physicist from Novorrossisk, has posed the following interesting questions:

(1) How exactly does an E by H slab (or em radiation or photons or photaenos) manage to follow say copper, including bends etc.

(2) How exactly does a change in impedance (or a termination) give partial (or full) reflexion.

(3) How is a good conductor a good obstructor? (there is a contradiction deep inside that, especially re heat losses). 

Here is my answer sent to Mac today:

My view:

Imagine a messy fall of water – like rain falling in all directions (something like we see around Niagara Falls). That’s energy current ExH. The drops of rain can happily move where they can easily penetrate. In EM terms this means very low epsilon and mu. If you put some relatively porous material (i.e. epsilon still being lowish but not very low) around, the drops will still relatively easily penetrate but partly reflect back. Imagine you put a gutter made of hard water proof material (i.e. very high epsilon). Drops will start concentrating near and along the gutter because they can’t penetrate the gutter’s material. Most likely the density of rain around the gutter will be much higher than further away from the gutter.

Let’s now turn to Mac’s questions:

1) How exactly does an E by H slab (or em radiation or photons or photaenos) manage to follow say copper, including bends etc.

In the view of the above model, ExH slab will follow copper as its impenetrable gutter.

(2) How exactly does a change in impedance (or a termination) give partial (or full) reflexion.

In the view of the above model, change in impedance will give partial or full reflection. Low impedance means high epsilon and no penetration but following the gutter. Termination (high impedance) means no more good gutter to follow, hard to disperse near the terminator, reflect the energy stream back and inscrease pressure on the gutter (e.g. higher E). If the impedance is low (higher epsilon), part of the stream is reflected part is guttered.

(3) How is a good conductor a good obstructor? (there is a contradiction deep inside that, especially re heat losses). 

In the view of the above model, we should actually “reverse” (cf. “We reverse this …” as per Heaviside) the terminology when we move from electric current to energy current. From the energy current point of view we must call highly permitting material (high epsilon) low permitting material (like copper). Sponge is should have higher permittivity than copper. Copper is a gutter. Every time part of the rain that hits the gutter it loses energy. When it hits the sponge it still penetrates.

What we have in EM now is a mess. Everything is defined from the point of view of imaginary ‘electric current’ being seen as a promoter of energy propagation. The notion of materials with respect to their names like permittivity is made to serve the  imaginary world rather than material world.

So, you are quite right:  here is a contradiction deep inside that



Looking at various ‘paradoxes’ from the energy current standpoint

There are many puzzling questions around the relationship between Maxwell’s equations and elementary particles. Many of them form unresolved paradoxes. Perhaps, physicists can explain them in one or another way, but these explanations are often fairly complex and difficult to understand by a more practically minded person.

For example:

I saw the following paragraph in the paper “A derivation of Maxwell’s equations using the Heaviside notation” by Damian P. Hampshire in 376, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

“We suggest investigating an Einstein–Podolsky–Rosen experiment [17]. Typically, an entangled electron–positron pair is mixed and prepared as a superposition of states with equal and opposite magnetic moments (or spins). The charges are separated and the magnetic moment or spin of the electron is measured. The well-known instantaneous collapse of the wavefunction occurs, so that the positron ends up with the opposite magnetic moment (or spin) to the electron. The appearance of the moment of the positron is triggered by entirely quantum mechanical effects—no direct electromagnetic communication occurs between the electron and positron. Indeed, one can think about the two charges as a single entity. However, one can argue that we do not really know how the information that leads to the positron producing a magnetic moment of opposite sign is instantaneously received— beyond asserting it is part of the fabric of quantum mechanics, or part of the nature of a macroscopic wavefunction. We suggest that while the moment of the positron is being created (rather than excited), the production of the B-field associated with its magnetic moment may not be coupled to the production of any E-field at all. So, one could measure (∂E/∂t)r and (∂B/∂t)r in the wavefront of the positron, hoping to find B-fields with E-fields that are inconsistent with
Maxwell’s equations.”

I hypothesise that if we consider the (only!) postulate of Energy current ExH captured in finite space – pretty much as we do it in a section of Tx line, we should be able to explain the above entanglement of electron-positron as a superposition of states. My previous blogs about Wakefield 4 experiment exactly talk about the effects of this vein. We trap EM energy into a cap (Tx line section) and then short-circuit it. We have the effect of the “two-faced Janus” – where the same object switches between electron and positron states.



A two-faced Janus, this electron!

Let’s think a bit more about the Wakefield 4 experiment.

We have already observed how it can be that in some points of the same Tx like we experience different behaviour of essentially the same object.

The point near the shortened (ON state) switch has 0V potential, so basically the E component of the Poynting vector ExH is destroyed here, but we have a full H component all the time here.

In point B – in the middle of the Tx line, we have a full swing oscillation between +7V and -7V with equal intervals of time being in each of these two states. So here we don’t have H component active but only full E component swapping its positive state to negative and back.

In points A and C we have a mix of both E and H dominated intervals.

What sort of conclusions we may make out of this?

Well, one of such conclusions is that a Tx line, first charged and then short-circuited, is something that can appear in its different spatial sides, either like a swinging capacitor (only E field is visible), or inductor (only H field visible), or a bit of both – a time-division multiplexed cap-inductor.

My hypothesis is now, is electron a tiny Tx line that behaves like that two-faced Janus?

Why not?! The entire world, as I used to hypothesise in my “Energy current and computing” paper is discretised or granulated into Tx lines where we have substances (or lack of them)  with characteristic epsilon and mu, and these “cocktails of epsilon and mu of particular values” form our matter, and behave in EM fields accordingly, turning to us with their points A, B, C etc … depending where we touch them with our instruments!


More comments on Wakefield 4 experiment – energy hides as magnetic near the switch!

Incidentally, I forgot to say in my comments to drawings for Wakefield 4 that wherever and whenever (spatially and temporally) we have a zero potential in the waveforms, it doesn’t mean the energy current drops to zero there and then. No! Energy current there is in magnetic H rather than electric E form. For example, when the switch is being connected short, the voltage drops from 7V to 0V. That’s where and when energy current changes it’s presence from E to H. But it carries on without loss (we assume lossless Tx line).

So, the above fact does only enhance the statement that the content of the capacitor is dynamic. We have spatio-temporal changes of the entire ExH content.
Here is a hopefully useful analogy:
Energy current is just like a rotating ribbon which in some points in time passes some points in space vertically but then turns to the horizontal plane and vice versa. When this cap (Tx line) was charged uniformly to 7V the entire ribbon was moving vertically. When we jammed it by short-circuiting the switch, it had to turn to horizontal in order to pass through our jam. In other points (A,B,C)  it began to vacillate from horizontal to vertical and back.
But in order to vacillate it had to move and rotate like a conveyor ribbon inside the cap. With the switch being turned ON, we don’t push the ribbon forward we only jam it! It doesn’t need to be pushed. It has its own drive at the speed of light in the medium!
An interesting experiment would be to modulate the switch with some ON/OFF duty cycle!

Wakefield 4 experiment – causal picture in energy current

Ivor Catt reminded yesterday about the Wakefield 4 experiment, the description can be found here:

I have sketched the space-time cause-effect diagram for it:


It can be seen that in the point next to the switch after we short-circuit the cable to ground, although the normal state of potential is 0V, we have transfer of energy vi the reflection of steps with a coefficient (-1). Energy comes to this point from both sides independently. This means that as soon as say a falling step of -7V arrives from the side of point A, it gets reflected into a rising step +7V and hence the overall level of potential is unchanged.

The mid-point B oscillates with the overall amplitude of 14V from +7V to -7V.

If you look at the experimental traces from the above link

especially with the compressed time resolution, we have an interesting effect of sine-like wave oscillation at the level of steps (rather than conventional sine-wave as an envelope in condensed circuit-theoretic LC circuits) which takes place in the middle of the loop – point B, and this oscillation is totally ‘hidden’ from the view at say point of the switch, which ‘thinks’ that the cable is in static 0V state. This is a wonderful effect from the distributed-capacitor of the Tx line. The moving energy current is invisible to the outside world from the terminal point but it actually stays in the Tx line for a a fairly long time even with the unmitigated natural losses of the real-copper cable. The system has hidden memory to perform energy transfer!

Is energy current and speed of light vector or scalar?

Yesterday, Ivor Catt effectively questioned whether it matters at all whether energy current, and its velocity, is a vector or scalar:

” ExH: “ExH was the primitive … By definition, I think it is a scalar. It cannot be a vector because there is no directional information inherent in it. It is a single constant for the medium.” Malcolm Davidson 

Perhaps Malcolm made a mistake here. None of us is perfect.
I prefer to not talk about vectors and scalars. In our book, we never mentioned them. . I have never mentioned them in any of my articles, including these;
I wonder if anyone will venture that in that case, our writings (and seminars) are valueless. I suggest none of us three ever mentioned vectors and scalars in the seminars we gave for ten years. . We earned a lot of money for ten years not mentioning vectors or scalars.
Seminars called “Digital Electronic Design Seminars” and no mention of vectors or scalars! One of us is called Dr. David Walton! Earlier he worked in Dublin with Nobellist Walton (no relative).
I should finally mention that I do not object to anyone using the words. The above is just a smokescreen for my co-author Malcolm. The three of us spent decades designing digital electronics, and became quite sophisticated designing high speed digital systems.. None of the engineers I worked with in the UK and USA ever mentioned vectors or scalars.
When  I was Principal Lecturer in a college, I never mentioned vectors or scalars. Nobody else did either.
Ivor Catt

Today Malcolm Davidson wrote in response to Stephen Crothers:

Hi Stephen,
You stated;

ExH: “ExH was the primitive … By definition, I think it is a scalar. It cannot be a vector because there is no directional information inherent in it. It is a single constant for the medium.” Malcolm Davidson

If Malcolm means by ExH a cross product, the cross product only has meaning when operating on vectors. Both and H

are vectors. The cross-product of two vectors is a vector that is orthogonal to both vectors, such as and H, and, by the

right-hand rule, points in the direction of one’s thumb when one’s fingers curl from E to H. The magnitude of the cross 

product of two vectors A and B is AB times sine of the angle between A and B, the angle between A and B being that which 

is less than or equal to 90 degrees, A and B being the magnitudes of the vectors A and B respectively. Hence the magnitude 

of the Poynting vector is EH, since and H are orthogonal to one another.

I agree with everything you state above, within the conventional way we were taught, however these days I look at it from a slightly different perspective. (and of course) for all these years we discussed energy current using the phrase “The Poynting Vector!

Because we can know only the value at any one point, the direction being defined by the inherent Tx Line itself isn’t the idea of a vector superfluous? Of course I immediately think of a microwave line of site Tx & Rx system. Given that the ExH “vector” is moving at speed c and  has it’s own coordinate system built in I cannot think of where we would apply the basic rules of vector algebra.

Any thoughts? Also is it useful ? If I take two microwave beams . at say 45o angle how do the E & H fields add up to the scalar value that could be measured. anyway thanks for keeping me honest!



My response to that was:

Hi Malcolm,

I agree with you, likewise I agree with Stephen. The notions of scalar and vector are the the kind of mathematical quagmire which puts some pseudo-strictness on our way of thinking. Of course we can take the line of reasoning where we can say, well, a scalar value is a special case of vector value, where we don’t show the direction because the direction is given by an out-of-model assumption. In that way, your reasoning  about the Tx giving us such a direction removes from us the liability for (not) talking about the direction. All this is a mathematical maze in which we can be easily trapped. And here, I am on Ivor’s side – does it really matter if it is vector or scalar? The point is about the Occam – it just makes us see the everlasting ability of universe to transfer energy at c. One might ask, in what direction? the direction is defined by the way how the EM (or gravitational …) field is perturbed in some point in space and the materials (including vacuum) surrounding this point.

Kind regards


Towards computing on energy current

My further discourse with Ed and Ivor last night has resulted in the following message from Ed.



your yesterday message to Ivor makes me consider once again the mathematical equations Ivor introduces in his paper “The Heaviside Signal”. I concentrate on the equation c(dE/dx) = dE/dt in Appendix 1 (ignoring the ” – ” on the right side). This formula is constructed by interpreting the propagation of a voltage step in space (first diagram) and in time (second diagram) as a velocity v of motion in the direction of time, v = dx/dt, that is, a vector quantity. In the diagrams, this velocity v is symbolized by the letter c. With dx/dt = v the equation c(dE/dx) = dE/dt results in c = dx/dt = v. This asserted equality of c and v then insinuates that the voltage step would really “travel” in space and time with a vector velocity v = c in the direction of time. As I see things, the equation v = c is mistaken, since v is a vector quantity, which c is not. c is a scalar, as is proven by the Poynting vector E when put over p, resulting in c: E/p = c. Vector E over vector p = mv; v is a vector, c is a scalar quantity. Q.e.d. So c does not symbolize some velocity of some motion in the direction of time. What else does this factor mean, as it is  undoubtedly a quotient of a quantity of space, L, over a quantity of time, T, c [L/T] ? If you draw a diagram of Cartesian coordinates, space L in the vertical axis, time T in the horizontal, and you put the elements of space, L, over the elements of time, T, you get the constant quotient [L/T] to characterize the diagram as its parameter, or “grating constant”. Evidently this constant is not a vector, but a scalar. This shows that not every quotient dx/dt [L/T] represents a velocity of propagation in time, as the vector v does it. Rather it may be the case that such a quotient, the more if it is a constant (!), just represents the parameter of the space-time frame of reference wherein an observed phenomenon like the above-mentioned voltage step takes place. And this takes us into the middle of our finding that Poynting’s energy vector E = pxc differs from the classical scalar energy E = mv^2/2. This difference, as we can see now, is a consequence of not distinguishing between velocity v (vector, and variale), and the scalar constant c. Do you agree?


To which I have replied the following:


I am afraid I disagree with your conclusion that c coming out of the analysis of c(dE/dx) = dE/dt in Appendix 1  is a vector!

You raise a metaphysical story around it I am afraid.

This c is a constant coefficient. c is not a vector – its scalar.We have vectors around it dE/dt and dE/dx. These are vectors – one is force (or power in modern terms) and the other is momentum (as we agreed with you before). 

Full stop here!

Then, we should acknowledge the fact that there is a physical element behind this c – and this is energy current, which emanates in the universe from its Big Bang!

This is the carrier of interactions in Nature. This wasn’t known to Artistotle, Galileo, Newton, Maxwell … Ivor was and is the first to give this carrier the appropriate place.

If you had known a bit of the physics of communication (I recommend you to read Hans Schantz papers and book for that) you’d realise that its completely natural for communication (or interaction for that matter) to have a carrier – and this carrier for ExH does not need massed matter – it can perfectly well live in vaccuum.

That’s my take on this. You played an important role in this discourse. We have identified what momentum is in Catt’s theory and Heaviside Signal. Eureka!

What I have also discovered thanks to Ivor is the potential way for future computing – which is based NOT on envelope characteristics – such as exponentials and sines, but on discretised steps – this can potentially give way to improving the speed of computations by 2-3 orders of magnitude – we just need appropriate devices to support this speed and react to changes transmitted in energy current. I am already working on this!!!