The work of Max Planck enables the visualization of how mass and energy are connected.

Max Planck (shown at the left) made the fundamental scientific breakthrough that energy comes in quanta (I believe with a little help from Einstein).  His equation  E=hf  can be put in terms of wavelength  E=hc/λ.  Less famously Max Planck proposed the Planck Units (Planck length, Planck mass, etc) in 1899.  These Planck Units have physical dimensions (length, distance, etc) that are derived from the physical constants (G, h, c) and not from physical measurements.  Planck units are also known as the natural units. See this web site

Some play with fundamental equations:
1.  Max Planck’s equation E=hc/λ will be written as ELight=hc/λLight.  This is because the equation was created for photons, aka light.
2.  I will extend this result to apply to mass and say that  EMass=hc/λCompton.  This is just a hunch (yea, another one) because mass and energy are closely related and the symmetry points in this direction.

3.  We can test the above hunch by noting that the Compton wavelength is λCompton = h/(mc).  See Section 16 to see a derivation of the Compton wavelength.
4.  When we substitute λCompton = h(mc) into the equation EMass=hc/λCompton we get EMass=mc2.  This is enough of a prof for me to say that the equation EMass=hc/λCompton is reasonable.
5.   If mass is quantized like light is quantized, what is it that hops?  For that matter what is it that hops in a photon of light.  Here are two equations:  ELight=hc/λLight.  EMass=hc/λCompton
I will assume that the energy in both cases is mc
2
Then: m
Light = (h/c)(1/λLight) and mCompton = (h/c)(1/λCompton)
The conclusion is that light and mass differ only because of their wavelength, they both contain the fundamental quantity h/c that during a hop turns into a photon or a mass.
6.  The deBroglie wavelength λdeBroglie = h/(mv) which describes how mass moves, is very much like a light wavelength. It gives the electron, light like qualities which make the electron microscope possible. The electron mass itself (as demonstrated above) is associated with the Compton wavelength.

Further on in this section, a particle (like an electron) will be visualized as a λ-hopping dance consisting of two wavelengths, a Compton wavelength and a deBroglie wavelength. Both of these wavelengths have a source which I will call a Planck Instant. This Planck Instant has the magnitude  h/c and exists for a Planck length.

Postulate: All wavelengths associated with light and mass are created by “Planck Instances” that λ-hop.  If the wavelength of the hop is a light wavelength the result will be photon (light) energy.  If the wavelength of the hop is a Compton wavelength the result will be a mass (mass energy).

Light and Particles

1.  The Planck Instance is the source of light and matter. It is the thing that appears in space-time between the λ-hops.  If the Planck Instance hops continuously in one direction it becomes a light wave. If it hops back and forth it becomes a mass (see diagrams below).  The Planck Instance is the fundamental constituent of light and particles.  In Section 35 the Planck Instance postulated here will be associated with the elusive ghost particle the “neutrino”.

2.   A Graphical Definition for Light Energy, where light has a wavelength span from about 10-12     meters (gamma rays) to the diameter of the universe about 1026 meters.

The Planck Instant (three black dots) has only a momentary existence at every zero velocity node.
In between these nodes it does not exist.  Also note that the velocity of the various wavelengths of
light are less than “c” the max speed of space-time (see next Section 26 for more detail).

3. A Graphical Definition for Particle Mass: Particle mass also is constructed from Planck     Instances (see diagram below).  Like light, particle mass consists of a sequence of Planck     Instances, only now the sequence of λ-hops are not uniformly in one direction and have what can be considered as spin.

Mass looks as if a Planck Instance, instead of monotonically moving in one direction does a turn     around (like a tennis ball that has spin).  This also looks like a Planck Instant hitting a mirror.  I do     not postulate any mechanisms to account for this motion since there are no mirrors associated with an electron. It is just the nature of Planck Instances when they have particular Compton wavelengths. On the bounce back there is a momentum transfer. This mechanism acts to invert the Planck Instant and pin it to the velocity of space-time c, instead of it being pined to “0” velocity as in the case of light energy.  The velocity profile of a particle is shown as the heavy black outline in the  above diagram.        Even though the particle is defined by Planck instances that λ-hop, it can be thought of as a “hard” particle with a Compton wavelength that is λ-hopping (see above diagram) with a deBroglie    wavelength.

4.  The Derivation of the deBroglie Wavelength

We can calculate average particle velocity v as: v = c λComptondeBroglie
a.   v = c λComptondeBroglie     This is seen as the average velocity in the above diagram.
b.   λdeBroglie = c λCompton/v     Rearranging the equation in “a” above
c.   λCompton is by definition equal to h/mc (this relationship can also be derived, see Section 16).
d.  If we substitute h/mc (the Compton wavelength) into the equation in step “b” we get

λdeBroglie = c (h/mc)/v
e.  Simplifying the above equation we get the result that λdeBroglie = h/mv
We have the remarkable result that the deBroglie wavelength can be derived from the Compton
wavelength and the velocity of the particle.

5.  The Mechanics of Why Mass Increases with Velocity
(the result of Special Relativity):
Whenever velocity increases all the Planck Instance spacings become shorter. This is due to the     relativistic shortening of length as velocity increases. If the Compton wavelength of the particle     becomes shorter, the mass of the particle must increase by definition. The result is that particle     mass is not a constant, but is a function of the velocity of the particle. We usually do not notice this because particle velocities are usually much slower than the speed c.

6.  The Explanation of Mass Increase with Velocity:
Take another look at the diagram above (The Nature of Particle Mass) and notice that the “energy”     of the particle is inversely related to the deBroglie wavelength. And the mass of the particle is     inversely related to the Compton wavelength. The complete particle always consists of both the     deBroglie component and the Compton component. If the deBroglie wavelength component     changes due to velocity (relativistic shortening), the Compton component has to compensate to     keep the total mass energy balance. This is the fundamental reason why mass changes with     velocity.  And it can now be visualized!

7.  The Relation Between Mass and Velocity:
In the FQXi.org essay (the Elephant in the Room) I make the case that Einstein’s special relativity     equation m = m0 / (1-v2/c2)1/2 cannot be used to say that the mass of a particle approaches infinity     as v approaches c.  The equation is perfect in form, but it has theoretical limits that need to be considered. For the visualization above, note that at the maximum velocity possible for the particle occurs when the Compton wavelength [λCompton = h/(mc)] is as short as it can be. This shortest value is the Planck Length PL.  We can substitute PL for λCompton and solve for m (see Section 31).  The result is that m is equal to the Planck mass PM. This result indicates that all quantum mechanical particles have a maximum mass (as a function of velocity) that is the Planck mass (quite a bit less than infinity).  Check out the essay (the Elephant in the Room) for the details.

8.  Why Look at Quantum Mechanical Phenomena in Terms of Wavelengths?
Answer: To get another viewpoint on physical phenomena. The standard viewpoint is that of matter. The matter is electrons, protons (quarks), atoms ... etc. This matter is mostly summarized in the periodic table of the elements. The viewpoint presented in this “tale of two wavelengths” is that of energy, and it has two components a neutrino and the way it moves (hops). Both viewpoints represent the same territory. This is interesting! See below for an more on mass.

9.  The Reason Why Gravitational Mass Is Equal to Inertial Mass:
Answer: Both operate via gravitons see section21.

10. What Is Mass?

This is a great question......I am working on it!