Unveiling Pythagoras: How Circle-Based Area Transformations Reveal the Truth

I recently had a discussion about the Pythagorean theorem and how to understand it better. I realized that the role of the circle as a fundamental element is often underrated. You can hardly find any proof where the right-angled triangle is placed within a circle to support the reasoning. Instead, the triangles are usually presented in a plane without reference. However, when you position the triangle inside a circle, it becomes much easier to visualize how the relationship holds as the right-angle vertex moves along the circle.

The key idea to understand is that in the following image, for a unit square, the green and blue areas are always equal to cos²(2α), while the orange and red areas correspond to sin²(2α). It is evident that the square, with an area of 1, is the sum of its left and right sections.

By opening the GeoGebra resource [Pythagoras proof in a circle (opens a new tab)], you can slide the right-angle vertex (red point) anywhere along the upper half of the circle. 

The meaning of the Hubble constant: a Universe in Rotation

In the 1910-1920s, various astronomers discovered that the spectral lines frequencies in the emission spectra of galaxies were prevalently redshifted. The shift in frequency was interpreted as a Doppler shift, meaning that galaxies moving away are redshifted, and galaxies moving towards us are blueshifted. The higher the relative velocity between us and the galaxy, the larger the Doppler shift.

In 1922, Carl Wilhelm Wirtz (1876-1939) noticed that dimmer galaxies generally had a larger redshift. Which suggested that the farther away the galaxy, the higher its recessional velocity.

Edwin Hubble (1889-1953) tried to infer the distance d of the galaxies by correlating it with the brightness of pulsating stars in each of those galaxies. A rough fit in a graph of the recessional velocity v against the inferred distance showed a proportionality v =K*d, where K seemed to be a constant proportionality factor [Hubble, 1929]. In homage to Hubble, we later called named this as Hubble’s law and wrote K as the Hubble constant H0. On all later measurements of distant galaxies, Hubble’s law has been successfully confirmed.

Now, a physicist should ask himself what this law means for the dynamics of the Universe. During the 1920s, interpretational work on the early theory of general relativity saw Hubble’s law as a confirmation for the expansion of the Universe, where the rate of expansion was higher the farther the galaxies. This “Expansion of the Universe” interpretation has fueled almost all subsequent work in Cosmology, giving unexpected challenges about the cosmological constant, dark matter and dark energy. I won’t discuss this, as I am not a Cosmology specialist at all.

I just wanted to point out an overseen fact that Hubble’s law could also be interpreted as a confirmation of the rotation of the Universe, as a whole, around an axis, at a constant angular velocity, without further ado. For this, we should understand that in such a Universe, the perceived emitted/received electro-magnetic radiation between ALL objects that orbit at constant angular velocity is ALWAYS redshifted. The Doppler shift in such a system is proportional to the radial component vr of the relative velocity, see Figure. This radial component is directly proportional to the distance between both objects, wherever the object is situated on its orbit. The perceived relative velocity for each object is therefore vr = K*d, following a Hubble law. Of course, there should be some corrections that would need to be included in this rule of thumb law, as we can’t expect each body in the Universe to rotate at the same angular velocity, but the general behavior would follow a Hubble law.

A Universe in rotation, as a whole, is in my opinion intellectually more satisfying than a Universe in expansion, as we see orbiting objects all over the Universe. To my knowledge, this interpretation of a rotating Universe has never got the deserved attention, or has it?

Figure: The perceived relative velocity between 2 objects that orbit about a common center at constant angular velocity is always a recessional velocity. It is the difference vr1-vr2 between the radial velocities. The Doppler shift is therefore always a redshift. This perceived velocity v is proportional to the distance between both objects, according to a law v = K*d, where d is the distance between both objects.