While many readers find my direct challenge to Einstein’s theory of relativity insightful, others quickly reject (often without examination) ideas that contradict one of their most deeply–heald convictions. Fortunately, I believe that scientists are innately curious and will examine any idea that piques his or her interest. This series is designed to do exactly that: it’s designed to make you say “Hmmm?”

In Part 3 of the Unleash Your Curiosity series, we’re going to look at circles and spheres, shapes many of us learned about in primary school. We’ll begin with a definition:

*A circle (two–dimensional) or a sphere (three–dimensional) is defined as the collection of all points that are equidistant from a common center point*.

Mathematically, this definition is represented by the equations:

*x ^{2} + y^{2} = r^{2}*

and

*x ^{2} + y^{2} + z^{2} = r^{2}*

for a circle and sphere, respectively.

You are now asked to answer two questions. First, consider Figure 1 and determine which shape(s) satisfies the *textual definition* of a circle. Of course, the only answer that satisfies the textual definition is the one in the upper right corner, labeled “Circle”.

**Figure 1**

Second, reconsider Figure 1 and answer the question: Which shape(s) satisfy the *mathematical definition* of a circle. Before you answer that question, consider how well the mathematical definition of a circle aligns with the textual definition by examining the collection of points for the line in Figure 2.

**Figure 2**

Interestingly, the collection of points that represent a straight line also satisfy the mathematical equation for a circle. Figure 2 reveals an important and extremely subtle distinction between the mathematical and textual definitions of a circle. In the textual definition, the requirement for a constant radius is *explicit*, while in the mathematical definition, it is *implicit*. This subtle but important difference means that when you have not explicitly confirmed that the radius for each point is the same, every collection of points when represented as *<x, y, r> *or *<x, y, z, r>* tuples will satisfy the mathematical equation for a circle or sphere, regardless of the actual shape that the points represent.

Answering the question from above: a failure to explicitly check for a constant radius means that each shape in Figure 2 will satisfy the mathematical equation for a circle or sphere, as long as the radius *r* is included in the tuple. Simply said, without confirming a constant radius, one can incorrectly conclude that a collection of points is a circle or sphere when in actuality it is a straight line, a triangle, or squiggle, or any other shape for that matter.

Perhaps at this point, you’re crying foul, asserting: “Of course a circle has a common radius! Everyone knows that!” While I hope “everyone knows that”, this discussion has raised two important points.

- First, adherence to the mathematical equations alone cannot confirm that a shape is a circle or a sphere if you have not also explicitly confirmed that the radius for each point is the same.
- Second, when the constant distance of the radius is not examined and it is included in each point’s tuple, one can incorrectly conclude that a collection of points is a circle or sphere when in actuality it is not.

Please keep this discussion in mind as you (re)read section 3 of Einstein’s 1905 paper: *On the Electrodynamics of Moving Bodies*. Specifically, examine Einstein’s spherical wave proof found in that section, which is included below as Figure 3 for your convenience. As you work through the proof, draw the original and transformed shapes. Pick a sufficiently large velocity so that you can visually see the problem. While the finding is mathematically true at all non–zero velocities, you will most easily see it when using larger velocities.

**Figure 3**

When you work through the proof using real numbers and draw the corresponding shapes, you’ll find that Einstein’s transformed shape is not a spherical shape because each point does not share the same radius. Notice that because no one knew to check for a constant radius, this mistake has gone undiscovered for more than a century. What are the implications of Einstein incorrectly concluding that his “principles are compatible”?

Unsurprisingly, some will reject the assertion that anything is wrong with Einstein’s work. They will ignore or dismiss the constant radius requirement and insist that the second shape is a spherical shape due to the relativistic perspective of the observer in the moving system. This defense fails to recognize the importance and significance of Einstein’s mathematical proof, where relativity does not exist prior to the completion of the proof’s final sentence. Mathematically and logically, one cannot use a conclusion resulting from the proof as evidence that the proof is correct. Otherwise, we simply would conclude *relativity is right because it’s relativity*.

Once again, I hope that this series has piqued your curiosity and that you’ve read something that made you say “Hmmm?”

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**Steven B. Bryant** is a futurist, researcher, and author who investigates the innovative application and strategic implications of science and technology on society and business. He is the author of * DISRUPTIVE: Rewriting the rules of physics*, which is a thought–provoking book that shows where relativity fails and introduces Modern Mechanics, a unified model of motion that fundamentally changes how we view modern physics.

**is available at Amazon.com, BarnesAndNoble.com, and other booksellers!**

*DISRUPTIVE*Images courtesy of Pixabay.