This page shows the need for, meaning of, and construction of Whole PI. See Dimensions of Space for its application.
A New Understanding of the Pattern Behind the World Starts Here.
Figure 1. Most people have never objectively thought about the Unit Definition behind the Unit Circle. It is currently based on either of TWO defining units. The Unit Circle based on a Unit Radius has a circumference of 2·π. The Unit Circle based on a Unit Diameter has a circumference of π. Thus, the Unit Circle is either 2·π or π, depending on the Unit Definition. To differentiate these, the defining unit must always accompany the symbol for pi. Otherwise, if you just see the symbol pi, you can't tell whether it's half a circle based on a Unit Radius or a whole Unit Circle based on a Unit Diameter.
According to the Merriam Webster Dictionary, pi is the ratio of the circumference of a circle to its diameter. Notice that there is no mention of radius, even though the equations we learned as kids are based on radius: C = 2·π·r , A = π·r2 , V = 4/3·π·r3 .
See more on the Dimensions of Space page.
The definition of pi from the commonly used equations based on radius would be: "pi is the ratio of half of a circle to its radius".
Figure 2. Since there there are two Unit Circles based on different units (radius or diameter), the circumference for a Unit Circle is either 2·π = 6.28318530... or π = 3.14159265..., depending on the Unit Definition, r = 1 or d = 1 Since these two circles have the exact same circumference (one was copied and pasted to be the other), based on two different units, the circumference = 2·π and also circumference = π, so 2·π = π .
Figure 3. If a whole Unit Circle is 2·π (based on a unit radius, r1), then one π is half of a circle. And the value for π, 3.14159265..., is HALF a circle. If a whole Unit Circle is π (based on a unit diameter, d1), then one π is a whole circle. And the value for π, 3.14159265..., is the WHOLE circle. So again, The definition of π from the commonly used equations based on radius would be: "pi is the ratio of half of a circle to its radius". The graphics above, which both represent π, are not equal. In both cases above, π has the same number, 3.14159265..., and the same symbol, but two different graphical meanings (either HALF a unit circle or a WHOLE unit circle).
Figure 4. Thus, the same number, 3.14159265... represents either HALF of the circumference of a circle based on radius, 3.14159265 = 1/2 · (2·π)·r1 = 1/2 · (6.28318530...) or the number 3.14159265... = π·d1. Of course, these equations scale to any value of r or d.
Figure 5. For radius-based circles: π = (C/2)/rx. For example, if r = 5 then C = 2·π·r5 or C = 10·π, so π = (C10π/2)/r5, or π = (10π/2)/5 = 5π/5 = π. For diameter-based circles: π = C/dy. For example, if d = 10, C = π·d10 or C = 10·π, so π = C10π/d10, or π = 10π/10 = π. In this example, one circle is 10·π based on radius, which is the same circumference as the second circle, which is 10·π based on diameter. Since they are the same circumference, but based on two different units (r or d), there is no way to tell which units were used if you just see "10·π".
Figure 6. It was shown above that 2·π = π, based on two different unit definitions. How can 2·π = π, and 6.28318530... = 3.14159265...? Because the same symbol is used with two different unit definitions.
The same symbol represents either HALF a Unit Circle or a WHOLE Unit Circle. This is The Pi Paradox. One symbol represents two different things. It is the same symbol, with the same numeric value, but with two different meanings.
Figure 7. The solution to The Pi Paradox is simple: Use two different symbols for the two different meanings of π. By using two different symbols for the two different meanings, the symbol for each meaning indicates the meaning that accompanies the symbol. Thus, you no longer have a paradox such as 2·π = π, you now have 2·π = PI. In this way, π can be referred to as Classic Pi, and the new symbol is referred to as Whole PI (since it refers to a whole circle based on a whole diameter). When written in text, instead of as a symbol, Classis Pi is sometimes written as pi, and sometimes written as Pi, depending on whether it is in the middle of a sentence or at the beginning of a sentence, or in a title. To differentiate from this, Whole PI is always capitalized, so PI is always different from pi or Pi.
Figure 8. The symbol for Whole PI was constructed to contain the components that Whole PI is based on: three whole diameters (representing the 3 dimensions) and a whole circle. The three diameters are arranged in a ratio of 1:2, in the same way that the ratio of a radius to a diameter is 1:2. Thus, the ratio of the diameters in the symbol is the same as the ratio of a unit radius to a unit diameter. Furthermore, the three diameters show a value of 3 in the symbol, which is the integer value of the symbol. Last, but certainly not least, the symbol contains a whole circle, meaning the symbol reflects the circularity of the entity it represents.
The complete article describing Whole PI, and how it relates to Space, Mass, and the Periodic Table is entitled:
"The Case for Whole PI and Alternative Equations for Space, Mass, and The Periodic Table" at http://dx.doi.org/10.13140/RG.2.1.3348.6968/2
The Whole PI True Type Font for use in documents, presentations, graphics, etc. is available here:
Figure 9. These are the radius-based equations for the three dimensions that we all learned as children. Can you see the pattern behind the three dimensions? Actually, you have to learn integral calculus to learn how to go from one dimension to the next. Where did the 4/3 come from? Why? Again, learn integral calculus to see where this came from. The pattern is not obvious at first examination. See the page on Dimensions of Space to see how clear and obvious the pattern is if you use Whole PI.
Figure 10. Again, you can clearly see the origin of the symbol for Whole PI, which contains the three diameters of our three-dimensional (3-D) world plus the whole circle on which it is based. The symbol reflects the meaning of the symbol.
Here is an animated version of Figure 10, which shows the construction of the symbol for Whole PI.
Figure 11. Here are the new Equations for the Dimensions of Space based on Whole PI. The core of all equations is PI·dp/2p . The First Dimension is unique, because it has a "2" in the front to signify it as the first. The new Whole PI equation for the First Dimension is 2PI·dp/2p. All equations have the same core, only the First Dimension is unique. The "d" is the diameter length, "p" is the dimension number (1,2,3), and the "2p" in the denominator expresses the duality of all dimensions and equals the number of radii in that dimension. See more at: www.DimensionsOfSpace.com.