3-D Ideas

the shape of thoughts

simple ideas are like a point ... self-defined, self-referential 

straight forward ideas are an ordered collection of points (numeral, sequential)

logical ideas are 2-D network of Logic Paths

complex ideas are 3-D ... polygon "many faces" ... with ... one core central node ... AND ... one encompassing shell

 
good ideas ... are definable in 2-D terms
but
great ideas ... are definable ONLY in 3-D terms

Euclidean geometry vs. non-Euclidean geometry: hyperbolic and elliptic line geometry

Behavior of lines with a common perpendicular in each of the three types of geometry

# In Euclidean geometry the lines remain at a constant distance from each other, and are known as parallels.
# In hyperbolic geometry they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultraparallels.

non-Euclidean geometry is characterized by a non-vanishing Riemann curvature tensor. 

Greek and Roman drachm are related by the approximate ratio 25 : 32 ~ ℨ

The dram (archaic spelling drachm; apothecary symbol ) was historically both a coin and a weight.

Modern unit of mass

In the avoirdupois system, the dram is the mass of

1256 pound or

116 ounce.

So the dram weighs 87532 grains or exactly 1.771 845 195 3125 grams.

The dram (symbol: ʒ) is also the mass of 196 pound (℔) or 18 ounce (℥) in the apothecaries' system that survived until the middle of the 20th century in English-speaking countries. It is equal to 3 scruples (℈) or 60 grains (G). Thus, it is equal to exactly 3.887 9346 grams.

Unit of volume

The fluid dram is defined as 18 of a fluid ounce, which means it is exactly equal to

In the United Kingdom, a teaspoon was formerly defined as 3/2 fluid dram.

Dram is also used informally to mean a small amount of liquid, especially Scotch whisky.

File:English mass units graph.svg


MINA

The mina (also mna, Ancient Greek μνᾶ) is an ancient Near Eastern unit of weight equivalent to 50 shekels. The mina, like the shekel, was also a unit of currency; in ancient Greece it was equal to 100 drachmae. The Greek word mna was borrowed from Semitic; compare Hebrew māneh, Aramaic mĕnē, Syriac manyā, Ugaritic mn, and Akkadian manū.

From earliest Sumerian times, a mina was a unit of weight. At first, talents and shekels had not yet been introduced. By the time of Ur-Nammu, the mina had a value of 1/60 talents as well as 60 shekels. The value of the mina is calculated at 1.25 pounds

Ezekiel refers to a mina ('maneh' in the King James Version) as sixty shekels[4].
Jesus Christ tells the "parable of the ten minas" in Luke 19:11-27.
From the Akkadian period, 2 mina was equal to 1 sila of water

mna  μνᾶ

man  μᾶν

Images

(math) hypocycloid = fixed point on a circle rolls within a larger circle

hypocycloid is a special plane curve generated by the trace of a fixed point on a small circle that rolls within a larger circle. It is comparable to the cycloid but instead of the circle rolling along a line, it rolls within a circle.

If k is a rational number, say k = p/q expressed in simplest terms, then the curve has p cusps.

If k is an irrational number, then the curve never closes, and fills the space between the larger circle and a circle of radius R − 2r.

Hypocycloid Examples

The hypocycloid is a special kind of hypotrochoid, which are a particular kind of roulette.

A hypocycloid with three cusps is known as a deltoid.

A hypocycloid curve with four cusps is known as an astroid.

 

 

Delta - Toid
The red curve is a hypocycloid traced as the smaller black circle rolls around inside the larger blue circle (parameters are R=3.0, r=1.0, and so k=3), giving a deltoid.

R=k ??

waht if
R=3.0, r=2.0, and so k=??? ... parabolic?

Derived curves

The evolute of a hypocycloid is an enlarged version of the hypocycloid itself, while the involute of a hypocycloid is a reduced copy of itself.  

The pedal of a hypocycloid with pole at the center of the hypocycloid is a rose curve.

The isoptic of a hypocycloid is a hypocycloid.

taxicab numbers (math)

 I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable omen.

"No", he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two [positive] cubes in two different ways."

\operatorname{Ta}(1) = 2 = 1^3 + 1^3
\begin{matrix}\operatorname{Ta}(2)&=&1729&=&1^3 + 12^3 \\&&&=&9^3 + 10^3\end{matrix}
\begin{matrix}\operatorname{Ta}(3)&=&87539319&=&167^3 + 436^3 \\&&&=&228^3 + 423^3 \\&&&=&255^3 + 414^3\end{matrix}

In mathematics, the n-th cabtaxi number, typically denoted Cabtaxi(n), is defined as the smallest positive integer that can be written as the sum of two positive or negative or 0 cubes in n ways. Such numbers exist for all n (since taxicab numbers exist for all n); however, only 10 are known (sequence A047696 in OEIS):

\begin{matrix}\mathrm{Cabtaxi}(1)&=&1&=&1^3 \pm 0^3\end{matrix}
\begin{matrix}\mathrm{Cabtaxi}(2)&=&91&=&3^3 + 4^3 \\&&&=&6^3 - 5^3\end{matrix}
\begin{matrix}\mathrm{Cabtaxi}(3)&=&728&=&6^3 + 8^3 \\&&&=&9^3 - 1^3 \\&&&=&12^3 - 10^3\end{matrix}
\begin{matrix}\mathrm{Cabtaxi}(4)&=&2741256&=&108^3 + 114^3 \\&&&=&140^3 - 14^3 \\&&&=&168^3 - 126^3 \\&&&=&207^3 - 183^3\end{matrix}
\begin{matrix}\operatorname{Ta}(4)&=&6963472309248&=&2421^3 + 19083^3 \\&&&=&5436^3 + 18948^3 \\&&&=&10200^3 + 18072^3 \\&&&=&13322^3 + 16630^3\end{matrix}

Big Numbers


one SQUARE = one
10 SQARED = 100

thousand SQUARED = Million  ~ZEROS6 ~1 illi on
Million SQUARED = Trillion  ~ ZEROS12 ~ 3 illi on
Trillion SQUARED = septillion ~ ZEROS 24 ~ 7 illi on
septillionSQ  = quindecillion ~ ZEROS 48 ~   15 illi on


Number of zeros



3 thousand
6 million
9 billion
12 trillion
15 quadrillion
18 quintillion
21 sextillion
24 septillion
27 octillion
30 nonillion
33 decillion
36 undecillion
39 duodecillion
42 tredecillion
45 quattuordecillion
48 quindecillion
51 sexdecillion
54 septendecillion
57 octodecillion
60 novemdecillion
63 vigintillion
66 - 120  
303 centillion
600