Ф
and ψ and the Mysteries of Geometric Form
The
Mysteries of Geometric Form.
When we look to star forms, we see
fascinating numeric relationships begin to reveal themselves.
Ф:
the golden mean.
Patterns of form and number are
readily found to be a part of this irrational value. It's known that the Fibonacci sequence (0, 1, 1, 2, 3, 5, …)
tends toward the phi proportion, as does any sequence of numbers generated in
the same fashion. Further, by tracing
around a 5-pointed star, it is fun to discover that the exponential values of
Ф provide us with the Fibonacci sequence. A circuitous route!
Together with students we have opened
doors to greater understanding of phi patterns and forms. For example:
|
Root
|
Ф
relation
|
Number
|
Ф
relation
|
|
√2
|
√(Ф3
+ 1)/Ф
|
1
|
Ф1 - 1/Ф
|
|
√3
|
√(Ф4
+ 1)/Ф
|
3
|
Ф2 + 1/Ф2
|
|
√5
|
√(Ф5
+ 2)/Ф
|
4
|
Ф3 - 1/Ф3
|
|
√6
|
√(Ф6
- √5)/Ф
|
7
|
Ф4 + 1/Ф4
|
|
√7
|
√(Ф8
+ 1)/Ф2
|
11
|
Ф5 - 1/Ф5
|
|
@Paul Stang
|
|
18
|
Ф6 + 1/Ф6
|
|
Radii
|
|
|
|
|
dodecahedron
|
√(Ф4
+ 1)/2
|
√5
|
Ф1 + 1/Ф
|
|
icosahedron
|
√(Ф2
+ 1)/2
|
2
|
Ф1 + 1/Ф2
|
|
Radical value
|
Equivalent
|
Radical value
|
Equivalent
|
|
(1
+ √5)
2
|
Φ1
|
(1
- √5)
2
|
1/Φ
|
|
(3
+ √5)
2
|
Φ2
|
(3
- √5)
2
|
1/Φ2
|
|
(4
+ 2√5)
2
|
Φ3
|
(4
- 2√5)
2
|
1/Φ3
|
|
(7
+ 3√5)
2
|
Φ4
|
(7
- 3√5)
2
|
1/Φ4
|
|
(11
+ 5√5)
2
|
Φ5
|
(11
- 5√5)
2
|
1/Φ5
|
Notice from the tables the second
"phi" sequence of 1,3,4, 7…
Ψ
- another proportion
By
selecting '1' as an inner radius, we can calculate to obtain the various
measures shown in the forms which follow.
There are many more unique relations to √2 and √3 that are
suspected; awaiting confirmation. Other
number relations remain elusive with regard to their possible √x
values.
As can be seen in the 9-pointed star
there is a proportion (I call it ψ
- psi) which is uniquely reminiscent of phi geometries and calculation.
The diagram shows
geometries contained within the 9-pointed star. This spiral is created just as that which can be used to build
the 5-pointed star. This psi proportion
is found throughout the geometries of the 9-pointed star.
The ψ-proportion
shares similar properties with its "cousin":
ψ +
1/ψ2 = 3, ψ3
- 3ψ2 + 1 =
0, and √(ψ3 + 1)/ψ = √3.
Ф2 + 1/Ф2 = 3, Ф3 - 2Ф2 + 1 = 0,
and √(Ф4
+ 1)/Ф = √3
My initial investigations, inspired by
a question from one of my students, suggest that the 7-pointed and perhaps all
odd-numbered stars have their own unique proportions. What is significant about the 9-pointed star is that its
geometries give way readily to equilateral triangles and their mathematic
relations, whole number division of the circle, trisection, cubic roots, and
parallels to phi.
The Sacred Wisdom of Mathematics
Each
day, I encourage the non-use of calculators in the classroom. As a result, a student or I will see some
pattern develop as we work with trigonometric ratios, phi, or graphing
parabolas; for examples. Many times,
mathematics is referred to as the "universal" language. What I find as I really work the problems,
without a technological assist, is that amazing relationships exist and there
are in fact universal terms like; √2 and √3.
The
Pythagoreans were right to study number, but the extent to which they could
have used algebra and other processes would arguably have been limited compared
to ours. Today, we have centuries of
development to use in the classroom since the Classical Greeks and their
Egyptian forebears. Unfortunately, when
we jump to the calculator and obtain decimal answers, rounded off, it is nearly
impossible to see patterns develop, and hence math tends to be dead and
dull.
Let's
briefly look at some amazing cases, and see if we don't come away thinking that
there might be some inherent 'Wisdom' buried within mathematics.
Wisdom found in Algebra, Geometry and
Trigonometry:
We
often use the Pythagorean Theorem, graphing of lines, and trigonometric ratios
associated with 30°, 60°, and 90° in mathematics. Let us look in detail at a sampling of these.
The
slope of the line.
When
bisecting the angle formed by a line, an interesting process is to take lines
whose slope corresponds with Pythagorean triangles like 3-4-5, 5-12-13
etc. If we encourage our students to
use fractions, and identify fractions from decimal values, a very interesting
pattern develops as shown in the left columns of the following table. This leads us to look further, for possible
relations, which the table proceeds to show do in fact exist!
|
|
Line
slope
|
Bisector
slope
|
|
Line
slope
|
Bisector
slope
|
|
Pytha-
gorean
Triangles
Reciprocals
|
3/4
|
1/3
|
Patterns in √2,√3 and Ф
Radicals and primes
|
1
|
(√2-1)/1
|
|
5/12
|
1/5
|
2
|
(√5-1)/2 =Ф - 1
|
|
7/24
|
1/7
|
3
|
(√10 - 1)/3
|
|
9/40
|
1/9
|
4
|
(√17 - 1)/4
|
|
|
|
x
|
(√(x + 1)2 -
1)/x
|
|
|
|
2(1+√3/3)
|
√3 - 1
|
|
4/3
|
1/2
|
√2/2
|
√3 - √2
|
|
12/5
|
2/3
|
√3/3
|
2 - √3
|
|
24/7
|
3/4
|
1/2
|
1/Φ3
|
|
40/9
|
4/5
|
√2
|
(√3 - 1)/√2
|
|
@Paul
Stang
|
|
|
2√2
|
√2/2
|
|
√3 or √3/1
|
√3/3
|
|
√5/2
|
√5/5
|
|
√7/3
|
√7/7
|
|
√11/5
|
√11/11
|
|
√13/6
|
√13/13
|
Trigonometric
ratios.
Using
the calculator, we find there are no whole number relationships between angles
and sides in a triangle. Our 3-4-5
friend, and others, give angles such as 36.86°. So,
we assume we must use the calculator in all of our calculations. Interestingly enough, if we use the hexagon
as our guide, this useful figure shows us that the radius and sides are equal
in length, divide the circle into 6ths (and 4ths and 12ths), and provide nice
challenges in finding √2/2 and √3/3.
30°, 60°, 45° . Trigonometric ratios via diagrams.
√0 √1 √2 √3 √4 √4 √3 √2 √1 √0
2
2 2 2 2 2 2 2 2
2
First quadrant sine values Second quadrant sine
values
Identities.
While there
are no whole number relationships, there are square-root-whole-number
connections all relating to 360-based counting, and fractional division of the
circle. When we apply half angle
formulas, we find further interesting developments from our friends √2
and √3.
|
Radians
|
Revolutions
|
Angle
|
SIN
|
COS
|
TAN
|
|
π/24
|
1/48
|
7.5°
|
√(2-√(2+√3))
2
|
√(2+√(2+√3))
2
|
|
|
2π/24, π/12
|
2/48,
1/24
|
15°
|
√(2-√3)
2
|
√(2+√3)
2
|
2-√3
|
|
4π/24, π/6
|
4/48,
1/12
|
30°
|
√1
2
|
√3
2
|
√3
3
|
|
|
@Paul
Stang
|
|
|
|
