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tweak language in chebyshev.1 and stereo.2

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queue-miscreant 2026-09-21 12:55:49 -05:00
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25
posts/math/chebyshev/1/index.qmd
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@ -259,18 +259,18 @@ $$
= (z^2 - z - 1) (z^2 + z - 1) = (z^2 - z - 1) (z^2 + z - 1)
$$ $$
According to how we derived this series, when $z = 2\cos(\theta)$, the roots of this polynomial According to how we derived this polynomial, when $z = 2\cos(\theta)$, the roots
correspond to when $\sin(5\theta) / \sin(\theta) = 0$. correspond to when $\sin(5\theta) / \sin(\theta) = 0$.
This relation itself is true when $\theta = \pi / 5$, since $\sin(5 \pi / 5) = 0$. This relation itself is true when $\theta = \pi / 5$, since $\sin(5 \pi / 5) = 0$.
One of the factors must therefore be the minimal polynomial of $2\cos(\pi / 5 )$. One of the factors must therefore be the minimal polynomial of $2\cos(\pi / 5 )$.
The former happens to be correct correct, since $2\cos( \pi / 5 ) = \varphi$, the golden ratio. The former happens to be the correct choice, since $2\cos( \pi / 5 ) = \varphi$, the golden ratio.
Note that the second factor is the first evaluated at -*z*. Note that the second factor is the first evaluated at -*z*.
#### Heptagons #### Heptagons
An example of where constructability fails is for $2\cos( \pi / 7 )$. An example where constructability fails is for $2\cos( \pi / 7 )$.
$$ $$
\begin{align*} \begin{align*}
@ -289,7 +289,8 @@ But there are no (nondegenerate) cubics that one can produce via compass and str
#### Decagons #### Decagons
One might think the same of $2\cos(\pi /10 )$ One might think the same of $2\cos(\pi /10 )$.
The relevant polynomial is:
$$ $$
\begin{align*} \begin{align*}
@ -300,16 +301,17 @@ $$
\end{align*} \end{align*}
$$ $$
This expression also contains the polynomials for $2\cos( \pi / 5 )$. This expression also contains the polynomials for $2\cos( \pi / 5 )$,
This is because a regular decagon would contain two disjoint regular pentagons, since connecting every other vertex of a regular decagon produces
produced by connecting every other vertex. two disjoint regular pentagons.
![ ![
   
](./decagon_divisible.png) ](./decagon_divisible.png)
The polynomial which actually corresponds to $2\cos( \pi / 10 )$ is the quartic, The polynomial which actually corresponds to $2\cos( \pi / 10 )$ is the quartic,
which seems to suggest that it will require a fourth root and somehow decagons are not constructible. which seems to suggest that it will require a fourth root and imply that decagons
are somehow not constructible.
However, it can be solved by completing the square... However, it can be solved by completing the square...
$$ $$
@ -496,12 +498,12 @@ $$
If $z = 1$, these two generating functions are equal. If $z = 1$, these two generating functions are equal.
The same can be said for $z = 2$ with the generating function of the Pell numbers, The same can be said for $z = 2$ with the generating function of the Pell numbers,
and so on for higher recurrences (corresponding to metallic means) for higher integral *z*. and so on for higher recurrences (corresponding to metallic means) for higher integral *z*.
In fact, it is possible to find the Fibonacci numbers (Pell numbers, etc.)
in Pascal's triangle, which you can read more about
[here](http://users.dimi.uniud.it/~giacomo.dellariccia/Glossary/Pascal/Koshy2011.pdf).
In terms of the Chebyshev polynomials, this series manipulation removes the alternation in In terms of the Chebyshev polynomials, this series manipulation removes the alternation in
the coefficients of $U_n$, restoring Pascal's triangle to its nonalternating form. the coefficients of $U_n$, restoring Pascal's triangle to its nonalternating form.
Related to the previous point, it is possible to find the Fibonacci numbers (Pell numbers, etc.)
in Pascal's triangle, which you can read more about
[here](http://users.dimi.uniud.it/~giacomo.dellariccia/Glossary/Pascal/Koshy2011.pdf).
Manipulating the Series Manipulating the Series
@ -674,6 +676,7 @@ Though they were present in the earlier Chebyshev table,
```{python} ```{python}
#| echo: false #| echo: false
#| label: tbl-cosinepolynomials
#| classes: plain #| classes: plain
def poly_to_rising_power_list(poly, var): def poly_to_rising_power_list(poly, var):

14
posts/math/stereo/2/index.qmd
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@ -190,7 +190,7 @@ For comparison, the complex exponential (as it is a parallel construction) has a
Since the exponential function is its own derivative, the expression acquires Since the exponential function is its own derivative, the expression acquires
an imaginary coefficient through the chain rule. an imaginary coefficient through the chain rule.
[^1]: This is forgoing the fact that complex derivatives require more care than their real counterparts. [^1]: This is forgoing the extra care that complex derivatives require beyond their real counterparts.
It matters slightly less in this case since this function is complex-valued, but has a real parameter. It matters slightly less in this case since this function is complex-valued, but has a real parameter.
$$ $$
@ -249,12 +249,12 @@ $$
= 2\pi i = 2\pi i
$$ $$
In this example, Γ is a counterclockwise curve parametrized by γ which loops once around In this example, *Γ* is a counterclockwise curve parametrized by *γ* which loops once around
the pole at *z* = 0. the pole at *z* = 0.
More loops will scale this by a factor according to the number of loops. More loops will scale this by a factor according to the number of loops.
Normally this equality is demonstrated with the complex exponential, but will $o_1$ work just as well? Normally this equality is demonstrated with the complex exponential, but will $o_1$ work just as well?
If Γ is the unit circle, the integral is: If *Γ* is the unit circle, the integral is:
$$ $$
\oint_\Gamma {1 \over z} dz \oint_\Gamma {1 \over z} dz
@ -281,9 +281,9 @@ $$
\end{gather*} \end{gather*}
$$ $$
It is certainly possible to perform these contour integrals along straight lines; It is certainly possible to perform these contour integrals along straight lines
in fact, integrating along lines from 1 to *i* to -1 to -*i* deals with a in the complex plane; in fact, making *Γ* a diamond-shaped contour from
similar integral involving arctangent. 1 to *i* to -1 to -*i* produces a similar integral involving arctangent.
However, the best one can do to construct more loops with lines is to count each line However, the best one can do to construct more loops with lines is to count each line
multiple times, which isn't extraordinarily convincing. multiple times, which isn't extraordinarily convincing.
@ -304,7 +304,7 @@ $$
$$ $$
This has an additional benefit: using the series form of $1 / (1 + t^2)$ and integrating, This has an additional benefit: using the series form of $1 / (1 + t^2)$ and integrating,
one obtains the series form of the arctangent. one obtains [the series form of arctangent](https://en.wikipedia.org/wiki/Arctangent_series).
This series converges for $-1 \le t \le 1$, which happens to match the bounds of integration. This series converges for $-1 \le t \le 1$, which happens to match the bounds of integration.
The convergence of this series is fairly important, since it is tied to formulas for π, The convergence of this series is fairly important, since it is tied to formulas for π,
in particular [Leibniz's formula](https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80). in particular [Leibniz's formula](https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80).