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extra extra revisions to polycount

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queue-miscreant 2025-02-24 03:06:14 -06:00
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5 changed files with 26 additions and 12 deletions
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_freeze/polycount/3/index/execute-results/html.json
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_freeze/polycount/5/index/execute-results/html.json
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polycount/3/index.qmd
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@ -13,10 +13,22 @@ categories:
- phinary - phinary
- binary - binary
- python - python
execute: code-fold: true
code-fold: true
--- ---
<style>
.cell-output-display .figure {
text-align: center;
}
/* #long-trunc-figures .figure-img { */
.figure-img {
max-width: 512px;
object-fit: contain;
height: 100%;
}
</style>
```{python} ```{python}
#| echo: false #| echo: false
@ -581,7 +593,7 @@ $$
$$ $$
This neatly ties repeating spacings in with carries. This neatly ties repeating spacings in with carries.
^[Recall that when we naively computed ten in base phi, we got "10100.0100101010101010101". ^[Recall that when we naively computed ten in base *φ*, we got "10100.0100101010101010101".
After a certain point, this expansion alternates between 0 and 1. Assuming that this is true repetition After a certain point, this expansion alternates between 0 and 1. Assuming that this is true repetition
and applying $10_\varphi = 1.\underline{01}_\varphi$, one obtains "10100.0101", which is canonical. ] and applying $10_\varphi = 1.\underline{01}_\varphi$, one obtains "10100.0101", which is canonical. ]

6
polycount/4/index.qmd
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@ -446,7 +446,7 @@ Unfortunately, this is not the case.
All of the expansions above are identically four, and manipulating the digits directly would just All of the expansions above are identically four, and manipulating the digits directly would just
give a different number. give a different number.
Instead, we could try representing "4" in a more direct manner Instead, we could try representing "4" in a more direct manner.
There are a few other options available. There are a few other options available.
- Add the nonrepeating part of the mixed balanced expansion of 4 with the - Add the nonrepeating part of the mixed balanced expansion of 4 with the
@ -615,7 +615,7 @@ plt.title("DFT of $4_{\\kappa}$ after mapping $\\bar{1}$ to $1$")
plt.plot(abs(np.fft.fft([abs(i) for i in cendree_adic_4[:256]]))[:129]) plt.plot(abs(np.fft.fft([abs(i) for i in cendree_adic_4[:256]]))[:129])
``` ```
If you look closely, you'll notice that this plot is a mirror of the other. If you look closely, you can notice that this plot is a mirror of the other.
### Longer Truncations ### Longer Truncations
@ -677,7 +677,7 @@ The [next post](../5) will return to integral sequences, and the patterns produc
Recall that $3 = 10.02_{\kappa}$ and $4 = 11.02_{\kappa}$. Recall that $3 = 10.02_{\kappa}$ and $4 = 11.02_{\kappa}$.
We can add two expansion together to produce a new valid expansion. We can add two expansions together to produce a new valid expansion.
Hence, Hence,
:::: {.row layout-ncol="2"} :::: {.row layout-ncol="2"}

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polycount/5/index.qmd
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@ -266,7 +266,7 @@ Because this operation is commutative, the shapes of the rectangles must agree a
The shape of the rectangular blocks is somewhat odd. The shape of the rectangular blocks is somewhat odd.
We can assess the number of terms the series "hangs" before progressing by looking at the We can assess the number of terms the series "hangs" before progressing by looking at the
[*run lengths*](https://en.wikipedia.org/wiki/Run-length_encoding)). [*run lengths*](https://en.wikipedia.org/wiki/Run-length_encoding).
:::: {.row layout-ncol="1"} :::: {.row layout-ncol="1"}
::: {.row} ::: {.row}
@ -365,7 +365,9 @@ $$
In this mapping, for $n = 2$, zero goes to red and one goes to cyan. In this mapping, for $n = 2$, zero goes to red and one goes to cyan.
The following is a 100x100 image of the multiplication table of $\oplus_\text{Zeck}$ from zero to ninety-nine. The following is a 100x100 image of the multiplication table of $\oplus_\text{Zeck}$ from zero to ninety-nine.
![](fibonacci_deficiency_mod_2.png){.image-wide} ![
&nbsp;
](fibonacci_deficiency_mod_2.png){.image-wide}
### Anima Moduli ### Anima Moduli