fixes for sitewide display

.gitignore

@@ -1,2 +1,3 @@
-**/*_files
+_freeze/
-*.html
+_site/
+/.quarto/

polycount/1/index.qmd

@@ -1,13 +1,15 @@
 ---
+title: "Polynomial Counting 1: A primer"
 format:
   html:
     html-math-method: katex
 jupyter: python3
+date: "2021/02/03"
+categories:
+  - algebra
+  - phinary
 ---
 
-Polynomial Counting: A primer
-=============================
-
 The single most common method of representing numbers in the modern world is in *positional numeral system*.
 Despite being taught in early grade school, it is the result of millennia of mathematical thought.
 
@@ -120,7 +122,7 @@ Naively, we might use the following "greedy" algorithm to derive the base-$b$ ex
 (also called a [β-expansion](https://en.wikipedia.org/wiki/Non-integer_base_of_numeration)) of a number $x$:
 
 ```{python}
-from math import log
+from math import log, floor
 
 def beta_expand_greedy(
   x: float,
@@ -130,7 +132,7 @@ def beta_expand_greedy(
   ret = {}
 
   while x > tol:                    # While we're not precise enough
-    p = int(log(x, b))              # Get the place value p
+    p = int(floor(log(x, b)))       # Get the place value p
     digit, new_x = divmod(x, b**p)  # Get the quotient and remainder from
                                     #   dividing by this place value
     ret[p] = int(digit)             # Place the digit in place value p
@@ -138,7 +140,12 @@ def beta_expand_greedy(
 
   return ret
 
-def as_digits(digits: dict[int, int]):
+```
+
+As a demonstration, this algorithm, when run on a decimal number gives the same value:
+
+```{python}
+def as_digits(digits: dict[int, int]) -> str:
   '''Convert a dictionary from `beta_expand_greedy` to a sequence of digits'''
   return "".join(
     str(digits.get(i, 0)) + ("." if i == 0 else "")
@@ -160,7 +167,8 @@ There are three problems with this.
     Due to the nature of the approximation, the result can also appear in an unexpected form:
     ```{python}
     phi = (5**0.5 + 1) / 2
-    # print(as_digits(beta_expand_greedy(10, phi)))
+    print("Expected:", "10100.0101")
+    print("Got:     ", as_digits(beta_expand_greedy(10, phi)))
     ```
 
 2. It relies on a transcendental function, the logarithm.
@@ -170,7 +178,7 @@ There are three problems with this.
    However, if `p` is always positive, we can instead use integer arithmetic, which is more precise.
    Generally, we need some form of fractional arithmetic.
 
-Modern FPUs are make the last two items somewhat trivial, but they are necessarily approximate.
+Modern FPUs are make the last two items somewhat trivial, but they necessarily make the calculation approximate.
 Fortunately for phinary, there is a direct method which remedies all these issues and produces exact results without floating-point operations.
 
 
@@ -363,8 +371,8 @@ Expansions of the integers up to 10 are:
 
 These are known as *Zeckendorf expansions*.
 
-These representations very similar to the phinary strings above.
+These representations seem very similar to the phinary strings above.
-Not only that, but this sequence is also the sequence of all binary strings that do not contain two consecutive 1's ([OEIS A014417](https://oeis.org/A014417)).
+Not only that, but this sequence is also the sequence of all binary strings that do not contain two consecutive "1"s ([OEIS A014417](https://oeis.org/A014417)).
 This representation is exactly as arbitrary as preferring the greedy phinary representation;
 instead, this is the "greedy series expansion" of an integer in the Fibonacci numbers.
 
@@ -395,7 +403,7 @@ $$
 
 In the expansion of 2, the rightmost "1" seems to underflow.
 However, in the expansion of 4, we must consider the second "1" in the negative first place value.
-It acts as a sort of temporary storage which is immediately transferred into place value 0.
+It acts as a sort of temporary storage which is immediately transferred into place value zero.
 
 
 Synthesis: Generalizing Phinary
@@ -517,4 +525,4 @@ Closing
 -------
 
 With these restrictions in mind, I wrote a simple Haskell library to help explore these systems (found [here](https://github.com/queue-miscreant/GenBase)).
-The [next post]() will discuss quadratic polynomials with larger coefficients than 1, and problems not discussed with higher expansions.
+The [next post](../2) will discuss quadratic polynomials with larger coefficients than 1, and problems not discussed with higher expansions.

polycount/2/index.qmd

@@ -3,9 +3,13 @@ title: "Polynomial Counting: Binary and beyond"
 format:
   html:
     html-math-method: katex
+date: "2021/02/05"
+categories:
+  - algebra
+  - binary
 ---
 
-This post assumes you have read [the one prior](1), which introduces generalized polynomial counting.
+This post assumes you have read [the one prior](../1), which introduces generalized polynomial counting.
 
 Before I start, I'd like to introduce some shorthand.
 Since we prefer (monic) polynomials which may correspond to linear recurrences, it is useful to use an ordered tuple of their coefficients.
@@ -329,5 +333,5 @@ This polynomial belongs to a family called
   which have complex roots of 1 as their roots.
 I'll have more to say about cyclotomic polynomials in the future, but they cannot be carries by themselves.
 
-Since higher numbers appearing in carries are somewhat scary, the [next post]() will
+Since higher numbers appearing in carries are somewhat scary, the [next post](../3) will
   focus on two more generalizations of the Fibonacci numbers and exceptions to the above rules.