Author: IMC

Polygonal numbers are a mainstay of recreational and school mathematics, providing a nice bridge between numbers and shapes. The diagrams above show some of the hexagonal numbers.

Some examples of two-dimensional polygonal numbers are:

the triangular numbers: 1, 3, 6, 10, 15, …
the square numbers: 1, 4, 9, 16, 25, …
the pentagonal numbers: 1, 5, 12, 22, 35,…
the hexagonal numbers: 1, 6, 15, 28, 45, …

Comparing the listing for the hexagonal numbers with the diagrams above, you can see how the sequences are built diagrammatically. In general, beginning with a single dot, k-sided polygons are built by adding layers (called gnomons) consisting of k-2 segments, with each segment of the gnomon having one more dot than the segments of the previous layer. In this way, the nth gnomon consists of segments each n dots long, but with k-3 dots shared by adjoining segments (the corners).

The description above can lead you to a recursive formula for k-polygonals, writing p_k,n for the nth k-polygonal number:

Unwinding the recursion gives you a summation formula for k-polygonals:

Knowing a little about sums gives you the direct formula for k-polygonals:

Coming a little out of left-field is this combinatorial formula for k-polygonals:

This last formula expresses two ideas: that the triangular numbers correspond to the r=0 column of Pascal’s triangle, and that every polygonal number can be “triangulated”:

The combinatorial formula for p_kn can be generalized to higher-dimensional polygonal numbers (pyrimidal numbers, etc.).

The recreation here lies in showing that the various formulas for p_k,n are really the same, and then exploring the relationships between the different k-polygonals. A great resource is J.H. Conway and R.K. Guy’s The Book of Numbers.a

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to dan.mackinnon*

Sonobe and phizz modular origami units are assembled into polyhedral models in similar ways, but the models that they produce are “dual” to each other (vertices in one correspond to faces in the other). Assembling models out of sonobe and phizz units with this duality in mind provides a nice way of exploring duality and the relationships between edges, faces, and vertices.

The sonobe and phizz modules are both examples of edge modules that come together in groups of 3. In both cases, when the units come together in their groups of 3 they meet in a small triangular pyramid. These pyramids in turn come together in clusters of 3 to 6. The essential difference between the phizz and sonobe modules is in how these clusters form. In the phizz, there is a gap between the groups, so the resulting cluster seems to form the edges of a polygon. In the sonobe, the gap between the groups is small, so the cluster seems to form around a point. Consequently, in the phizz, the center of the cluster takes on the role of a polygonal face, while in the sonobe it is takes on the role of a vertex. Meanwhile, the pyramids formed by the groups of three units become raised vertices in the phizz unit, while in the sonobe they become cumulated faces of the resulting polyhedron.

We naturally interpret what phizz modules generate as polyhedral skeletons, while we see the shapes generated by sonobe as cumulated (or augmented) polyhedra. The fact that this different interpretation is based on the size of the gap that forms in the center of the module clusters suggests that seeing an origami model as a particular polyhedron or its dual is, to some extent, a matter of perception and interpretation.

In phizz models, the fact that the modules come together in groups of 3 dictates that the finished models have vertices of degree (or valence) 3, while in the sonobe case, the 3 units come together to make triangular faces. Dual models have the same number of edges, which corresponds to the number of modules required to build the model.

Some of the models that you can build are summarized below. The picture at the top of the post shows the 30-unit phizz dodecahedron and its dual, the 30-unit sonobe icosahedron.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to dan.mackinnon*

Looking at the last few digits that appear in the numbers that form the sequence b^0, b^1, b^2, b^3, … for b a positive integer, you’ll notice that the digits will always begin to repeat after a certain point. For example, looking at the last digit of the sequences for b = 2, 3, and 4 we have the sequences

b = 2: 1, 2, 4, 8, 6, 2, 4, 8, 6, …
b = 3: 1, 3, 9, 7, 1, 3, 9, 7, …
b = 4: 1, 4, 6, 4, 6, 4, 6, …

If we look at the sequence of last two digits of these sequence where b =2 we have

b = 2: 1, 2, 4, 8, 16, 32, 64, 28, 56, 12, 24, 48, 96, 92, 84, 68, 36, 72, 44, 88, 76, 52, 4, …

This sequence then repeats the loop that began at 4.

We can describe these sequences as T_b,d(n) = (b^n)mod 10^d. Recursively, T_b,d(n) = (T_b,d(n-1)*b)mod 10^d

These sequences are always eventually periodic. Although these sequences are simple to understand and calculate, there are several interesting ways of describing them.

For example, you can think of the elements of T_b,d as a commutative monoid, with multiplication defined as a*b = (a*b)mod 10^d. They form a monoid since 1 is always a member, and you can show that T_b,d is closed under the * operation. It turns out that for some values of b, and d, T_b,d is a group.

You can also think of this set as a finite state machine or graph, where each element is a node and the transition from one node to the next is defined by the operation *b mod 10^d. This provides a nice way of displaying the sequences. The pictures in this post were created by writing a short program to calculate the sequences, and then formatting the output to draw a di-graph in SAGE. The graph at the top of the post is for b=8, d=1, while the graph below is for b=2, d=2. The graph at the bottom of the page is for b=7, d=1.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to dan.mackinnon*

The third diagonal column in Pascal’s Triangle (r = 2 in the usual way of labeling and numbering) consists of the triangular numbers (1, 3, 6, 10, …) – numbers that can be arranged in 2-dimensional triangular patterns. The fourth column of Pascal’s triangle gives us triangular-based pyramidal numbers (1, 4, 10, 20, …), built by stacking the triangular numbers. The columns further out give “higher dimensional” triangular numbers that arise from stacking the triangular numbers from the previous dimension.

It is not by coincidence that the triangular and higher-dimensional triangular numbers appear in Pascal’s Triangle. If you think about layering of polygonal numbers in terms of equations, you get

In the above equation p^d_(k,n) is the nth k-polygonal number of dimension d. Triangular numbers are the 3-polygonal numbers of dimension 2, square numbers are the 4-polygonal numbers of dimension 2, “square based pyramidal numbers” would be denoted as p^3_(4,n).
from the sum above, you can obtain this equation:

Which looks very much like the Pascal Identity C(n,r) = C(n-1,r-1) + C(n-1,r), except for some translation of the variables. To be precise, if we consider the case where k=3 and use r = d and n‘ = n+d-1 we can translate the triangular numbers into the appropriate positions in Pascal’s Triangle.

Along with the definitions for the end columns, the Pascal Identity allows us to generate the whole triangle. This suggests the following strategy for calculating the higher k-Polygonal numbers: create a modified Pascal’s Triangle whose first column is equal to k-2 (instead of 1), and whose last column is equal to 1 (as usual). This modified Pascal’s Triangle is generated using these initial values and the usual Pascal Identity.

Here is an example with k=5, which sets the first column values equal to 3 (except for the top value, which we keep as 1) and yields the pentagonal numbers (column 3) and the higher pentagonal numbers.

The formula for these modified Pascal Triangles is given by this equation:

If we apply the change of variables mentioned above, we can obtain this general formula for the higher polygonal numbers in terms of combinations:

This formula illustrates how polygonal numbers are built out of triangular numbers. It says that the nth d-dimensional k-polygonal number is equal to the nth d-dimensional triangular number, plus (k-3) copies of the n-1 d-dimensional triangular number. This is a little easier to understand when you forget about the higher-dimensions and look at the regular 2-dimensional polygonal number.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to dan.mackinnon*

 

John Sims has created a number of pi-related art works. One, the Pi Color Map, can be recreated effectively using TinkerPlots. The image above is one such Pi Color Map, using 2281 digits of pi.

Here are some instructions for creating a Pi Color Map in Tinkerplots.

1. Obtain a listing of the digits of pi – up to a reasonable number. You can get the digits from several sites, including the pi day site.

2. Paste your listing to a text document, and get them arranged into a single column. One strategy for doing this is and use the find/replace feature of a word-processor to replace each number with the number itself plus a line-break(e.g. in Word, replace 2 with 2^l, etc.).

3. If you’ve included the decimal point, remove it. For the first line of your document, provide a heading like pi_expansion. This will be your TinkerPlots attribute.

3. Import the text file into TinkerPlots using the File>Import menu.

4. Create a new attribute called digit whose formula is digit=concat(“”,pi_expansion). This creates a categorical data type that TinkerPlots won’t treat numerically. This is what you will use as your color key. Using the pi_expansion attribute gives a spectrum of color, rather than distinct colors for each number.

5. Create a new attribute called place, whose formula is place=caseIndex. This is what you will order your plot by.

6. Create a new plot, lock the color key on the digit attribute. Select the place attribute and press the Order By button.

7. Change your icon type to small squares, and stack the cases.

You can play with different options to get different effects for your color map.

One nice thing about doing this in TinkerPlots is that you can investigate the data further. The color map plot highlights the apparent randomness of the pi expansion, but you can also create other attributes and plots to investigate things like the running average of the digits, occurrences of consecutive digits, and the overall distribution of the digits (it should be uniform).

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to dan.mackinnon*

 

If You take a look at the square numbers (n^2, n a positive integer), you’ll notice plenty of patterns in the digits. For example, if you look at just the last digit of each square, you’ll observe the repeating pattern 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, … If you construct a graph of “last digit” vs n (like the one below, built with Falthom), the symmetry and period of this digit pattern is apparent.

Why does this happen? The periodic nature of the pattern is easy to understand – when you square a number, only the digit in the ones place contributes to ones place of the product. For example, 22*22 and 32*32 are both going to have a 4 as their last digit – the values in the tens place (or any other place other than the ones) do not affect what ends up as the last digit.

The reason for the symmetry about n=5 is a little less obvious. To see what is going on, it is helpful to use modular arithmetic and to realize that ” last digit of n” is the same as “n mod 10”. Considering what 10-n looks like mod 10 after it is squared, we have the equation below.

This tells us that the last digit of (10-n)^2 is the same as the last digit of n^2, because everything else that is different about these two numbers is divisible by 10.

If you look at the last two digits of the square numbers, you see another repeating pattern that has similar symmetries.

This is a nice looking graph – the period is 50 with a line of symmetry at n=25. You can think about it in the same way as the one-digit case, this time the symmetry is understood by looking at (50-n)^2 mod 100. (Looking at numbers mod 100 tells us their last two digits.)

If you decide to investigate patterns in cubes or higher powers, you’ll see somewhat similar results. Using the binomial theorem and modular arithmetic, you can see why even powers give symmetry similar to the n^2 case, while odd powers do not (although all are periodic).

This graph shows the pattern in the last digit of n^3.

This last graph shows the pattern for the last two digits of n^4.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to dan.mackinnon*

Although creating origami sonobe units is likely to be considered by most to be an essential part of making sonobe polyhedra, a cut-out version of the sonobe module can be used to help beginners learn how to weave the units together.

The unit pictured here can be reproduced multiple times in a document (a PowerPoint slide works well, since you don’t have to worry about margins), and printed onto card stock. Printing onto card stock gives a solid unit to work with, while regular paper will likely be too flimsy.

After printing, cut out each unit and cut slits into them along the bold horizontal lines (a utility blade or exacto-knife works well). Depending on whether you want to hide the printing or not, you can mountain-fold the diagonal line and vally-fold the dotted vertical lines, or vice-versa (all modules should be folded the same).

You will need 6 units for a cube, 12 for an “augmented octahedron”, and 30 for an augmented icosahedron. Models can also be assembled from 3 units (a triangular di-pyramid) 9 units (two fused cubes), and other combinations.

The main idea in creating this module was to create a set of reusable units that could be used in professional development workshops for teachers learning modular origami for the first time. After seeing how the units hold together, the next step is to learn how to fold the units from paper.

To construct smaller modules, simply connect less than 5 units around. A 4 unit base will form the pattern for the augmented octahedron (which will take a total of 12 units). A 3 unit base will form the pattern for the cube (which will take a total of 6 units).

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to dan.mackinnon*

The diagram above (known as the tetractys) shows the first four triangular numbers (1, 3, 6, 10, …). Although there is a simple formula for calculating these numbers directly, t(n) = 1/2(n(n+1)), constructing them by these layered-triangle diagrams helps to show their geometric and recursive properties.

More generally, polygonal numbers arise from counting arrangements of dots in regular polygonal patterns. Larger polygons are built from smaller ones of the same type by adding additional layers of dots, called gnomons. Beginning with a single dot, k-sided polygons are built by adding gnomons consisting of k-2 segments, with each segment of the gnomon having one more dot than the segments of the previous layer. In this way, the nth gnomon consists of segments each n dots long, but with k-3 dots shared by adjoining segments (the corners).

This post describes how you can draw figures that illustrate the polygonal numbers and explore the polygonal numbers in general (triangular, square, pentagonal, hexagonal, etc.) using either TinkerPlots or Fathom. Both TinkerPlots and Fathom work well, but TinkerPlots creates nicer pictures, and allows for captions directly on the graph.
Without describing the details of how you create Fathom or TinkerPlot documents, here are the attributes that you will want to define in order to draw diagrams like the ones shown.

Required attributes
Create a slider k. This will allow you to set what kind of polygonal number you want to draw (k=3 gives triangular numbers, k=4 gives square numbers, etc.)
Define the following attributes:

n The number itself. This is a natural number beginning at 1 and continuing through the number of cases.

gnomon This states which “layer” or gnomon the number belongs in. It is calculated based on a number of other attributes.

g_index This is the position of the number within the gnonom – it ranges from 1 up until the next k-polygonal number is hit.

s_index Each gonom is broken up into sections or sides – what is the position within the side? Each side is of length equal to the gonom number. The first gonom has sides of length 1, the second has length 2, etc.

corner This keeps track of whether or not the number is a “corner” or not. This is based primarily on the s_index attribute.

c_index This keeps track of how many corners we have so far. There are only k-1 corners in a gnomon (the first number n=1 is the remaining corner). So, when we hit the last corner, we know we are at a polygonal number.

k_poly Records whether or not the number n is k-polygonal. It does this by checking to see if it is the last corner of a gnonom.
The attributes listed above are required for finding the position of each number within the figure; th following attributes are used in actually drawing the figures.

angle The base corner angle for the polygon is determined by k. This is the external angle for each corner.

current_angle We have to add to the base angle at each corner as we turn at each corner. This attribute is used to keep track of the total current angle.

dx This is the x-component of the unit direction vector that we are travelling in. Each new dot moves one dx over in the x-direction. It is given by the cosine of the current angle.

dy This is the y-component of the unit direction vector that we are travelling in. Each new dot moves one dy over in the y-direction. It is given by the sine of the current angle.

prev_g_1_x This is the x-coordinate of the first dot in the previous gonom layer. We need to know this because it will be the starting point for the next layer – each layer starts back at the “beginning” of the figure.

prev_g_1_y This is the y- coordinate of the first dot in the previous gonom layer.

x This is the x-coordinate of the current dot, calculated either from the previous dot or from the first dot in the previous layer.

y This is the y-coordinate of the current dot, calculated either from the previous dot or from the first dot in the previous layer.

caption Used to display the number on the plot (TinkerPlots only)
Below are the formulas for each attribute, written in “ascii” math. They are presented without a full explanation, in the hopes that if you try to implement this you will think about and explore each using the formulas and the descriptions above as a guide. Alternate methods for drawing the diagrams are possible, and you might find other formulas that achieve the same goals. Note that there are nested if() statements in several formulas.

n = caseIndex

gnomon = if(n=1){1, if(prev(k_poly)){prev(gnomon)+1, prev(gnomon)

g_index = if(n=1){ 1, if(prev(k_poly){1, prev(g_index) +1

s_index = if(n=1){ 1, if(prev(k_poly){1, if(prev(s_index) = gnomon){2,prev(s_index)

corner = (s_index=1) or (s_index=gnomon)

c_index= if(g_index=1){1, if(corner){prev(c_index)+1, prev(c_index)

k_poly = if(n=1){true, (c_index=k-1)

prev_g_1_x = if (n=1){0, if(g_index=2){prev(x), prev(prev_g_1_x)

prev_g_1_y = if(n=1){0, if(g_index=2){prev(y), prev(prev_g_1_y)

angle = pi-((k-2)*(pi/k))

current_angle = if(g_index =1) {pi-angle, if(pref(corner)){prev(current_angle)-angle, prev(current_angle)

dx = cos(current_angle)

dy = sin(current_angle)

x= if(n=1){0, if(g_index=1){prev_g_1_x +dx, prev(x) +dx

y =if(n=1){0, if(g_index=1){prev_g_1_y +dy, prev(y) +dy

caption = if(k_poly){n,””

To actually draw the diagrams, create a new plot with the x and y attributes as the horizontal and vertical axis, respectively. Add cases to the collection to populate the diagram. Optionally you can show connecting lines, and (in TinkerPlots) add a legend using the caption attribute.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to dan.mackinnon*

Alan Turing: still casting a long shadow over mathematics

One hundred and fifty years of mathematics will be proved wrong if a new computer program stops running. Thankfully, it’s unlikely to happen, but the code behind it is testing the limits of the mathematical realm.

The program is a simulated Turing machine, a mathematical model of computation created by codebreaker Alan Turing. In 1936, he showed that the actions of any computer algorithm can be mimicked by a simple machine that reads and writes 0s and 1s on an infinitely long tape by working through a set of states, or instructions. The more complex the algorithm, the more states the machine requires.

Now Scott Aaronson and Adam Yedidia of the Massachusetts Institute of Technology have created three Turing machines with behaviour that is entwined in deep questions of mathematics. This includes the proof of the 150-year-old Riemann hypothesis – thought to govern the patterns of prime numbers.

Turing’s machines have long been used to probe such questions. Their origins lie in a series of philosophical revelations that rocked the mathematical world in the 1930s. First, Kurt Gödel proved that some mathematical statements can never be proved true or false – they are undecidable. He essentially created a mathematical version of the sentence “This sentence is false”: a logical brain-twister that contradicts itself.

No proof of everything

Gödel’s assertion has a get-out clause. If you change the base assumptions on which proofs are built – the axioms – you can render a problem decidable. But that will still leave other problems that are undecidable. That means there are no axioms that let you prove everything.

Following Gödel’s arguments, Turing proved that there must be some Turing machines whose behaviour cannot be predicted under the standard axioms – known as Zermelo-Fraenkel set theory with the axiom of choice (C), or more snappily, ZFC – underpinning most of mathematics. But we didn’t know how complex they would have to be.

Now, Yedidia and Aaronson have created a Turing machine with 7918 states that has this property. And they’ve named it “Z”.

“We tried to make it concrete, and say how many states does it take before you get into this abyss of unprovability?” says Aaronson.

The pair simulated Z on a computer, but it is small enough that it could theoretically be built as a physical device, says Terence Tao of the University of California, Los Angeles. “If one were then to turn such a physical machine on, what we believe would happen would be that it would run indefinitely,” he says, assuming you ignore physical wear and tear or energy requirements.

No end in sight

Z is designed to loop through its 7918 instructions forever, but if it did eventually stop, it would prove ZFC inconsistent. Mathematicians wouldn’t be too panicked, though – they could simply shift to a slightly stronger set of axioms. Such axioms already exist, and could be used to prove the behaviour of Z, but there is little to be gained in doing so because there will always be a Turing machine to exceed any axiom.

“One can think of any given axiom system as being like a computer with a certain limited amount of memory or processing power,” says Tao. “One could switch to a computer with even more storage, but no matter how large an amount of storage space the computer has, there will still exist some tasks that are beyond its ability.”

But Aaronson and Yedidia have created two other machines that might give mathematicians more pause. These will stop only if two famous mathematical problems, long believed to be true, are actually false. These are Goldbach’s conjecture, which states that every even whole number greater than 2 is the sum of two prime numbers, and the Riemann hypothesis, which says that all prime numbers follow a certain pattern. The latter forms the basis for parts of modern number theory, and disproving it would be a major, if unlikely, upset.

Practical benefits

Practically, the pair have no intention of running their Turing machines indefinitely in an attempt to prove these problems wrong. It’s not a particularly efficient way to attack that problem, says Lance Fortnow of the Georgia Institute of Technology in Atlanta.

Expressing mathematical problems as Turing machines has a different practical benefit: it helps to work out how complex they are. The Goldbach machine has 4888 states, the Riemann one has 5372, while Z has 7918, suggesting the ZFC problem is the most complex of the three. “That would match most people’s intuitions about these sorts of things,” Aaronson says.

Yedidia has placed his code online, and mathematicians may now compete to reduce the size of these Turing machines, pushing them to the limit. Already a commenter on Aaronson’s SaiBlog claims to have created a 31-state Goldbach machine, although the pair have yet to verify this.

Fortnow says the actual size of the Turing machines are irrelevant. “The paper tells us that we can have very compressed descriptions that already go beyond the power of ZFC, but even if they were more compressed, it wouldn’t give us much pause about the foundations of math,” he says.

But Aaronson says shrinking Z further could say something interesting about the limits of the foundations of maths – something Gödel and Turing are likely to have wanted to know. “They might have said ‘that’s nice, but can you get 800 states? What about 80 states?’” Aaronson says. “I would like to know if there is a 10-state machine whose behaviour is independent of ZFC.”

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Jacob Aron*

Let’s lay down some definitions:

• Let FF be a field and let EE be an extension of FF. An element α∈Eα∈E is algebraicover FF if there is a polynomial f(x)f(x) in F[x]F[x] such that f(α)=0f(α)=0 (i.e. α∈Eα∈E is an algebraic element if it is the root of some polynomial with coefficients in FF).
• Let FF be a field and let EE be an extension of FF. If K⊂EK⊂E is a subset of EE which contains all elements which are algebraic over FF, then KK is actually a subfieldof EE and an algebraic extension of FF. We call KK the algebraic closure of FF and denote it by ¯¯¯¯FF¯. [1]

It’s a fact that an algebraic closure ¯¯¯¯FF¯ exists for every field FF (and is actually unique up to isomorphism). So we can draw a containment picture like the one above.

Those familar with some topology and/or analysis will notice that such a “field tower” is suggestive of a vaguely analogous result: given a topological space XX we can always (assuming some conditions about XX, namely it being locally-compact Hausdorff) stick an open set VV and its closure ¯¯¯¯VV¯ between a certain compact set KK and open set UU:  K⊂V⊂¯¯¯¯V⊂U.K⊂V⊂V¯⊂U.

Now, don’t buy too much into the analogy. I only mention this topological result to motivate the fact that the closure of a set and the algebraic closure of a field do indeed convey the same concept: wholeness. It seems then that we can view algebraic elements as the mathematical cousins of limit points of sequences of real numbers. Why? Because, topologically speaking, what is the closure of a set? The collection of limit points of that set, right? So in particular, when we let our topological space be RR, the set of real numbers (with the usual topology) and consider the subset {xn}∞n=1{xn}n=1∞ – some sequence of real numbers,

• we say x∈Rx∈R is a limit pointof {xn}{xn} if for every ϵ>0ϵ>0 there is an n∈Nn∈N such that |xn−x|<ϵ|xn−x|<ϵ.

Then in light of our comments above, we can make the analogous statement for a subset of polynomials {fn(x)}⊂F[x]⊂E[x]{fn(x)}⊂F[x]⊂E[x]:

• We say α∈Eα∈E is an algebraic element(over FF) if there is an n∈Nn∈N such that fn(α)=0fn(α)=0**.

Notice there’s no need for an approximation by ϵϵ in the second bullet. Why? Well, imagine placing a “metric” dd on E[x]E[x] by d:E[x]×E[x]→Ed:E[x]×E[x]→E via***

(So intuitively, f(x)f(x) is far away from αα if αα is not a root, but if αα is a root of f(x)f(x), then f(x)f(x) and αα are just as close as they can be.) In this way, the distance between an algebraic element and its corresponding polynomial is precisely 0. So in this case there’s no need to approximate a distance of zero by an arbitrarily small ϵϵ-ball – we have zero exactly!

And thus we have stumbled upon another insight into one of the main differences between analysis and algebra: you know the adage –

Analysts like inequalies; algebraists like equalities!

Digging Deeper

It would be interesting to see if there’s something in the language of category theory which allows one to see that closure of an algebraic field and closure of a topological set really are the same. Now I don’t know much about categories, but as one of my classmates recently suggested, we might want to look for a functor from the category of fields to the category of topological spaces such that the operation of closure is equivalent in each. In this case, perhaps it’s more appropriate to relate an algebraic closure to the completion of a topological space, as opposed to its closure. Admittedly, I’m not sure about all the details, but I think it’s worth looking into!

Footnotes:

* This is actually a bit deceiving. How we measure “closeness” really depends on the topology of the space we’re working on. For example, we can place the ray topology on RR so that the open sets are intervals of the form (a,∞)(a,∞) for a∈Ra∈R. Then in the strict definition of a limit point we see that -763 is a limit point of the interval (0,1)(0,1) even though it’s “far away”!

** Okay okay… there’s no reason to assume an arbitrary collection of polynomials is countable. I really should write FF for some family of polynomials in which case this statement would read “…if there is some f∈Ff∈F such that f(α)=0f(α)=0.” But bear with me for analogy’s sake.

*** I put “metric” in quotes here because as defined dd is not a metric in the strict sense of the word. Indeed, we don’t have the condition d(f(x),α)=0d(f(x),α)=0 if and only if “f(x)=αf(x)=α” since the latter is like comparing apples and oranges! But it would be interesting to see if we could place looser version of a metric on a polynomial ring. For instance, the way I’ve defined dd here, an open ball centered at αα would correspond to all polynomials in E[x]E[x] which have αα as a root! This idea seems to be related to Hilbert’s Nullstellensatz and the Zariski topology.

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*Credit for article given to Tai-Danae Bradley*