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The Beauty of Roots
John Baez, Dan Christensen and Sam Derbyshirewith lots of help from Greg Egan
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Definition. A Littlewood polynomial is a polynomial whosecoefficients are all 1 and -1.
Let’s draw all roots of all Littlewood polynomials!
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Certain regions seem particularly interesting:
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The hole at 1:
Note the line along the real axis: more Littlewood polynomialshave real roots than nearly real roots.
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The holes at i and eiπ/4:
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This plot is centered at the point 45 :
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This is centered at the point 45 i :
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This is centered at 12e i/5:
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Can we understand these pictures? Let
D = {z ∈ C : z is the root of some Littlewood polynomial}
Theorem 1. D ⊆ {1/2 < |z| < 2}
Proof. Suppose z is a root of a Littlewood polynomial. Then
1 = ±z ± z2 ± · · · ± zn
If |z| < 1 then
1 ≤ |z|+ |z|2 + · · ·+ |z|n <|z|
1− |z|
so |z| > 1/2. Since z is the root of a Littlewood polynomial ifand only if z−1 is, D is contained in the annulus 1
2 < |z| < 2.
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Can we understand these pictures? Let
D = {z ∈ C : z is the root of some Littlewood polynomial}
Theorem 1. D ⊆ {1/2 < |z| < 2}
Proof. Suppose z is a root of a Littlewood polynomial. Then
1 = ±z ± z2 ± · · · ± zn
If |z| < 1 then
1 ≤ |z|+ |z|2 + · · ·+ |z|n <|z|
1− |z|
so |z| > 1/2. Since z is the root of a Littlewood polynomial ifand only if z−1 is, D is contained in the annulus 1
2 < |z| < 2.
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Can we understand these pictures? Let
D = {z ∈ C : z is the root of some Littlewood polynomial}
Theorem 1. D ⊆ {1/2 < |z| < 2}
Proof. Suppose z is a root of a Littlewood polynomial. Then
1 = ±z ± z2 ± · · · ± zn
If |z| < 1 then
1 ≤ |z|+ |z|2 + · · ·+ |z|n <|z|
1− |z|
so |z| > 1/2. Since z is the root of a Littlewood polynomial ifand only if z−1 is, D is contained in the annulus 1
2 < |z| < 2.
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Theorem 2. {2−1/4 ≤ |z| ≤ 21/4} ⊆ D.
Proof. This was proved by Thierry Bousch in 1988. We won’tprove it here.
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Theorem 2. {2−1/4 ≤ |z| ≤ 21/4} ⊆ D.
Proof. This was proved by Thierry Bousch in 1988. We won’tprove it here.
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The closure D is easier to study than D. For example:
Theorem 3. D is connected.
Proof. This was proved by Bousch in 1993. Let’s sketch howthe proof works. It’s enough to show D ∩ {|z| < r} is connectedwhere r is slightly less than 1.
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The closure D is easier to study than D. For example:
Theorem 3. D is connected.
Proof. This was proved by Bousch in 1993. Let’s sketch howthe proof works. It’s enough to show D ∩ {|z| < r} is connectedwhere r is slightly less than 1.
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Definition. A Littlewood series is a power series all of whosecoefficients are 1 or −1.
Littlewood series converge for |z| < 1.
Lemma 1. A point z ∈ C with |z| < 1 lies in D if and only ifsome Littlewood series vanishes at this point.
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Definition. A Littlewood series is a power series all of whosecoefficients are 1 or −1.
Littlewood series converge for |z| < 1.
Lemma 1. A point z ∈ C with |z| < 1 lies in D if and only ifsome Littlewood series vanishes at this point.
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Definition. A Littlewood series is a power series all of whosecoefficients are 1 or −1.
Littlewood series converge for |z| < 1.
Lemma 1. A point z ∈ C with |z| < 1 lies in D if and only ifsome Littlewood series vanishes at this point.
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A Littlewood polynomial is not a Littlewood series! But anyLittlewood polynomial, say
P(z) = a0 + · · ·+ adzd
gives a Littlewood series having the same roots with |z| < 1:
P(z)
1− zd+1 = a0+· · ·+adzd +a0zd+1+· · ·+adz2d+1+a0z2d+2+· · ·
Thus D ⊆ R, where R is the set of roots of Littlewood series.
Our job is to show D = R.
To do this, let’s show that R is closed and D is dense in R.
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A Littlewood polynomial is not a Littlewood series! But anyLittlewood polynomial, say
P(z) = a0 + · · ·+ adzd
gives a Littlewood series having the same roots with |z| < 1:
P(z)
1− zd+1 = a0+· · ·+adzd +a0zd+1+· · ·+adz2d+1+a0z2d+2+· · ·
Thus D ⊆ R, where R is the set of roots of Littlewood series.
Our job is to show D = R.
To do this, let’s show that R is closed and D is dense in R.
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A Littlewood polynomial is not a Littlewood series! But anyLittlewood polynomial, say
P(z) = a0 + · · ·+ adzd
gives a Littlewood series having the same roots with |z| < 1:
P(z)
1− zd+1 = a0+· · ·+adzd +a0zd+1+· · ·+adz2d+1+a0z2d+2+· · ·
Thus D ⊆ R, where R is the set of roots of Littlewood series.
Our job is to show D = R.
To do this, let’s show that R is closed and D is dense in R.
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A Littlewood polynomial is not a Littlewood series! But anyLittlewood polynomial, say
P(z) = a0 + · · ·+ adzd
gives a Littlewood series having the same roots with |z| < 1:
P(z)
1− zd+1 = a0+· · ·+adzd +a0zd+1+· · ·+adz2d+1+a0z2d+2+· · ·
Thus D ⊆ R, where R is the set of roots of Littlewood series.
Our job is to show D = R.
To do this, let’s show that R is closed and D is dense in R.
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Let L be the set of Littlewood series. L ∼= {−1, 1}N, so with theproduct topology it’s homeomorphic to the Cantor set.
Choose 0 < r < 1. Let M be the space of finite multisets ofpoints with |z| ≤ r , modulo the equivalence relation generatedby S ∼ S ∪ {p} when |p| = r .
Lemma 2. Any Littlewood series has finitely many roots in thedisc {|z| ≤ r}. The map ρ : L → M sending a Littlewood seriesto its multiset of roots in this disc is continuous.
Since L is compact, the image of ρ is closed. From this we canshow that R, the set of roots of Littlewood series, is closed.
Since Littlewood polynomials are densely included in L and ρ iscontinuous, we can show that D, the set of roots of Littlewoodpolynomials, is dense in R.
It follows that D = R.
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Let L be the set of Littlewood series. L ∼= {−1, 1}N, so with theproduct topology it’s homeomorphic to the Cantor set.
Choose 0 < r < 1. Let M be the space of finite multisets ofpoints with |z| ≤ r , modulo the equivalence relation generatedby S ∼ S ∪ {p} when |p| = r .
Lemma 2. Any Littlewood series has finitely many roots in thedisc {|z| ≤ r}. The map ρ : L → M sending a Littlewood seriesto its multiset of roots in this disc is continuous.
Since L is compact, the image of ρ is closed. From this we canshow that R, the set of roots of Littlewood series, is closed.
Since Littlewood polynomials are densely included in L and ρ iscontinuous, we can show that D, the set of roots of Littlewoodpolynomials, is dense in R.
It follows that D = R.
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Let L be the set of Littlewood series. L ∼= {−1, 1}N, so with theproduct topology it’s homeomorphic to the Cantor set.
Choose 0 < r < 1. Let M be the space of finite multisets ofpoints with |z| ≤ r , modulo the equivalence relation generatedby S ∼ S ∪ {p} when |p| = r .
Lemma 2. Any Littlewood series has finitely many roots in thedisc {|z| ≤ r}. The map ρ : L → M sending a Littlewood seriesto its multiset of roots in this disc is continuous.
Since L is compact, the image of ρ is closed. From this we canshow that R, the set of roots of Littlewood series, is closed.
Since Littlewood polynomials are densely included in L and ρ iscontinuous, we can show that D, the set of roots of Littlewoodpolynomials, is dense in R.
It follows that D = R.
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Let L be the set of Littlewood series. L ∼= {−1, 1}N, so with theproduct topology it’s homeomorphic to the Cantor set.
Choose 0 < r < 1. Let M be the space of finite multisets ofpoints with |z| ≤ r , modulo the equivalence relation generatedby S ∼ S ∪ {p} when |p| = r .
Lemma 2. Any Littlewood series has finitely many roots in thedisc {|z| ≤ r}. The map ρ : L → M sending a Littlewood seriesto its multiset of roots in this disc is continuous.
Since L is compact, the image of ρ is closed. From this we canshow that R, the set of roots of Littlewood series, is closed.
Since Littlewood polynomials are densely included in L and ρ iscontinuous, we can show that D, the set of roots of Littlewoodpolynomials, is dense in R.
It follows that D = R.
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Let L be the set of Littlewood series. L ∼= {−1, 1}N, so with theproduct topology it’s homeomorphic to the Cantor set.
Choose 0 < r < 1. Let M be the space of finite multisets ofpoints with |z| ≤ r , modulo the equivalence relation generatedby S ∼ S ∪ {p} when |p| = r .
Lemma 2. Any Littlewood series has finitely many roots in thedisc {|z| ≤ r}. The map ρ : L → M sending a Littlewood seriesto its multiset of roots in this disc is continuous.
Since L is compact, the image of ρ is closed. From this we canshow that R, the set of roots of Littlewood series, is closed.
Since Littlewood polynomials are densely included in L and ρ iscontinuous, we can show that D, the set of roots of Littlewoodpolynomials, is dense in R.
It follows that D = R.
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Let L be the set of Littlewood series. L ∼= {−1, 1}N, so with theproduct topology it’s homeomorphic to the Cantor set.
Choose 0 < r < 1. Let M be the space of finite multisets ofpoints with |z| ≤ r , modulo the equivalence relation generatedby S ∼ S ∪ {p} when |p| = r .
Lemma 2. Any Littlewood series has finitely many roots in thedisc {|z| ≤ r}. The map ρ : L → M sending a Littlewood seriesto its multiset of roots in this disc is continuous.
Since L is compact, the image of ρ is closed. From this we canshow that R, the set of roots of Littlewood series, is closed.
Since Littlewood polynomials are densely included in L and ρ iscontinuous, we can show that D, the set of roots of Littlewoodpolynomials, is dense in R.
It follows that D = R.
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Lemma 3. The set R is connected.
Proof. It’s enough to show R ∩ {|z| < r} is connected, where2−1/4 < r < 1.
Suppose U ⊆ R ∩ {|z| < r} is closed and open in the relativetopology. We want to show U is empty.
Let LU be the set of Littlewood series with a root in U. LU is aclosed and open subset of L. Thus, we can determine whetherf ∈ L lies in LU by looking at its first d terms:
f (z) = a0 + a1z + · · ·+ ad−1zd−1 + adzd + · · ·
for some d . Choose the smallest d with this property.
We will get a contradiction if U, and thus LU , is nonempty! We’llshow d − 1 has the same property.
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Lemma 3. The set R is connected.
Proof. It’s enough to show R ∩ {|z| < r} is connected, where2−1/4 < r < 1.
Suppose U ⊆ R ∩ {|z| < r} is closed and open in the relativetopology. We want to show U is empty.
Let LU be the set of Littlewood series with a root in U. LU is aclosed and open subset of L. Thus, we can determine whetherf ∈ L lies in LU by looking at its first d terms:
f (z) = a0 + a1z + · · ·+ ad−1zd−1 + adzd + · · ·
for some d . Choose the smallest d with this property.
We will get a contradiction if U, and thus LU , is nonempty! We’llshow d − 1 has the same property.
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Lemma 3. The set R is connected.
Proof. It’s enough to show R ∩ {|z| < r} is connected, where2−1/4 < r < 1.
Suppose U ⊆ R ∩ {|z| < r} is closed and open in the relativetopology. We want to show U is empty.
Let LU be the set of Littlewood series with a root in U. LU is aclosed and open subset of L. Thus, we can determine whetherf ∈ L lies in LU by looking at its first d terms:
f (z) = a0 + a1z + · · ·+ ad−1zd−1 + adzd + · · ·
for some d . Choose the smallest d with this property.
We will get a contradiction if U, and thus LU , is nonempty! We’llshow d − 1 has the same property.
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Lemma 3. The set R is connected.
Proof. It’s enough to show R ∩ {|z| < r} is connected, where2−1/4 < r < 1.
Suppose U ⊆ R ∩ {|z| < r} is closed and open in the relativetopology. We want to show U is empty.
Let LU be the set of Littlewood series with a root in U. LU is aclosed and open subset of L. Thus, we can determine whetherf ∈ L lies in LU by looking at its first d terms:
f (z) = a0 + a1z + · · ·+ ad−1zd−1 + adzd + · · ·
for some d . Choose the smallest d with this property.
We will get a contradiction if U, and thus LU , is nonempty! We’llshow d − 1 has the same property.
![Page 34: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/34.jpg)
Lemma 3. The set R is connected.
Proof. It’s enough to show R ∩ {|z| < r} is connected, where2−1/4 < r < 1.
Suppose U ⊆ R ∩ {|z| < r} is closed and open in the relativetopology. We want to show U is empty.
Let LU be the set of Littlewood series with a root in U. LU is aclosed and open subset of L. Thus, we can determine whetherf ∈ L lies in LU by looking at its first d terms:
f (z) = a0 + a1z + · · ·+ ad−1zd−1 + adzd + · · ·
for some d . Choose the smallest d with this property.
We will get a contradiction if U, and thus LU , is nonempty! We’llshow d − 1 has the same property.
![Page 35: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/35.jpg)
Suppose f ∈ LU :
f (z) = a0 + a1z + · · ·+ ad−1zd−1 + adzd + · · ·
Suppose g has the same first d − 1 terms:
g(z) = a0 + a1z + · · ·+ ad−1zd−1 + bdzd + · · ·
We’ll show g ∈ LU too. Multiplying by −1 if necessary, we canassume a0 = 1:
f (z) = 1 + a1z + · · ·+ ad−1zd−1 + adzd + · · ·
There are two cases, ad = 1 and ad = −1. We’ll just do thefirst, since the second is similar. So:
f (z) = 1 + a1z + · · ·+ ad−1zd−1 + zd + · · ·
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Suppose f ∈ LU :
f (z) = a0 + a1z + · · ·+ ad−1zd−1 + adzd + · · ·
Suppose g has the same first d − 1 terms:
g(z) = a0 + a1z + · · ·+ ad−1zd−1 + bdzd + · · ·
We’ll show g ∈ LU too.
Multiplying by −1 if necessary, we canassume a0 = 1:
f (z) = 1 + a1z + · · ·+ ad−1zd−1 + adzd + · · ·
There are two cases, ad = 1 and ad = −1. We’ll just do thefirst, since the second is similar. So:
f (z) = 1 + a1z + · · ·+ ad−1zd−1 + zd + · · ·
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Suppose f ∈ LU :
f (z) = a0 + a1z + · · ·+ ad−1zd−1 + adzd + · · ·
Suppose g has the same first d − 1 terms:
g(z) = a0 + a1z + · · ·+ ad−1zd−1 + bdzd + · · ·
We’ll show g ∈ LU too. Multiplying by −1 if necessary, we canassume a0 = 1:
f (z) = 1 + a1z + · · ·+ ad−1zd−1 + adzd + · · ·
There are two cases, ad = 1 and ad = −1. We’ll just do thefirst, since the second is similar. So:
f (z) = 1 + a1z + · · ·+ ad−1zd−1 + zd + · · ·
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Suppose f ∈ LU :
f (z) = a0 + a1z + · · ·+ ad−1zd−1 + adzd + · · ·
Suppose g has the same first d − 1 terms:
g(z) = a0 + a1z + · · ·+ ad−1zd−1 + bdzd + · · ·
We’ll show g ∈ LU too. Multiplying by −1 if necessary, we canassume a0 = 1:
f (z) = 1 + a1z + · · ·+ ad−1zd−1 + adzd + · · ·
There are two cases, ad = 1 and ad = −1. We’ll just do thefirst, since the second is similar. So:
f (z) = 1 + a1z + · · ·+ ad−1zd−1 + zd + · · ·
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f (z) = 1 + a1z + · · ·+ ad−1zd−1 + zd + ad+1zd+1 + · · ·
has a root in U, and we want to show the same for any g ∈ Lwith the same first d − 1 terms, say
g(z) = 1 + a1z + · · ·+ ad−1zd−1 + bdzd + bd+1zd+1 + · · ·
We may assume g differs from f in its d th term:
g(z) = 1 + a1z + · · ·+ ad−1zd−1 − zd + bd+1zd + · · ·
It suffices to show that g̃ has a root in U:
g̃(z) =(
1 + a1z + · · ·+ ad−1zd−1)
/(
1 + zd)
since this has the same first d terms as g. Since f has a root inU, so does
f̃ (z) =(
1 + a1z + · · ·+ ad−1zd−1)
/(
1− zd)
since this has the same first d terms as f .
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f (z) = 1 + a1z + · · ·+ ad−1zd−1 + zd + ad+1zd+1 + · · ·
has a root in U, and we want to show the same for any g ∈ Lwith the same first d − 1 terms, say
g(z) = 1 + a1z + · · ·+ ad−1zd−1 + bdzd + bd+1zd+1 + · · ·
We may assume g differs from f in its d th term:
g(z) = 1 + a1z + · · ·+ ad−1zd−1 − zd + bd+1zd + · · ·
It suffices to show that g̃ has a root in U:
g̃(z) =(
1 + a1z + · · ·+ ad−1zd−1)
/(
1 + zd)
since this has the same first d terms as g. Since f has a root inU, so does
f̃ (z) =(
1 + a1z + · · ·+ ad−1zd−1)
/(
1− zd)
since this has the same first d terms as f .
![Page 41: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/41.jpg)
f (z) = 1 + a1z + · · ·+ ad−1zd−1 + zd + ad+1zd+1 + · · ·
has a root in U, and we want to show the same for any g ∈ Lwith the same first d − 1 terms, say
g(z) = 1 + a1z + · · ·+ ad−1zd−1 + bdzd + bd+1zd+1 + · · ·
We may assume g differs from f in its d th term:
g(z) = 1 + a1z + · · ·+ ad−1zd−1 − zd + bd+1zd + · · ·
It suffices to show that g̃ has a root in U:
g̃(z) =(
1 + a1z + · · ·+ ad−1zd−1)
/(
1 + zd)
since this has the same first d terms as g.
Since f has a root inU, so does
f̃ (z) =(
1 + a1z + · · ·+ ad−1zd−1)
/(
1− zd)
since this has the same first d terms as f .
![Page 42: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/42.jpg)
f (z) = 1 + a1z + · · ·+ ad−1zd−1 + zd + ad+1zd+1 + · · ·
has a root in U, and we want to show the same for any g ∈ Lwith the same first d − 1 terms, say
g(z) = 1 + a1z + · · ·+ ad−1zd−1 + bdzd + bd+1zd+1 + · · ·
We may assume g differs from f in its d th term:
g(z) = 1 + a1z + · · ·+ ad−1zd−1 − zd + bd+1zd + · · ·
It suffices to show that g̃ has a root in U:
g̃(z) =(
1 + a1z + · · ·+ ad−1zd−1)
/(
1 + zd)
since this has the same first d terms as g. Since f has a root inU, so does
f̃ (z) =(
1 + a1z + · · ·+ ad−1zd−1)
/(
1− zd)
since this has the same first d terms as f .
![Page 43: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/43.jpg)
g̃(z) =(
1 + a1z + · · ·+ ad−1zd−1)
/(
1 + zd)
andf̃ (z) =
(1 + a1z + · · ·+ ad−1zd−1
)/
(1− zd
)so
g̃(z) =
(1− zd
1 + zd
)f̃ (z)
Since f̃ has a root in U, so does g̃. QED!
![Page 44: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/44.jpg)
Here is the key to understanding the beautiful patterns in theset D. Define two functions from the complex plane to itself,depending on a complex parameter q:
f+q(z) = 1 + qzf−q(z) = 1− qz
When |q| < 1 these are both contraction mappings, so byHutchinson’s theorem on iterated function systems there’s aunique nonempty compact set Dq ⊆ C with
Dq = f+q(Dq) ∪ f−q(Dq)
We call this set a dragon.
Here’s the marvelous fact: the portion of D in a smallneighborhood of q ∈ C tends to look like Dq.
![Page 45: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/45.jpg)
Here is the key to understanding the beautiful patterns in theset D. Define two functions from the complex plane to itself,depending on a complex parameter q:
f+q(z) = 1 + qzf−q(z) = 1− qz
When |q| < 1 these are both contraction mappings, so byHutchinson’s theorem on iterated function systems there’s aunique nonempty compact set Dq ⊆ C with
Dq = f+q(Dq) ∪ f−q(Dq)
We call this set a dragon.
Here’s the marvelous fact: the portion of D in a smallneighborhood of q ∈ C tends to look like Dq.
![Page 46: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/46.jpg)
Here is the key to understanding the beautiful patterns in theset D. Define two functions from the complex plane to itself,depending on a complex parameter q:
f+q(z) = 1 + qzf−q(z) = 1− qz
When |q| < 1 these are both contraction mappings, so byHutchinson’s theorem on iterated function systems there’s aunique nonempty compact set Dq ⊆ C with
Dq = f+q(Dq) ∪ f−q(Dq)
We call this set a dragon.
Here’s the marvelous fact: the portion of D in a smallneighborhood of q ∈ C tends to look like Dq.
![Page 47: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/47.jpg)
For example, here’s the set D near q = 0.375453 + 0.544825i :
![Page 48: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/48.jpg)
And here’s the dragon Dq for q = 0.375453 + 0.544825i :
![Page 49: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/49.jpg)
Let’s zoom in on the set of roots of Littlewood polynomials ofdegree 20. When we zoom in enough, we’ll see it’s a discreteset!
Then we’ll increase the degree and see how the set ‘fills in’.
Then we’ll switch to a zoomed-in view of the correspondingdragon, and then zoom out.
![Page 50: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/50.jpg)
Let’s zoom in on the set of roots of Littlewood polynomials ofdegree 20. When we zoom in enough, we’ll see it’s a discreteset!
Then we’ll increase the degree and see how the set ‘fills in’.
Then we’ll switch to a zoomed-in view of the correspondingdragon, and then zoom out.
![Page 51: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/51.jpg)
Let’s zoom in on the set of roots of Littlewood polynomials ofdegree 20. When we zoom in enough, we’ll see it’s a discreteset!
Then we’ll increase the degree and see how the set ‘fills in’.
Then we’ll switch to a zoomed-in view of the correspondingdragon, and then zoom out.
![Page 52: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/52.jpg)
center 0.42065 + 0.48354i , height .62508, degree 20
![Page 53: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/53.jpg)
center 0.42065 + 0.48354i , height .31304, degree 20
![Page 54: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/54.jpg)
center 0.42065 + 0.48354i , height .15652, degree 20
![Page 55: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/55.jpg)
center 0.42065 + 0.48354i , height .07826, degree 20
![Page 56: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/56.jpg)
center 0.42065 + 0.48354i , height .03913, degree 20
![Page 57: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/57.jpg)
center 0.42065 + 0.48354i , height .019565, degree 20
![Page 58: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/58.jpg)
center 0.42065 + 0.48354i , height .0097825, degree 20
![Page 59: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/59.jpg)
center 0.42065 + 0.48354i , height .0048912, degree 20
![Page 60: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/60.jpg)
center 0.42065 + 0.48354i , height .0024456, degree 20
![Page 61: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/61.jpg)
center 0.42065 + 0.48354i , height .0024456, degree 21
![Page 62: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/62.jpg)
center 0.42065 + 0.48354i , height .0024456, degree 22
![Page 63: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/63.jpg)
center 0.42065 + 0.48354i , height .0024456, degree 23
![Page 64: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/64.jpg)
center 0.42065 + 0.48354i , height .0024456, degree 24
![Page 65: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/65.jpg)
center 0.42065 + 0.48354i , height .0024456, degree 25
![Page 66: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/66.jpg)
center 0.42065 + 0.48354i , height .0024456, degree 26
![Page 67: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/67.jpg)
center 0.42065 + 0.48354i , height .0024456, degree 27
![Page 68: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/68.jpg)
dragon for 0.42065 + 0.48354i , height .0024456
![Page 69: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/69.jpg)
dragon for 0.42065 + 0.48354i , height .0048912
![Page 70: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/70.jpg)
dragon for 0.42065 + 0.48354i , height .0097825
![Page 71: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/71.jpg)
dragon for 0.42065 + 0.48354i , height .019565
![Page 72: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/72.jpg)
dragon for 0.42065 + 0.48354i , height .03913
![Page 73: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/73.jpg)
dragon for 0.42065 + 0.48354i , height .07826
![Page 74: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/74.jpg)
dragon 0.42065 + 0.48354i , height .15652
![Page 75: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/75.jpg)
dragon for 0.42065 + 0.48354i , height .31304
![Page 76: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/76.jpg)
dragon for 0.42065 + 0.48354i , height .62508
![Page 77: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/77.jpg)
The set D is the set of roots of all Littlewood series. The set Dqis the set of values of all Littlewood series at the point q:
Theorem 4. For |q| < 1, Dq = {f (q) : f ∈ L}.
This is easy to show.
But why does D near q tend to resemble Dq?
![Page 78: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/78.jpg)
The set D is the set of roots of all Littlewood series. The set Dqis the set of values of all Littlewood series at the point q:
Theorem 4. For |q| < 1, Dq = {f (q) : f ∈ L}.
This is easy to show.
But why does D near q tend to resemble Dq?
![Page 79: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/79.jpg)
The set D is the set of roots of all Littlewood series. The set Dqis the set of values of all Littlewood series at the point q:
Theorem 4. For |q| < 1, Dq = {f (q) : f ∈ L}.
This is easy to show.
But why does D near q tend to resemble Dq?
![Page 80: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/80.jpg)
Each Littlewood series f maps q to a point f (q) ∈ Dq. For pnear q,
f (p) ≈ f (q) + f ′(q) (p − q)
Thus, we expect f (p) = 0 when
p − q ≈ − f (q)
f ′(q)
If this reasoning is good, this formula approximately givespoints p ∈ D near q from points f (q) ∈ Dq.
So, we should expect that near q, the set D will approximatelylook like a somewhat distorted copy of the dragon Dq, orsometimes a union of such copies.
We’re working on stating this precisely and proving it.
![Page 81: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/81.jpg)
Each Littlewood series f maps q to a point f (q) ∈ Dq. For pnear q,
f (p) ≈ f (q) + f ′(q) (p − q)
Thus, we expect f (p) = 0 when
p − q ≈ − f (q)
f ′(q)
If this reasoning is good, this formula approximately givespoints p ∈ D near q from points f (q) ∈ Dq.
So, we should expect that near q, the set D will approximatelylook like a somewhat distorted copy of the dragon Dq, orsometimes a union of such copies.
We’re working on stating this precisely and proving it.
![Page 82: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/82.jpg)
Each Littlewood series f maps q to a point f (q) ∈ Dq. For pnear q,
f (p) ≈ f (q) + f ′(q) (p − q)
Thus, we expect f (p) = 0 when
p − q ≈ − f (q)
f ′(q)
If this reasoning is good, this formula approximately givespoints p ∈ D near q from points f (q) ∈ Dq.
So, we should expect that near q, the set D will approximatelylook like a somewhat distorted copy of the dragon Dq, orsometimes a union of such copies.
We’re working on stating this precisely and proving it.
![Page 83: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/83.jpg)
Each Littlewood series f maps q to a point f (q) ∈ Dq. For pnear q,
f (p) ≈ f (q) + f ′(q) (p − q)
Thus, we expect f (p) = 0 when
p − q ≈ − f (q)
f ′(q)
If this reasoning is good, this formula approximately givespoints p ∈ D near q from points f (q) ∈ Dq.
So, we should expect that near q, the set D will approximatelylook like a somewhat distorted copy of the dragon Dq, orsometimes a union of such copies.
We’re working on stating this precisely and proving it.
![Page 84: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/84.jpg)
D near q = 45 i :
![Page 85: John Baez, Dan Christensen and Sam Derbyshire with lots of](https://reader034.vdocuments.us/reader034/viewer/2022050115/626d140b45163c77ab35ef55/html5/thumbnails/85.jpg)
union of distorted dragons for q = 45 i :