# Extended explanation of Orevkov’s paper on proper holomorphic embeddings of
complements of Cantor sets in $\mathbb{C}^{2}$ and a discussion of their
measure.

Giovanni D. Di Salvo

###### Abstract.

We clarify the details of a cryptical paper by Orevkov in which a construction
of a proper holomorphic embedding $\gamma\colon\mathbb{P}^{1}\setminus
C\hookrightarrow\mathbb{C}^{2}$ is presented; in particular, it is proved that
such a construction can be done to get the Cantor set $C$ to have zero
Hausdorff dimension.

##  1\. Introduction

###  1.1. The main result and motivation

A very relevant open problem in complex geometry is to investigate whether
every open Riemann surface $\mathcal{R}$ admits a proper holomorphic embedding
into $\mathbb{C}^{2}$. It has been suggested to search for a counterexample as
the complement of a ”thick” Cantor set $C$ inside the Riemann Sphere, that is
$\mathcal{R}=\overline{\mathbb{C}}\setminus C$. In paper [4], the author
proved the existence of a proper holomorphic embedding from the complement of
a Cantor set $C\subset\overline{\mathbb{C}}$ into $\mathbb{C}^{2}$.
Nevertheless no information dealing with the size of $C$ is given; moreover
such a paper is quite cryptical. The motivations for the present paper are
therefore two: to write in detail the construction of both the Cantor set
$C\subset\overline{\mathbb{C}}$ and the proper holomorphic embedding
$\gamma\colon\overline{\mathbb{C}}\setminus C\hookrightarrow\mathbb{C}^{2}$
originally presented in [4] and discussing the measure of $C$, proving in
particular that it can be constructed to have zero Hausdorff measure. From the
construction it will be rather clear that it is really hard to prove that this
$C$ could have any kind of positive measure. The main result of the present
paper, originally presented by S. Orevkov in [4] without the estimate on the
size of $C$, is the following

######  Theorem 1.1.

There exists a Cantor set $C\subset\overline{\mathbb{C}}$ with $\dim_{H}(C)=0$
and a proper holomorphic embedding $\gamma\colon\overline{\mathbb{C}}\setminus
C\hookrightarrow\mathbb{C}^{2}$.

It worth mentioning that in [5] has been proved that the complement of a thick
Cantor set inside $\overline{\mathbb{C}}$ does not constitute a
counterexample, as it is therein provided the construction of a proper
holomorphic embedding $\overline{\mathbb{C}}\setminus
C\hookrightarrow\mathbb{C}^{2}$ where $C\subset\overline{\mathbb{C}}$ is a
Cantor set of arbitrarily large measure. Namely: such $C$ is realized as a
subset of a square whose side has length $2$ and for every $\epsilon>0$ it can
be built of Lebesgue measure greater than $4-\epsilon$.

##  2\. Definitions and preliminary results

###  2.1. Cantor sets

The Cantor set is defined as the intersection of the decreasing family of
subsets of the unit interval of the real line $E_{0}=[0,1]\subset\mathbb{R}$
defined inductively by removing at each step the open middle third from each
interval of the previous set.
Thus $E_{1}=[0,\frac{1}{3}]\cup[\frac{2}{3},1]$,
$E_{2}=[0,\frac{1}{9}]\cup[\frac{2}{9},\frac{1}{3}]\cup[\frac{2}{3},\frac{7}{9}]\cup[\frac{8}{9},1]$
and so on. Finally one defines

$E:=\bigcap_{n\geq 0}E_{n}\subset[0,1].$

Given a topological space $X$, we define _Cantor set_ any subset $C\subset X$
homeomorphic to $E$. Hence a Cantor set is characterized by the topological
properties of $E$; thus, according to Brouwer’s theorem in [2], $C\subset X$
is a Cantor set if and only if it is

  1. (i)C

not-empty ;

  2. (ii)C

perfect (i.e. closed and with no isolated points) ;

  3. (iii)C

compact ;

  4. (iv)C

totally disconnected (i.e. every its connected component is a one-point set) ;

  5. (v)C

metrizable .

###  2.2. Sperichal Distance

The one point compactification of $\mathbb{R}^{n}$ is homeomorphic to
$\mathbb{S}^{n}=\\{(x_{1},\dots,x_{n+1})\in\mathbb{R}^{n+1}\;:\;x_{1}^{2}+\cdots+x_{n+1}^{2}=1\\}$
via stereographic projection:

$\displaystyle\varphi_{n}$
$\displaystyle\colon\overline{\mathbb{R}^{n}}\to\mathbb{S}^{n}\;,$
$\displaystyle\varphi_{n}(x)$
$\displaystyle=\frac{2}{\|x\|^{2}+1}\left(x,\frac{\|x\|^{2}-1}{2}\right),\;\;x\neq\infty\;;$
$\displaystyle\varphi_{n}(\infty)$ $\displaystyle=(0,\dots,0,1)\;.$

This allows us to define a notion of distance on any such
$\overline{\mathbb{R}^{n}}$ as

$d_{n}(x,y):=\arccos(\varphi_{n}(x)\cdot\varphi_{n}(y)),\;\;\;x,y\in\overline{\mathbb{R}^{n}}\;\;,$

(see e.g. [1], Theorem 5 pag. 444) where $\cdot$ denotes the scalar product in
$\mathbb{R}^{n+1}$ . The spaces $\overline{\mathbb{C}}$ and
$\overline{\mathbb{C}^{2}}$ are just the case $n=2$ and $n=4$ respectively, so
we have a notion of distance on them, allowing us to talk about diameter of
their subsets. Namely, for a connected subset
$\Omega\subseteq\overline{\mathbb{R}^{n}}$ we denote its diameter as
$|\Omega|:=\sup\\{d_{n}(x,y)\;:\;x,y\in\Omega\\}$ and if
$\Omega=\bigcup_{j}\omega_{j}$, where $\omega_{j}$ are pairwise disjoint
connected subsets of $\overline{\mathbb{R}^{n}}$ then we define
$|\Omega|:=\sup_{j}|\omega_{j}|$.

###  2.3. Definitions and technical results

We will consider the Riemann Sphere
$\overline{\mathbb{C}}:=\mathbb{C}\cup\\{\infty\\}$ and the one-point
compactification of $\mathbb{C}^{2}$ as
$\overline{\mathbb{C}^{2}}:=\mathbb{C}^{2}\cup\\{\infty_{2}\\}$.

Given a subset $B\subset\mathbb{C}^{2}$ and a positive number $\alpha>0$, we
define the open subset

$B(\alpha):=\\{z\in\mathbb{C}^{2}\;:\;\operatorname{dist}(z,B)<\alpha\\}\;.$

Given a subset $\Omega$ of $\overline{\mathbb{C}}$ (resp.
$\overline{\mathbb{C}^{2}}$), we say that $\Omega$ is _bounded_ if $\Omega\cap
U=\emptyset$ for some $U$ open neighborhood of $\infty$ (resp. of
$\infty_{2}$), _unbounded_ otherwise.

We consider $\\{\epsilon_{n}\\}_{n\geq 1},\\{R_{n}\\}_{n\geq 0}$ strictly
monotone sequences of positive real numbers, with $\epsilon\to 0$ and
$R_{n}\to+\infty$. Consider moreover $\\{a_{n}\\}_{n\geq 1}\subset\mathbb{C}$
such that $R_{n-1}<|a_{n}|<R_{n}$.
We will refer to these sequences as the _parameters_ of the construction.
Fixing these sequences, the sets

$\displaystyle D_{n}:=$ $\displaystyle\\{|z|<R_{n}\\}\subset\mathbb{C}\;\;$
$\displaystyle C_{n}^{v}:=$ $\displaystyle D_{n}\times\mathbb{C}\;\;$
$\displaystyle C_{n}^{h}:=$ $\displaystyle\mathbb{C}\times D_{n}\;\;$
$\displaystyle C_{n}:=$ $\displaystyle C_{n}^{v}\cup C_{n}^{h}\;\;$
$\displaystyle B_{n}:=$ $\displaystyle\overline{C_{n}^{v}\cap
C_{n}^{h}}=\overline{D}_{n}\times\overline{D}_{n}\;\;$

and the rational shears
$f_{n}\colon\mathbb{C}^{2}\rightarrow\overline{\mathbb{C}^{2}}$ ,

$\displaystyle f_{n}(x,y):=\begin{cases}\left(x,y+g_{n}(x)\right)\;\;\mbox{for
odd}\;\;n\;\;\\\ \left(x+g_{n}(y),y\right)\;\;\mbox{for
even}\;\;n\;\;,\end{cases}$

where

$g_{n}(z):=\frac{\epsilon_{n}}{z-a_{n}}\;,$

are defined. Finally define the sequence of continuous mappings
$\gamma_{n}\colon\overline{\mathbb{C}}\to\overline{\mathbb{C}^{2}}$ as

$\displaystyle\gamma_{n}(z)$ $\displaystyle:=f_{n}\circ\cdots\circ
f_{1}(z,0),\;\;z\neq\infty$ $\displaystyle\gamma_{n}(\infty)$
$\displaystyle:=\infty_{2}$

and consequently the sets

$A_{n}:=\gamma_{n}(\overline{\mathbb{C}})=f_{n}\circ\cdots\circ
f_{1}(\mathbb{C}\times\\{0\\})\cup\\{\infty_{2}\\}\;.$

The construction of the proper holomorphic embedding $\gamma$ and of the
Cantor set $C$ in Theorem 1.1 relies on the sets and functions just defined:
we will see that a suitable choice of the sequences
$\\{\epsilon_{n}\\}_{n},\;\\{R_{n}\\}_{n}$ and $\\{a_{n}\\}_{n}$ allows to
obtain $C$ as the closure of the set $\bigcup_{n}\gamma_{n}^{-1}(\infty_{2})$
and the limit $\lim_{n}\gamma_{n}$ to be a proper holomorphic embedding
$\gamma$ on $\overline{\mathbb{C}}\setminus C$ into $\mathbb{C}^{2}$; in
particular the faster $\\{R_{n}\\}_{n}$ diverges, the smaller the size of $C$,
so that it can be constructed to have zero Hausdorff dimension.

In the rest of this paper $\\{k_{n}\\}_{n\geq 0}$ will denote the Fibonacci
sequence, that is the sequence of natural numbers defined by $k_{0}=k_{1}=1$
and

$k_{n+2}=k_{n+1}+k_{n},\;\;\;\forall n\geq 0\;.$

The following remark gathers some known fact about the number of solutions of
rational equations (see 4.23–4.25, [3]), that will be fundamental in the proof
of the subsequent Lemma.

######  Remark 2.1.

Let $r\colon\overline{\mathbb{C}}\to\overline{\mathbb{C}}$ be a rational
function. There exists a natural number $k$, called _degree_ of $r$ and
denoted as $\deg r$, such that for every $a\in\overline{\mathbb{C}}$, the
equation $r(z)=a$ has $k$ solutions counted with their multiplicity, which are
multiple only for a finite number of $a\in\overline{\mathbb{C}}$.

  1. (1)

In particular, up to a slightly small perturbation of $a$, the zeros of
$r(z)-a$ are simple.

  2. (2)

The zeros of $r$ are as many as the poles, counted with their multiplicity,
therefore the poles of $r$ are all simple if and only if they are $k$
distinct.

  3. (3)

Let $t$ be another rational function and $w\in\overline{\mathbb{C}}$ fixed. Up
to perturb $a\in\mathbb{C}$ arbitrarily little, the equations $t=w$ and
$r-a=0$ have no common solutions, as it is easily seen observing that
$r^{-1}(a)\cap r^{-1}(a^{\prime})=\emptyset$ for any $a^{\prime}\neq a$.

######  Lemma 2.1.

Define a sequence of rational functions recursively:

$\displaystyle r_{0}(z)$ $\displaystyle:=z$ $\displaystyle r_{1}(z)$
$\displaystyle:=g_{1}(z)$ $\displaystyle r_{n+2}(z)$
$\displaystyle:=r_{n}(z)+g_{n+2}(r_{n+1}(z))=r_{n}(z)+\frac{\epsilon_{n+2}}{r_{n+1}(z)-a_{n+2}},\;\;n\geq
0.$ (2.1)

Then, up to perturbing the elements of any sequence
$\\{a_{n}\\}_{n}\subset\mathbb{C}$ arbitrarily little, the following hold:

  1. (i)

for every $n\geq 0$, $r_{n}$ and $r_{n+1}$ do not share any pole and
$r_{n}^{-1}(\infty)\subsetneq r_{n+2}^{-1}(\infty)$; in particular $\infty$ is
a pole only for $r_{n}$ with $n$ even;

  2. (ii)

the poles of $r_{n}$ are $k_{n}$ and all simple; therefore the equation

$r_{n}(z)=a_{n+1}$

has exactly $k_{n}$ solutions, all distinct.

###### Proof.

We see that $r_{n}^{-1}(\infty)\cap r_{n+1}^{-1}(\infty)=\emptyset$ for all
$n\geq 0$ by induction on $n$. The base case follows immediately. Assume the
claim true for $n$; if by contradiction $\exists w\in r_{n+1}^{-1}(\infty)\cap
r_{n+2}^{-1}(\infty)$, then by (2.1), it must be $r_{n}(w)=\infty$, so
$r_{n}^{-1}(\infty)\cap r_{n+1}^{-1}(\infty)\neq\emptyset$, against inductive
hypothesis.
The inclusion $r_{n}^{-1}(\infty)\subsetneq r_{n+2}^{-1}(\infty)$ for all
$n\geq 0$ follows by looking at (2.1) as from (3) in Remark 2.1, up to
slightly perturbing $a_{n+2}$, one gets
$\\{r_{n}=\infty\\}\cap\\{r_{n+1}-a_{n+2}=0\\}=\emptyset$.
Clearly $\infty$ is a pole for $r_{0}$, thus what said so far ensures that
$\infty$ is a pole for $r_{2k}$ and not a pole for $r_{2k+1}$, for all $k\geq
0$. So (i) is proved.

Let us prove (ii) by induction on $n$. This is trivial for $r_{0}$ and
$r_{1}$. Assume the claim true up to $n+1$. As observed in (3) in Remark 2.1,
up to a slightly small perturbation of $a_{n+2}$, the sets
$\\{r_{n}=\infty\\}$ and $\\{r_{n+1}-a_{n+2}=0\\}$ have empty intersection.
Hence, by (2.1), the poles of $r_{n+2}$ are precisely the poles of $r_{n}$
(which are $k_{n}$ and simple by inductive hypothesis) and the poles of
$\frac{\epsilon_{n+2}}{r_{n+1}-a_{n+2}}$ (which are $k_{n+1}$ and simple as
the zeros of the denominator are, up to a slightly small perturbation of
$a_{n+2}$ as pointed out in (1) in Remark 2.1, in fact $\deg r_{n+1}=k_{n+1}$
by inductive hypothesis). Therefore the poles of $r_{n+2}$ are
$k_{n}+k_{n+1}=k_{n+2}$ and all simple, hence by (2) in Remark 2.1 one has
$\deg r_{n+2}=k_{n+2}$. So up to a slightly small perturbation of $a_{n+3}$,
(1) guarantees that the equation $r_{n+2}(z)=a_{n+3}$ has $k_{n+2}$ solutions,
all distinct. So (ii) is proved. ∎

We will assume from now on to choose $R_{n}$ large enough, so that
$r_{n}(z)=a$ has $k_{n}$ distinct solutions for every $|a|\geq R_{n}$. The
following Lemma is used in Section 3.3:

######  Lemma 2.2.

Let $r$ be a rational function with a finite number of poles
$r^{-1}(\infty)=\\{w_{1},\dots,w_{N}\\}\subset\overline{\mathbb{C}}$, all
simple. Assume that $r(z)=a$ has $k$ distinct solutions for every $|a|\geq
T_{0}$, for some $T_{0}$. Then for any $T\geq T_{0}$ large enough, the set

$\Omega(r,T):=\\{z\in\overline{\mathbb{C}}\setminus
r^{-1}(\infty)\;:\;|r(z)|>T\\}$

is the disjoint union of $k$ connected sets, each of which homeomorphic to the
punctured open disk $\\{|z|<1\\}\setminus\\{0\\}$ and containing exactly one
solution of $r(z)=a$, for every $|a|>T$. Moreover, such $k$ sets can be taken
to have arbitrarily small diameters, provided that $T$ is large enough.

###### Proof.

Assume the poles of $r$ are all distinct; being them simple one has that
$N=k$. We express $r$ on a neighborhood $V_{j}\subset\overline{\mathbb{C}}$ of
a pole $w_{j}$ as $r(z)=h_{j}(z)(z-w_{j})^{-1}$ or $r(z)=h_{j}(z)z$,
accordingly with $w_{j}$ to be different or equal to $\infty$, where
$h_{j}\in\mathcal{O}(V_{j})$ is bounded and never vanishing. Let
$a\in\mathbb{C}$ with $|a|>T$. The larger $T$, the closer $z$ has to be to the
poles for $r(z)=a$ to be solved. In particular, up to enlarging $T$ we may
assume that the $V_{j}$ are pairwise disjoint, therefore $X_{j}:=\\{z\in
V_{j}\setminus\\{w_{j}\\}\;:\;|r(z)|>T\\}$ are pairwise disjoint and
connected. In particular for each $\epsilon>0$, there exists $T>0$
sufficiently large such that the diameter of $X_{j}$ is less than $\epsilon$.
Moreover the union of the $X_{j}$ is $\Omega(r,T)$ and each $X_{j}$ contains
precisely one of the $k$ solutions of the equation $r(z)=a$ for any $|a|>T$,
for $T$ sufficiently large: if not, there must be one of them, say $X_{1}$,
such that for any $T$ sufficiently large it contains at least two solutions of
$r(z)=a$ for some $|a|>T$, leading $w_{1}$ to be a multiple pole for $r$. ∎

Finally, it is easily seen by induction that

$\displaystyle\gamma_{n}=\begin{cases}\left(r_{n-1},r_{n}\right)\;\;\mbox{for
odd}\;\;n\;\;\\\ \left(r_{n},r_{n-1}\right)\;\;\mbox{for
even}\;\;n\;\;,\end{cases}$ (2.2)

in fact the case $n=1$ is immediate and for $n$ even, for example

$\gamma_{n+1}=f_{n+1}(\gamma_{n})=f_{n+1}(r_{n},r_{n-1})=(r_{n},r_{n-1}+g_{n+1}(r_{n}))=(r_{n},r_{n+1})\;.$

##  3\. General Properties

This section is devoted to the description of the objects introduced in
Section 2.3, which are the ingredients of the construction of the proper
holomorphic embedding $\gamma\colon\overline{\mathbb{C}}\setminus
C\hookrightarrow\mathbb{C}^{2}$ presented in [4]. In particular we will focus
on the choice of the three sequences $\\{\epsilon_{n}\\}_{n\geq
1},\\{R_{n}\\}_{n\geq 0},\\{a_{n}\\}_{n\geq 1}$ upon which the whole
construction is based.

###  3.1. On the set $\gamma_{n}^{-1}(\infty_{2})$ and the sequence
$\\{R_{n}\\}_{n\geq 0}$. Definition and properties of vertical and horizontal
components of $A_{n}$

It follows from (2.2) and (i) in Lemma 2.1 that

$\gamma_{n}^{-1}(\infty_{2})\subsetneq\gamma_{n+1}^{-1}(\infty_{2})$ (3.1)

holds true for all $n\geq 0$; moreover $\infty\in\gamma_{n}^{-1}(\infty_{2})$
for every $n\geq 0$, so we can write

$\gamma_{n}^{-1}(\infty_{2})=\\{\infty\\}\cup\\{t_{j}^{(n)}\\}_{j=1}^{b_{n}-1}$
(3.2)

for some $t_{j}^{(n)}\in\mathbb{C}$, where $b_{n}=k_{n+1}$. Observe that
$A_{n}=\gamma_{n}(\overline{\mathbb{C}})\subset\overline{\mathbb{C}^{2}}$ is
compact with $\infty_{2}\in A_{n}$; moreover
$A_{n}\setminus\\{\infty_{2}\\}=\gamma_{n}({\mathbb{C}})$ is a connected
complex curve in $\mathbb{C}^{2}$.

######  Proposition 3.1.

Assume $\epsilon_{1},\dots,\epsilon_{n}$, $a_{1},\dots,a_{n}$ and
$R_{1},\dots,R_{n-1}$ fixed, $|a_{j+1}|>R_{j}>|a_{j}|$ for $j=1,\dots,n-1$.
Then there exist $R_{n}>|a_{n}|$ such that what follows is true.

  1. (a)

$A_{n}\setminus(B_{n}\cup\\{\infty_{2}\\})$ has $b_{n}$ disjoint components.

  2. (b)

$A_{n}\subset C_{n}$.

We refer to the components of $A_{n}\setminus(B_{n}\cup\\{\infty_{2}\\})$
contained in $C_{n}^{h}$ (resp. in $C_{n}^{v}$) as the _horizontal_ (resp.
_vertical_) components of $A_{n}$.
Correspondingly we split $b_{n}$ as $h_{n}+v_{n}$. It turns out that

$\displaystyle\begin{cases}v_{n}=k_{n-1}\\\
h_{n}=k_{n}\end{cases}(n\;\;\text{even}),\hskip
28.45274pt\begin{cases}v_{n}=k_{n}\\\
h_{n}=k_{n-1}\end{cases}(n\;\;\text{odd}).$ (3.3)

###### Proof.

The components of $A_{n}\setminus(B_{n}\cup\\{\infty_{2}\\})$ are the image,
via $\gamma_{n}$ of punctured neighborhoods in $\overline{\mathbb{C}}$ of the
points (3.2) (which are defined once
$a_{n},\epsilon_{n},a_{j},\epsilon_{j},R_{j},\;\;j\leq n-1$ are fixed). So, by
Lemma 2.2, the bigger $R_{n}$, the smaller these neighborhoods; hence these
$b_{n}$ neighborhoods are disjoint for $R_{n}$ large enough, achieving (a), as
$\gamma_{n}$ is injective on
$\overline{\mathbb{C}}\setminus\gamma_{n}^{-1}(\infty_{2})$ (see section 3.2).
Looking at (2.2) it is clear that unbounded components of
$A_{n}=\gamma_{n}(\overline{\mathbb{C}})$ are due to the poles of $r_{n}$ and
$r_{n-1}$; being $r_{n}^{-1}(\infty)\cap r_{n-1}^{-1}(\infty)=\emptyset$ by
(i) in Lemma 2.1, it follows that when one component of $A_{n}$ becomes
unbounded, the other remains bounded. Hence, for $R_{n}$ large enough, we get
(b).
Finally, (3.3) follows directly from Lemma 2.1, once we express $\gamma_{n}$
as in (2.2). ∎

###  3.2. Definition of certain sets $\Delta_{n},\;K_{n}$ and some properties
of the mappings $\gamma_{n}$

Let us the define

$\Delta_{n}:=\gamma_{n}^{-1}(A_{n}\setminus B_{n})\;.$

Given $U\subset\overline{\mathbb{C}^{2}}$ open neighborhood of $\infty_{2}$,
$\gamma_{n}^{-1}(U)$ is a neighborhood of $\gamma_{n}^{-1}(\infty_{2})$; this
last set has $b_{n}$ points, therefore Lemma 2.2 guarantees that for $U$
sufficiently small $\gamma_{n}^{-1}(U)$ is the disjoint union of $b_{n}$ open
neighborhoods $U_{j}^{(n)}$ of the points (3.2) (meaning the points of $U$ are
sufficiently close to $\infty_{2}$ with respect to the spherical measure
introduced in section 2.2; equivalently $U$ is contained in the complement of
a closed ball in $\overline{\mathbb{C}^{2}}$ of sufficiently large radius). In
particular $\forall\epsilon>0\;\exists U\subset\overline{\mathbb{C}^{2}}$ open
neighborhood of $\infty_{2}$ such that $|U_{j}^{(n)}|<\epsilon$.
Since

$\Delta_{n}=\gamma_{n}^{-1}(\overline{\mathbb{C}^{2}}\setminus B_{n})$ (3.4)

(straightforward with double inclusion), we can always suppose to have $R_{n}$
large enough such that

$\Delta_{n}=\bigcup_{j=1}^{b_{n}}U_{j}^{(n)}$

is a disjoint union. In particular there exists $R_{n}$ so large that

$|U_{j}^{(n)}|<\frac{1}{b_{n}^{n}}\;,\;\;\;\;\forall j=1,\dots,b_{n}\;.$ (3.5)

Let us now check that

$\gamma_{n}\colon\overline{\mathbb{C}}\setminus\gamma_{n}^{-1}(\infty_{2})\to\mathbb{C}^{2}$

is a holomorphic embedding by induction on $n$. As
$\gamma_{1}\colon\overline{\mathbb{C}}\setminus\\{a_{1},\infty\\}\to\mathbb{C}^{2},\;\gamma_{1}(z)=(z,\frac{\epsilon_{1}}{z-a_{1}})$,
the base case trivially follows. Assume the claim true for $n$. Then
$\gamma_{n+1}=f_{n+1}\circ\gamma_{n}$ is a holomorphic embedding in the
intersection of the subset where $\gamma_{n}$ is such (that is
$\overline{\mathbb{C}}\setminus\gamma_{n}^{-1}(\infty_{2})$) and
$\\{z\in\overline{\mathbb{C}}\;:\;\gamma_{n}(z)\notin
f_{n+1}^{-1}(\infty_{2})\\}=\overline{\mathbb{C}}\setminus\gamma_{n+1}^{-1}(\infty_{2})$.
Such an intersection is

$\overline{\mathbb{C}}\setminus\gamma_{n}^{-1}(\infty_{2})\cap\overline{\mathbb{C}}\setminus\gamma_{n+1}^{-1}(\infty_{2})=\overline{\mathbb{C}}\setminus(\gamma_{n}^{-1}(\infty_{2})\cup\gamma_{n+1}^{-1}(\infty_{2}))=\overline{\mathbb{C}}\setminus\gamma_{n+1}^{-1}(\infty_{2})$

and we are done (the last equality follows from (3.1)).

Define then the following compacts in $\overline{\mathbb{C}}$:

$\displaystyle K_{n}$
$\displaystyle:=\overline{\mathbb{C}}\setminus\Delta_{n}=\gamma_{n}^{-1}(A_{n}\cap
B_{n})=\gamma_{n}^{-1}(B_{n}),\;\;n\geq 1$ $\displaystyle K_{0}$
$\displaystyle:=\emptyset\;.$

It is straighforward to see that
$K_{n}\subset\overline{\mathbb{C}}\setminus\gamma_{n}^{-1}(\infty_{2})$. We
will see in (3.7) that $K_{n}\subsetneq K_{n+1}^{\circ}$; therefore there
exists $\delta_{n}>0$ such that if
$h\colon\overline{\mathbb{C}}\setminus\gamma_{n}^{-1}(\infty_{2})\to\mathbb{C}^{2}$
is holomorphic with

$\|\gamma_{n}-h\|_{K_{n}}<\delta_{n}\;,$

then $h$ is an embedding on $K_{n-1}$. Set $\delta_{0}:=1/4$ and assume
without loss of generality $\\{\delta_{n}\\}_{n\geq 0}$ to be strictly
decreasing.

###  3.3. From $\Delta_{n}$ to $\Delta_{n+1}$: definition of pair of pants

We will prove that for a suitable choice of the parameters, the sequences
$\\{\Delta_{n}\\}_{n}$ and $\\{K_{n}\\}_{n}$ are strictly decreasing and
increasing respectively; in particular we will focus on the topological
behavior of the sequence $\\{\Delta_{n}\\}_{n}$, when passing from step $n$ to
step $n+1$.

######  Proposition 3.2.

Assume $\epsilon_{1},\dots,\epsilon_{n+1}$, $a_{1},\dots,a_{n+1}$ and
$R_{1},\dots,R_{n}$ fixed, with $|a_{j+1}|>R_{j}>|a_{j}|$ for $j=1,\dots,n$
and rename the $U_{j}^{(n)}$ to highlight which one comes from horizontal or
vertical components of $A_{n}\setminus B_{n}$ as

$\Delta_{n}=\bigcup_{j=1}^{h_{n}}H_{j}^{(n)}\;\;\cup\;\;\;\bigcup_{j=1}^{v_{n}}V_{j}^{(n)}=H_{n}\cup
V_{n}\;.$

Then, for any $R_{n+1}>|a_{n+1}|$ sufficiently large, we have that

$\displaystyle\Delta_{n+1}\subsetneq\Delta_{n}$ (3.6) $\displaystyle
K_{n}\subsetneq K_{n+1}^{\circ}\;$ (3.7)

and passing from $\Delta_{n}$ to $\Delta_{n+1}$, for $n$ even, $V_{j}^{(n)}$
shrinks to $V_{j}^{(n+1)}$, $j=1,\dots,v_{n}$, while $H_{j}^{(n)}$ splits
itself into $H_{j}^{(n+1)}$ and $V_{v_{n}+j}^{(n+1)}$, $j=1,\dots,h_{n}$,
creating the so called _pair of pants_. For $n$ odd, a similar conclusion
holds, swapping vertical and horizontal components.

###### Proof.

Let $n$ be even. From (2.1) and (2.2) we have

$\displaystyle\gamma_{n}$ $\displaystyle=(r_{n},r_{n-1})$
$\displaystyle\gamma_{n+1}$
$\displaystyle=(r_{n},r_{n+1})=(r_{n},r_{n-1})+\left(0,\frac{\epsilon_{n+1}}{r_{n}-a_{n+1}}\right)\;,$

from which

$\displaystyle H_{n}$ $\displaystyle=\\{|r_{n}|>R_{n}\\},\;\;$ $\displaystyle
V_{n}$ $\displaystyle=\\{|r_{n-1}|>R_{n}\\},$ $\displaystyle H_{n+1}$
$\displaystyle=\\{|r_{n}|>R_{n+1}\\}.$

Moreover, up to slightly move $a_{n+1}$, (3) in Remark 2.1 guarantees that
$r_{n-1}=\infty$ and $r_{n}=a_{n+1}$ have disjoint solutions, so for $R_{n+1}$
large enough we have

$V_{n+1}=\\{|r_{n-1}|>R_{n+1}\\}\cup\\{|r_{n}-a_{n+1}|<\epsilon_{n+1}/R_{n+1}\\}=:V^{\prime}_{n+1}\cup
V^{\prime\prime}_{n+1}\;,$

with $V^{\prime}_{n+1}\cap V^{\prime\prime}_{n+1}=\emptyset$. Since
$|a_{n+1}|>R_{n}$, it follows that $R_{n+1}$ sufficiently large implies
$V^{\prime\prime}_{n+1}\subseteq H_{n}$; the inclusions
$V^{\prime}_{n+1}\subseteq V_{n}$ and $H_{n+1}\subseteq H_{n}$ are trivial, so
we get (3.6) and (3.7), where the strict inclusions follow from (3.4) and
Lemma 2.2, which guarantees the $U_{j}^{(n+1)}$ shrunk arbitrarily little
around the point they are neighborhood of, provided $R_{n+1}$ to be large
enough. Taking $R_{n+1}$ large enough guarantees moreover that
$V^{\prime\prime}_{n+1}$ has $k_{n}$ components (as the equation
$r_{n}=a_{n+1}$ has $k_{n}$ simple solutions by (ii) in Lemma 2.1) and
$H_{n+1}\cap V^{\prime\prime}_{n+1}=\emptyset$ (as $R_{n+1}>|a_{n+1}|$). Now
$H_{n}$ has $k_{n}$ components, each of which splits into a component of
$H_{n+1}$ and a component of $V^{\prime\prime}_{n+1}$. So the statement on the
pair of pants follows.

For $n$ odd the argument is the same, just switching the role of vertical and
horizontal objects. ∎

###  3.4. The sequence $\\{\epsilon_{n}\\}_{n\geq 1}$

Describing constraints for $R_{n}$ in sections 3.1, 3.2 and 3.3, we assumed to
have already fixed $a_{1},\dots,a_{n}$, $R_{0},\dots,R_{n-1}$, and in
particular $\epsilon_{1},\dots,\epsilon_{n}$. Let us now see how to choose the
$\epsilon_{n}$.

######  Proposition 3.3.

There are sequences $\\{a_{n}\\}_{n}$, $\\{\epsilon_{n}\\}_{n}$ and
$\\{R_{n}\\}_{n}$ such that

$\|f_{n}-\operatorname{Id}\|_{\overline{B_{j}(\frac{1}{2})}}<\delta_{j}\cdot\frac{1}{2^{j+n}}\;\;,\hskip
56.9055pt0\leq j<n$ (3.8)

###### Proof.

Let us construct the three sequences inductively on $n$. Case $n=1$; take any
$a_{1}\in\mathbb{C}$ such that $|a_{1}|>1/2$, define $B_{0}$ as the origin
$(0,0)\in\mathbb{C}^{2}$. Then there exists $\epsilon_{1}>0$ such that
$\|f_{1}-\operatorname{Id}\|_{\overline{B_{0}(\frac{1}{2})}}<\delta_{0}\cdot\frac{1}{2^{1+0}}$
(note that $B_{0}(\frac{1}{2})$ is the $1/2$-ball in $\mathbb{C}^{2}$).
Assume (3.8) true for $n$. In particular $a_{i},\epsilon_{i},R_{i},\;i\leq
n-1$ and $a_{n},\epsilon_{n}$ are defined, so we can take $R_{n}$ to satisfy
all the constraints discussed in the previous sections; in particular in Lemma
2.1, Proposition 3.1 and 3.2, condition (3.5). So we can fix $a_{n+1}$ such
that $|a_{n+1}|>\sqrt{2}R_{n}+1/2$. Define then $\epsilon_{n+1}>0$ such that
$\|f_{n+1}-\operatorname{Id}\|_{\overline{B_{n}(\frac{1}{2})}}<\delta_{n}\cdot\frac{1}{2^{2n+1}}$.
Then (3.8) follows in the case $n+1$. ∎

With such a choice of parameters, we achieve

$\|f_{n}\circ\cdots\circ
f_{j}-\operatorname{Id}\|_{B_{j-1}}<\frac{1}{2}\;\;,\hskip 56.9055pt1\leq
j\leq n\;.$ (3.9)

The case $n=1$ follows from the base case of (3.8). Assume (3.9) true for $n$
and let us see it holds for $n+1$:

$\displaystyle\|f_{n+1}\circ\cdots\circ
f_{j}-\operatorname{Id}\|_{B_{j-1}}\leq$
$\displaystyle\|f_{n+1}-\operatorname{Id}\|_{f_{n}\circ\cdots\circ\
f_{j}(B_{j-1})}$ $\displaystyle+$
$\displaystyle\|f_{n}-\operatorname{Id}\|_{f_{n-1}\circ\cdots\circ\
f_{j}(B_{j-1})}+\cdots+\|f_{j}-\operatorname{Id}\|_{B_{j-1}}$
$\displaystyle\leq$
$\displaystyle\|f_{n+1}-\operatorname{Id}\|_{\overline{B_{j-1}(\frac{1}{2})}}$
(3.10) $\displaystyle+$
$\displaystyle\|f_{n}-\operatorname{Id}\|_{\overline{B_{j-1}(\frac{1}{2})}}+\cdots+\|f_{j}-\operatorname{Id}\|_{\overline{B_{j-1}(\frac{1}{2})}}$
$\displaystyle\leq$
$\displaystyle\frac{\delta_{j-1}}{2^{n+1+j-1}}+\cdots+\frac{\delta_{j-1}}{2^{j+j-1}}$
(3.11) $\displaystyle=$
$\displaystyle\delta_{j-1}\cdot\sum_{i=2j-1}^{n+j}\frac{1}{2^{i}}<\frac{1}{2}\;.$

Hence (3.9) holds for any $1\leq j\leq n+1$ as promised. Just observe that
(3.10) follows since

$f_{n}\circ\cdots\circ\ f_{j}(B_{j-1})\subset
B_{j-1}(1/2)\subset\overline{B_{j-1}(1/2)}$

holds true by inductive hypothesis for any $1\leq j\leq n$ and (3.11) follows
from Proposition 3.3.

######  Remark 3.1.

Since the three sequences underlying the whole construction depend one on the
other, it is important to highlight the order with which their elements have
to be taken. We have seen in sections 3.1, 3.2 and 3.3 that in order to define
$R_{n}$, the parameters

$a_{j},\epsilon_{j},R_{j},\;j\leq
n-1,\;\;\text{and}\;\;\;a_{n},\epsilon_{n}\;$

need to be already fixed; then we immediately fix $a_{n+1}$ such that
$|a_{n+1}|>\sqrt{2}R_{n}+1/2$. Similarly, in section 3.4 it turned out that in
order to define $\epsilon_{n}$,

$a_{j},\epsilon_{j},R_{j},\;j\leq n-1,\;\;\text{and}\;\;a_{n}$

need to be previously fixed; in particular, to choose $\epsilon_{1}$, some
$a_{1}\in\mathbb{C}$, $|a_{1}|>1/2$, has to be fixed. Namely, the order to
follow for choosing the sequences of parameters in order to perform our
construction is

$a_{1},\epsilon_{1},R_{1},a_{2},\epsilon_{2},R_{2},\dots\;.$

##  4\. The Cantor set $C$ and the proper holomorphic embedding $\gamma$

###  4.1. Definition of the Cantor set $C$

We are ready to define the object of our interest:

$C:=\bigcap_{n}\Delta_{n}\subset\overline{\mathbb{C}}.\\\ $ (4.1)

Let us see $C$ is indeed a Cantor set by proving that it fulfills all the
properties (i)C–(v)C stated in section 2. Being $\overline{\mathbb{C}}$
metrizable, we get (v)C. Since for every $n\in\mathbb{N}$ one has

$\gamma_{n}^{-1}(\infty_{2})\subset\gamma_{n+j}^{-1}(\infty_{2})\subset\Delta_{n+j}\;\;\;\;\forall
j\in\mathbb{N}\;,$

it follows that

$\bigcup_{n\geq 1}\gamma_{n}^{-1}(\infty_{2})\subseteq C\;,$ (4.2)

from which $C\neq\emptyset$ and thus we get (i)C. From the above construction,
every component of $\Delta_{n}$ eventually shrinks to a point (see e.g. (3.5))
and since the role of vertical and horizontal components switches at each
step, it follows that $\forall n\in\mathbb{N},\;\forall
j\in\\{1,\dots,b_{n}\\}$ there will be infinitely many pair of pants inside
each $U_{j}^{(n)}$, so there will be no isolated point. Moreover exploiting
(3.7) one gets

$\left(\overline{\mathbb{C}}\setminus
C\right)^{\circ}=\left(\bigcup_{n}K_{n}\right)^{\circ}=\bigcup_{n}K_{n}^{\circ}=\bigcup_{n}K_{n}=\overline{\mathbb{C}}\setminus
C$ (4.3)

from which $C$ is closed, and thus (ii)C is verified. Being $C$ closed and
$\overline{\mathbb{C}}$ compact, it follows that $C$ is compact itself, so we
have (iii)C. Finally, if any connected component of $C$ had more than one
point, then (3.5) would fail for $n$ big enough. So $C$ is totally
disconnected, that is we have (iv)C and so we have proved $C$ is indeed a
Cantor set.

###  4.2. Definition of the proper holomorphic embedding
$\gamma\colon\overline{\mathbb{C}}\setminus C\hookrightarrow\mathbb{C}^{2}$

From time to time we will use the following fact

######  Remark 4.1.

If $\\{K_{n}\\}_{n}$ is a normal exhaustion for a domain
$\Omega\subseteq\overline{\mathbb{C}}$ and $\\{z_{n}\\}_{n}\subset\Omega$
converges to $z_{0}\in\partial\Omega$, then we can suppose without loss of
generality that $z_{n}\in K_{n}\setminus K_{n-1}$.

Consider $\gamma_{n}$ as a mapping $K_{n+1}^{\circ}\to\mathbb{C}^{2}$; thus
the natural domain to define a limit mapping is $\bigcup_{n\geq
1}K_{n}^{\circ}$ which equals $\overline{\mathbb{C}}\setminus C$ from (4.3).
For every $n\geq k$ one has

$\displaystyle\|\gamma_{n+1}-\gamma_{n}\|_{K_{k}}$
$\displaystyle=\|(f_{n+1}-\operatorname{Id})\circ\gamma_{n}\|_{\gamma_{k}^{-1}(B_{k})}$
$\displaystyle=\|(f_{n+1}-\operatorname{Id})\circ(f_{n}\circ\cdots\circ
f_{k+1}\circ\gamma_{k})\|_{\gamma_{k}^{-1}(B_{k})}$
$\displaystyle\leq\|f_{n+1}-\operatorname{Id}\|_{f_{n}\circ\cdots\circ
f_{k+1}(B_{k})}$
$\displaystyle\leq\|f_{n+1}-\operatorname{Id}\|_{\overline{B_{k}(\frac{1}{2})}}$
by (3.9) $\displaystyle<\delta_{k}\cdot\frac{1}{2^{n+1+k}}$
$\displaystyle\mbox{by (\ref{eps})}\;\;,$

from which it follows that

$\sum_{n\geq k}\|\gamma_{n+1}-\gamma_{n}\|_{K_{k}}<\sum_{n\geq
k}\delta_{k}\cdot\frac{1}{2^{n+1+k}}<+\infty\;$

for every fixed $k\geq 1$; this proves that $\\{\gamma_{n}\\}_{n}$ converges
uniformly on compact subsets of $\overline{\mathbb{C}}\setminus C$, in fact
this last set is $\bigcup_{n}K_{n}$ and $\\{K_{n}\\}_{n}$ is a normal
exhaustion of $\overline{\mathbb{C}}\setminus C$. We have thus defined a
holomorphic mapping

$\gamma\colon\overline{\mathbb{C}}\setminus C\to\mathbb{C}^{2}\;.$

It is now easily seen that $\gamma$ is a local embedding (namely: an embedding
on every $K_{k-1}$) since for every $k\geq 1$ one has

$\|\gamma-\gamma_{k}\|_{K_{k}}\leq\sum_{n\geq
k}\|\gamma_{n+1}-\gamma_{n}\|_{K_{k}}<\sum_{n\geq
k}\delta_{k}\cdot\frac{1}{2^{n+1+k}}<\delta_{k}\;.$

Finally, global embedding property of $\gamma$ will follow automatically once
properness is proved, achieving thus that $\gamma$ is a proper holomorphic
embedding from the complement of a Cantor set into $\mathbb{C}^{2}$, as
wanted. Consider then $\\{z_{n}\\}_{n}\subset\overline{\mathbb{C}}\setminus
C=\bigcup_{n}K_{n}$ such that $z_{n}\to\partial(\overline{\mathbb{C}}\setminus
C)=\partial C$. By Remark 4.1 we can assume $z_{n}\in K_{n}\setminus
K_{n-1}\subseteq\Delta_{n-1}$, from which $|\gamma_{n-1}(z_{n})|\to+\infty$ as
$n\to+\infty$. Nevertheless this does not imply that
$|\gamma(z_{n})|\to+\infty$, as it just claims the divergence of
$\\{\gamma_{n-1}(z_{n})\\}_{n}$ and no information on how the sequence
$\\{\gamma_{n}\\}_{n}$ acts on the elements of $\\{z_{n}\\}_{n}$ is given. In
particular, we need to achieve $z_{n}$ to be set outside some $R_{n}$-ball
(say $B_{n-3}$) by a whole tail of $\\{\gamma_{n}\\}_{n}$ (say $\gamma_{n+j}$
for every $j\in\mathbb{N}$). In this way the possibility for $z_{n}$ to come
back inside a ball is prevented and we obtain properness of the limit function
$\gamma$. Hence properness will be achieved if, for instance, the following
condition is satisfied:

$\gamma_{n+j}(K_{n}\setminus K_{n-1})\subset\overline{\mathbb{C}}^{2}\setminus
B_{n-3}\hskip 56.9055pt\forall n\geq 3,\;j\in\mathbb{N}\;.$ (4.4)

As we assumed $z_{n}\in K_{n}\setminus K_{n-1}\;\;\forall n$, one has

$\gamma(z_{n})=\lim_{j}\gamma_{j+n}(z_{n})\stackrel{{\scriptstyle(\ref{proper})}}{{\in}}\overline{\mathbb{C}}^{2}\setminus
B_{n-3}\;,$

which implies $|\gamma(z_{n})|\geq R_{n-3}$ and thus
$\lim_{n}|\gamma(z_{n})|=+\infty$, as wanted. Notice that

$\displaystyle\gamma_{n}(\gamma_{n}^{-1}(B_{n}))=A_{n}\cap B_{n}\;$ (4.5)

holds true for every $n\in\mathbb{N}$ and this allows to prove that (4.4)
holds, in fact

$\displaystyle\gamma_{n+j}(K_{n}\setminus K_{n-1})$
$\displaystyle=f_{n+j}\circ\cdots\circ
f_{n+1}\circ\gamma_{n}\left(\gamma_{n}^{-1}(B_{n})\setminus\gamma_{n-1}^{-1}(B_{n-1})\right)$
$\displaystyle\mbox{by}\;\;(\ref{mid})$ $\displaystyle=f_{n+j}\circ\cdots\circ
f_{n+1}\left((A_{n}\cap B_{n})\setminus(f_{n}(B_{n-1})\cap A_{n})\right)$
$\displaystyle\subseteq f_{n+j}\circ\cdots\circ f_{n+1}\left(B_{n}\setminus
f_{n}(B_{n-1})\right)$ $\displaystyle\subseteq f_{n+j}\circ\cdots\circ
f_{n+1}\left(B_{n}\setminus B_{n-2}\right)$ by (3.9) $\displaystyle\subseteq
B_{n+1}\setminus B_{n-3}$ by (3.9)
$\displaystyle\subseteq\overline{\mathbb{C}}^{2}\setminus B_{n-3}\;,$

as promised.

###  4.3. On the Hausdorff dimension of $C$ and proof of Theorem 1.1

Fix $\epsilon>0$; the _Hausdorff $\epsilon$-measure_ of $C$ is, by definition

$\displaystyle\mathcal{H}^{\varepsilon}(C):$ $\displaystyle=\liminf_{r\to
0}\left\\{\inf\left\\{\sum_{i}|T_{i}|^{\varepsilon}\;:\;C\subseteq\bigcup_{i}T_{i},\;\;|T_{i}|<r\right\\}\right\\}\;.$

Since $C\subset\Delta_{n}=\bigcup_{j=1}^{b_{n}}U_{j}^{(n)}$ holds true for any
$n$ and since for every $r>0$ there exists $N_{r}$ such that $1/b_{n}^{n}<r$
for all $n\geq N_{r}$, exploiting (3.5), it trivially follows that

$\displaystyle\inf\left\\{\sum_{i}|T_{i}|^{\varepsilon}\;:\;C\subseteq\bigcup_{i}T_{i},\;\;|T_{i}|<r\right\\}$
$\displaystyle\leq\inf\left\\{\sum_{j=1}^{b_{n}}|U_{j}^{(n)}|^{\varepsilon}\;:\;\;n\geq
N_{r}\right\\}$ $\displaystyle\leq\inf\left\\{\frac{1}{b_{n}^{\epsilon
n-1}}\;:\;\;n\geq N_{r}\right\\}=0\;,$

hence $\mathcal{H}^{\varepsilon}(C)=0$ for any $\epsilon>0$ fixed. Therefore,
recalling that the _Hausdorff dimension_ of $C$ is defined as

$\dim_{H}(C):=\inf\\{d\geq 0\;:\;\mathcal{H}^{d}(C)=0\\}\;,$

we conclude that

$\dim_{H}(C)=0\;.$

This last observation, together with the construction of the proper
holomorphic embedding $\gamma:\overline{\mathbb{C}}\setminus
C\hookrightarrow\mathbb{C}^{2}$ made in section 4.2, completes the proof of
Theorem 1.1.

###  4.4. Two further characterizations of $C$

The original definition of the Cantor set $C$ given in [4] is

$C=\gamma^{-1}(\infty_{2})\;.$ (4.6)

Actually $\gamma$ is not defined on the Cantor set $C$, therefore, in order
for this last relation to make sense, we need to extend $\gamma$ on $C$. Being
$C$ a Cantor set in $\overline{\mathbb{C}}$, it is the boundary of
$\overline{\mathbb{C}}\setminus C$, domain on which $\gamma$ is proper,
therefore

$\lim_{w\to C,w\in\overline{\mathbb{C}}\setminus C}|\gamma(w)|=\infty\;,$

in fact if $z\in C$, being $C$ a Cantor set, $z$ is an accumulation point of
the complement of $C$, which is $\bigcup_{n}K_{n}$, thus $z$ can be hit by a
sequence $\\{z_{n}\\}_{n}\subset\bigcup_{n}K_{n}$; by Remark 4.1 let $z_{n}\in
K_{n}\setminus K_{n-1}$. On one hand it is clear that
$\lim_{j}\gamma_{n+j}(z_{n})=\gamma(z_{n})$ for every $n$, on the other hand
(4.4) implies $\lim_{n}\lim_{j}\gamma_{n+j}(z_{n})=\infty_{2}\;$. So we can
extend $\gamma$ on $C$ by continuity defining $\gamma\equiv\infty_{2}$ on $C$.
Since trivially $\gamma\neq\infty_{2}$ on $\overline{\mathbb{C}}\setminus C$,
we have that defining $C$ as in (4.6) matches with the definition (4.1) given
in Section 4.1. We conclude by characterizing $C$ in one last way:

$C=\operatorname{cl}\left(\bigcup_{n\geq
1}\gamma_{n}^{-1}(\infty_{2})\right)\;.$ (4.7)

Being $C$ closed, (4.2) implies that

$\operatorname{cl}\left(\bigcup_{n\geq
1}\gamma_{n}^{-1}(\infty_{2})\right)\subseteq C\;.$

Consider now the following inductive procedure: at step $n$ we have $b_{n}$
points, the elements of $\gamma_{n}^{-1}(\infty_{2})$, and $b_{n}$
neighborhoods $U_{j}^{(n)}$ of them. At step $n+1$ we have the old $b_{n}$
points and $k_{n}$ new points (that is, the elements of
$\gamma_{n+1}^{-1}(\infty_{2})$) and $b_{n+1}=b_{n}+k_{n}$ neighborhoods
$U_{j}^{(n+1)}$ of them, $b_{n}$ of which are just the $U_{j}^{(n)}$ shrunk
around the points of $\gamma_{n}^{-1}(\infty_{2})$ and the $k_{n}$ remaining
ones are neighborhoods of the new points
$\gamma_{n+1}^{-1}(\infty_{2})\setminus\gamma_{n}^{-1}(\infty_{2})$. Now
$\cup_{j=1}^{b_{n}}U_{j}^{(n)}=\Delta_{n}$ forms a decreasing sequence of open
sets, whose intersection defines $C$ and all the $U_{j}^{(n)}$ eventually
shrink to the point they are neighborhood of, hence $C$ is the limit of the
above inductive procedure. Therefore, given $w\in C$, either it appears at
some finite step of the procedure (hence $w\in\gamma_{n}^{-1}(\infty_{2})$ for
some $n$), or it appears in the limit: by construction, the points tend to
accumulate (in fact there is no isolated point, see (ii)C), that is
$w\in\operatorname{Acc}\left(\bigcup_{n\geq
1}\gamma_{n}^{-1}(\infty_{2})\right)\setminus\left(\bigcup_{n\geq
1}\gamma_{n}^{-1}(\infty_{2})\right)$. So
$w\in\operatorname{cl}\left(\bigcup_{n\geq
1}\gamma_{n}^{-1}(\infty_{2})\right)$ and we are done.

## References

  * [1] D. A. Brannan, M. F. Esplen and J. J. Gray; Geometry. Cambridge University Press, Second Edition (2012).
  * [2] L.E.J. Brouwer; On the structure of perfect sets of points. KNAW, Proceedings 12, (1909-1910), Amsterdam, 1910, pp. 785–794.
  * [3] O. Forster; Lectures on Riemann Surfaces. Springer–Verlag, New York Inc. 1981.
  * [4] S. Y. Orevkov; Proper analytic embedding of $\mathbb{C}\mathbb{P}^{1}$ minus Cantor set into $\mathbb{C}^{2}$. Uspekhi Mat. Nauk. 63 (2008), no. 1(379), 155–156 (in Russian) English transl.: Russian Math. Surveys, 63 (2008), 168–169.
  * [5] E. F. Wold and G. D. Di Salvo; Proper holomorphic embeddings of complements of large Cantor sets in $\mathbb{C}^{2}$.


# GENERATIVE PRE-TRAINED TRANSFORMERS FOR BIOLOGICALLY INSPIRED DESIGN

Qihao Zhu
Data-Driven Innovation Lab
Singapore University of Technology and Design
<EMAIL_ADDRESS>
Xinyu Zhnag
State Key Laboratory of Automotive Safety and Energy
Tsinghua University
<EMAIL_ADDRESS>
Jianxi Luo
Data-Driven Innovation Lab
Singapore University of Technology and Design
<EMAIL_ADDRESS>

(March 31, 2022)

###### Abstract

Biological systems in nature have evolved for millions of years to adapt and
survive the environment. Many features they developed can be inspirational and
beneficial for solving technical problems in modern industries. This leads to
a novel form of design-by-analogy called bio-inspired design (BID). Although
BID as a design method has been proven beneficial, the gap between biology and
engineering continuously hinders designers from effectively applying the
method. Therefore, we explore the recent advance of artificial intelligence
(AI) for a computational approach to bridge the gap. This paper proposes a
generative design approach based on the pre-trained language model (PLM) to
automatically retrieve and map biological analogy and generate BID in the form
of natural language. The latest generative pre-trained transformer, namely
GPT-3, is used as the base PLM. Three types of design concept generators are
identified and fine-tuned from the PLM according to the looseness of the
problem space representation. Machine evaluators are also fine-tuned to assess
the correlation between the domains within the generated BID concepts. The
approach is then tested via a case study in which the fine-tuned models are
applied to generate and evaluate light-weighted flying car concepts inspired
by nature. The results show our approach can generate BID concepts with good
performance.

_Keywords_ Bio Inspired Design $\cdot$ Computer Aided Conceptual Design
$\cdot$ Design Automation $\cdot$ Generative Design $\cdot$ Innovative Design
Methods

##  1 INTRODUCTION

Bio-inspired design (BID) is a relatively novel design methodology in the
engineering design field. It is based on the observation of biological natural
phenomena (ISO/TC266,, 2015) and aims to develop novel solutions for real-
world problems with analogies from nature (Helms et al., 2009). For millions
of years, creatures on earth managed to evolve features to survive the
environments and compete with other species, and some of them could be crucial
for solving the technical challenges we are facing today. In a bullet train
example (Linic et al., 2020), engineers observed that kingfishers are capable
of diving into the water from air with extremely high speed without making a
splash and took the kingfisher’s beak shape as an analogy to develop the head
of the bullet train. This improvement not only removed the sonic boom caused
by the train traveling at high speed through a tunnel, but also resulted in a
faster speed and lower energy consumption.

Although the method has been proven helpful and become popular among designers
and researchers, it is challenging for designers to find, recognize, and
understand biological information in design practice because they lack the
necessary knowledge and experience in the biology field (Vattam and Goel,
2013; Kruiper et al., 2018). For such challenges related to knowledge and
information process, a data-driven approach and knowledge-based artificial
intelligence can aid human designers for BID (Jiang et al., 2022). Recent
advances in deep learning-based show particular promises if we can properly
apply them to biological data and design problems.

In this paper, we apply one of the latest natural language processing (NLP)
techniques to help designers innovate with BID. We propose a novel approach
using the pre-trained language model (PLM) to automate the retrieval and
mapping of biological information through text generation. The generated
concepts can be evaluated through classifiers fine-tuned from the same base
PLM. This could generate and evaluate many BID concepts in a very short
period, and the natural language representation of the results are
understandable for designers. We further validate the approach in a flying car
design project and tested its performance to solve a real-world technical
problem together with a team of flying car experts.

##  2 BACKGROUND AND RELATED WORKS

###  2.1 Design-by-Analogy

Design-by-analogy (DbA) is a design methodology that innovates the design
target through the analogy with a different domain (Goel, 1997), which is
proven effective to help overcome design fixation (Linsey et al., 2010).
Analogy in design is generally defined as “illustration of an idea by means of
another familiar idea that is similar or parallel to it in some significant
features” (Hey et al., 2008). This indicates two underlying domains in analogy
reasoning. So far, researchers have identified these two domains in DbA, the
domain that contains the problem and needs to be understood is called the
target domain, while the one that provides a potential solution to the problem
is called the source domain (Goel, 1997; Hey et al., 2008; Linsey et al.,
2008).

Many prior studies have investigated the cognitive process and reasoning
behind DbA, and they basically agree on three important subphases: retrieval,
mapping, and evaluation (Hey et al., 2008; Linsey et al., 2008; Kokinov and
French, 2003; Hall, 1989). The retrieval of source for analogy depends on the
relational similarity between domains (Linsey et al., 2008; Gentner and
Markman, 1997; Verhaegen et al., 2011). This means that the retrieved source
domain should share a common relationship with the target domain regarding how
they are comprised by their components. Then, mapping between the target and
the retrieved source is established and evaluated. This is usually done by
comparing the elements and patterns between domains (Holyoak, 1996; Bhatta and
Goel, 1996).

In recent years, many data-driven design-by-analogy methods and tools have
been proposed. Readers may refer to Shuo et al. (Jiang et al., 2022) for the
latest and comprehensive review of data-driven design-by-analogy research. In
this study, we focus on bio-inspired design (BID), which is a special form of
DbA, and explore a pre-trained language model to automate the retrieval,
mapping, and evaluation of biological analogies for design concept generation.

###  2.2 Bio-Inspired Design

Bio-inspired design (BID) is an innovation method whereby designers take
inspiration from biological phenomena. Although some researchers argue that
BID goes beyond analogy (ISO/TC266,, 2015; Vincent et al., 2006; Shu et al.,
2011). The methodology has grown into an innovative design process in the
field of engineering design which leverages analogies to biological systems to
develop solutions for engineering challenges (Helms et al., 2009).

According to literatures, there are two main approaches for bio-inspired
design: problem-driven and solution-based (Fayemi et al., 2017; Badarnah and
Kadri, 2015). The problem-driven approach is a biomimetic development method
that aims to address a practical problem, with the problem serving as the
process’s starting point, while the solution-based approach seeks inspiration
from the observation of nature and allows the identification of a design
problem, and therefore the knowledge of a biological system of interest serves
as the basis for design. For this paper, we focus on the problem-driven
approach as it better fits the general design process model proposed by Pahl
et al. (2007).

Researchers have identified three domains in problem-driven BID: problem
domain, nature domain, and solution domain (Badarnah and Kadri, 2015).
Generally, the problem domain and solution domain represent the situations of
the target domain before and after analogy mapping, while the nature domain is
corresponding to the source domain which indicates the biological features
used as analogy source. However, it is universally acknowledged that the
transitions between the three domains are challenging due to the distinctions
between engineering and biology (Kruiper et al., 2018; Salgueiredo and
Hatchuel, 2016; Vincent and Mann, 2002). Therefore, many contributions in
recent years have explored methods and tools trying to bridge the three
domains in BID.

The transition from problem domain to nature domain includes the abstraction
of the engineering problem and the identification of the biological system
(Vincent and Mann, 2002; Wanieck et al., 2017). For abstraction, Nagel et al.
(2010) employ the functional model of the desired engineering system for
representing the essence of functionality, which is then used to explore
biological solutions for inspiration. Cheong et al. (2011) propose a natural
language processing-based process model that begins with the formulation of
original functional terms to represent a problem. For biological system
identification, AskNature (Deldin and Schuknecht, 2014) groups biological
strategies and innovations by function according to the Biomimicry Taxonomy,
so that it encodes biological information in engineering terms. Another
popular attempt on this issue is to search or push biology keywords based on
technical problems. Shu and Cheong (2014) explore a natural language approach
to search for biology keywords for analogy. Chen et al. (2021a) also proposed
a method to automatically push keywords based on Composite Correlation
Intension values.

The gap between nature domain and solution domain is believed to be the result
of the challenge of designers’ understandability (Kruiper et al., 2018). And,
the representations of biological systems are key to knowledge transfer in BID
process (Sartori et al., 2010). To bridge this gap, AskNature (Deldin and
Schuknecht, 2014) provides biomimicry strategies that help designers develop a
deeper understanding of how to apply the information within the nature domain.
Sartori et al. (2010) developed the SAPPhIRE (State-changes, Actions, Parts,
Phenomena, Inputs, oRgans and Effects) guideline to formulate four distinct
levels of abstraction at which the transfer between domains could take place.
Chen et al. (2021b) proposed a method combining dependency parsing and keyword
extraction to extract structure-function knowledge from nature domain.

##  3 RESEARCH METHOD

###  3.1 Pre-Trained Language Model (PLM)

To overcome the gaps, we propose a natural language generation approach to
bridge the gap between the three domains (i.e., nature, solution, and problem)
in BID and create more understandable textual concepts. This is accomplished
by the generation and evaluation of design concepts through the application of
the pre-trained language model (PLM).

PLMs are language models that have been trained with a large dataset of
textual information and can be applied to deal with specific language-related
tasks (Arslan et al., 2021). For example, the base model of BERT was trained
with Wiki and books data that contains over 3.3 billion tokens (Kenton and
Toutanova, 2019), and the largest PLM nowadays, namely GPT-3, was trained on a
500 billion tokens dataset (Brown et al., 2020). These huge pre-training
datasets offer PLMs not only the capability of understanding human language,
but also the knowledge and logic that come with it.

PLMs like BERT (Kenton and Toutanova, 2019) or GPT (Brown et al., 2020;
Radford et al., 2019) have been increasingly popular in the natural language
processing (NLP) field for their state-of-the-art performance in many
downstream NLP tasks (Duan et al., 2020) including text completion and text
classification. There are mainly two available mechanisms for a PLM to perform
these NLP tasks: fine-tuning and prompt-based learning. Fine-tuning, first
proposed by Hinton and Salakhutdinov (2006), is a technique to re-train the
pre-trained model with a small amount of task-specific dataset for the task of
interest. The parameters of the base model are updated in the process. Prompt-
based learning, on the other hand, makes no change to the base model but can
leverage what the model has already learned with simple prompts. This
mechanism leads to the few-shot learning technique which requires only a few
examples of the desired task as prompt (Brown et al., 2020).

In this paper, the PLM we use for concept generation and evaluation is the
latest version of generative pre-trained transformer (GPT), i.e., GPT-3 (Brown
et al., 2020), and we fine-tune the base model for different tasks in our
approach. The reasons that we choose to fine-tune GPT-3 over other models or
techniques are as follow: 1. The base model of GPT-3 was pre-trained on much
larger dataset than any other PLMs, which makes it more knowledgeable about
biology and nature; 2. Fine-tuning can better leverage the BID-related
knowledge and logic from the prepared dataset as we assume that previous
knowledge of BID is not included in the base PLMs; 3. A large dataset is
required to fine-tune a BERT or GPT-2 model which is unobtainable for the BID
task, while according to OpenAI (ope, ), GPT-3 only needs a few hundred of
high-quality samples for fine-tuning; 4. Masked language models like BERT are
generally weak at natural language generation, because they can only learn
contextual representation of words, namely natural language understanding, but
not organize and generate language (Duan et al., 2020), which makes them
unsuitable for our generative approach.

###  3.2 Dataset

The dataset we use to fine-tune the GPT-3 model is collected from the
innovation section of the AskNature website (Deldin and Schuknecht, 2014). The
dataset contains 221 excellent BID samples with each sample represented by
textual information of a benefits, applications, the challenge, innovation
details, and a biomimicry story (Table 1).

Table 1: Data composition of each innovation sample in the fine-tuning dataset Component |  Types of Data |  Description
---|---|---
Benefits |  Keywords |  The advantages of the innovation.
Applications |  Keywords |  The applications of the innovation.
The Challenge |  Paragraph |  A statement of problem in the target domain that is challenging to solve.
Innovation Details |  Paragraph |  Introduction of the innovation that aims to solve the challenge.
Biomimicry Story |  Paragraph |  A biological natural phenomenon that is applicable as analogy for solving the challenge.

The variety of information in this dataset offers great opportunities to
explore different fine-tuning strategies, and this can be done simply through
reformulating and customizing the textual data into input-output pairs. For
example, given the input-output pair of applications to innovation details,
after fine-tuning the model can generate new innovations based on any
applications of user’s interest. If we train it the other way around, i.e.,
use innovation details as input and applications as output, the model can do a
very different task that extract keywords of the applications from any input
innovation description. By customizing the AskNature dataset and combining
different fine-tuned models, different BID related tasks can be implemented.

###  3.3 Concept Generation and Correlation Evaluation

Based on both DbA and BID literatures, we propose a framework (Figure 1) to
address our BID concept generation approach. The generation process starts by
retrieving a biological system from nature domain based on a given problem
space, and the extracted biology information is then mapped to the solution
domain. This is done by fine-tuning the GPT-3 model for text completion task.
The evaluation process assesses the correlation between domains, i.e., if the
generated solution aiming to solve the given problem, and if it has taken
inspiration from the biological information. Separate text classifiers are
fine-tuned from the GPT-3 base model to assess each correlation.

Figure 1: Theoretical framework

For concept generation, we identify three types based on the looseness of the
problem space (Table 2). Type-1 generation has the minimum constraints
regarding how the problem is defined, it takes only the applications as input
and generate biomimicry and innovation based on it. This means that the
retrieval and mapping of the biological information only take the target’s
applications into consideration and leaves the problem space wide-open. This
type is suitable for scenarios where companies want to innovate a certain
product of interest, i.e., drone, but without a specific problem in mind. By
hypothesis, Type-1 generator should be able to provide the most diverse
results.

Type-2 constrains the problem space by introducing benefits, i.e., what
advantages is this innovation expected to bring. It can be seen as an
abstraction of a complex problem into a few keywords of benefits and
applications. This type using the abstract representation of the problem is
suitable when designers have only a general requirement based on the user’s
needs, e.g., a small and light-weighted drone.

Type-3 generation further constrains the problem space by using a paragraph of
challenge statement, which could contain three to four sentences that
describes a problem from the engineering perspective. As the strictest way of
problem definition that we can use based on the AskNature dataset, this type
could be useful for the engineering teams that may come across a specific and
detailed problem in design practice and wish to solve it by BID.

Table 2: Fine-tuned models for generation and evaluation Model |  Input |  Output |  Description
---|---|---|---
Generation
Type-1 Generation |  Applications |  Biomimicry Story, Innovation |  Generate concept with an open problem space
Type-2 Generation |  Benefits, Applications |  Biomimicry Story, Innovation |  Represent the problem in simple keywords of applications and benefits
Type-3 Generation |  Challenge Statement |  Biomimicry Story, Innovation |  Use the full challenge statement as problem space
Evaluation
Problem-Solution Correlation |  Benefits, Innovation |  Related/Unrelated |  Evaluate if the generated innovation may bring the given benefits, applicable to type-2 generation
Problem-Solution Correlation |  Challenge, Innovation |  Related/Unrelated |  Evaluate if the generated innovation aims to solve the given challenge, applicable to type-3 generation
Nature-Solution Correlation |  Biomimicry, Innovation |  Related/Unrelated |  Evaluate if the generated innovation takes inspiration from the biomimicry, applicable for all three types of generation

Due to the varied problem representation of the three types of generation,
different sets of classifiers are required to evaluate the correlation between
domains. This includes benefits-solution correlation, challenge-solution
correlation, and nature-solution correlation (Table 2). However, directly
learning relations of logic between texts is difficult for today’s NLP
techniques. Therefore, we transform these relation learning tasks into simpler
binary text classification tasks to predict if the two domains in the given
text information are correlated. To construct the dataset for fine-tuning a
classifier, we need negative samples that do not acquire the correlation of
interest but maintain the form and topic that are as close to the positive
samples as possible.

Figure 2 represents our method to construct such datasets. Assuming two
domains A and B are to be evaluated if they are correlated. The two domains
that form the positive samples are from the collected AskNature dataset, with
a label that indicates their relevancy. On the other hand, in negative ones we
replace only the B domain into random samples to create the irrelevance. Such
a sample is randomly generated by GPT-3. Some techniques are needed to control
its form and topic so that the only difference between the positive B domain
and the negative B domain is their relevancy to the domain A.

The technique to generate negative samples varies based on the domains to be
evaluated. For the problem-solution correlation evaluators, an extra generator
is fine-tuned from the GPT-3 base model that takes in applications as input
and generates random innovation details. This ensures the negative solution
samples ignore the problem, but still describe the same product application in
the same form as in AskNature. Similarly, the negative samples for nature-
solution correlation are also prepared by random generation, whereas this time
we select five samples of innovation details on AskNature that do not contain
any biological information and use them as examples for few-shot learning in
GPT-3. Negative samples prepared in this technique contain no biological
information. This is because we observed that the bad generation results
usually didn’t take the wrong nature domain as inspiration, but they do not
use analogy from nature at all.

Furthermore, as Soares et al. (2019) suggest for language model to learn
relations, marker tokens are added with all samples for fine-tuning the
classifiers. Marker tokens are simple tags that are added before and after the
sentences or paragraphs of interest. An example of how the marker tokens are
used is: “[Bio]Octopus tentacles have suckers that allow the organisms to hold
small objects. The suckers have small…[/Bio][Inno]The soft manipulator was
inspired by the suckers of octopus tentacles. It is made of a temperature-
responsive layer of…[/Inno]”. The tags that come with square brackets in the
above example are marker tokens.

Figure 2: Dataset construction method for fine-tuning the correlation
evaluators

##  4 RESULTS AND DISCUSSION

###  4.1 Fine-tuning Results

The three types of generators are fine-tuned from GPT-3’s base model of
Davinci, which is currently the largest base model that OpenAI provides via
their API. The model was fine-tuned for 4 epochs with the batch size of 1
(batch size is determined based on 0.2% of the size of training samples). The
gradually decreasing training loss of all three types illustrated in figure 3
shows the model’s capability of learning BID.

Figure 3: Training loss for fine-tuning three types of concept generation

To assess the models regarding their capability of generating novel design
concepts, we employ the metrics of Word Mover’s Distance (WMD) (Kusner et al.,
2015). WMD is widely applied in NLP field to measure the distance between two
text documents based on word embeddings and is popular in natural language
generation (NLG) tasks like image captioning (Kilickaya et al., 2017) and
machine translation (Chow et al., 2019) to assess the performance of the NLG
model. Such applications often aim for minimizing WMD as they want the
generated results to be as close to the ground truth as possible. However, we
want to do the opposite in our design concept generation task, i.e., assess
which model gives the larger WMD, because it means the model can generate more
diverse results. In design tasks, the diversity of concepts could lead to
novel solutions, and therefore we use the metrics to represent the model’s
capability of generating potentially novel concepts. Note that this approach
can only assess the performance of the generator model but not the novelty of
an individual concept generated with customized input because the ground truth
for similarity measurement is needed and it only exists in samples in the
dataset.

To measure the WMD, we select three samples of robotics from the AskNature
innovation dataset: an aerial robot, a ground robot, and an underwater robot.
Their applications, benefits, and challenges are used as input according to
the three types of generation, and 50 concepts are generated for each sample
based on each generator type. Then, we extract the word embeddings of the
innovation details of each original sample and generated ones, measure the
embeddings similarity using WMD. The word embedding extraction method we use
in this experiment is the pre-trained Word2Vec model provided by Gensim
(https://radimrehurek.com/gensim/). Figure 4 (a) (b) (c) shows the WMD
distribution of the 50 results of each sample and each type, figure 4 (d)
organizes all the 450 results based on types. Type-1 generation results in the
most diverse concepts but only slightly better than Type-2, while Type-3 gives
the least diverse ones. This is not surprising as Type-1 and Type-3 have the
loosest and strictest defined problem space as constraints, respectively.

Figure 4: WMD distribution of generated concepts

The evaluators are fine-tuned from GPT-3’s base model of Curie, which is not
as strong as Davinci but quite capable of complex text classification [38].
Figure 5 shows the classification accuracy when fine-tuning the evaluators.
The accuracy of the benefits-innovation evaluator and challenge-innovation
evaluator reach 67.4% and 79.8% respectively after 4 epochs, while the
biomimicry-innovation evaluator achieves 97.8% accuracy after only 1 epoch.

However, the 67.4% accuracy of benefits-innovation correlation evaluator is
acceptable but not very good, which may have negative influences on the
evaluation of Type-2 generation results. The authors will continue working on
improving the classification accuracy in future works.

Figure 5: Classification accuracy of the three correlation evaluators

###  4.2 Case Study

We have applied our BID concept generation approach in a real government-
funded project to design flying cars. Along with the acceleration of the
urbanization process, ground traffic in cities is crowded and shows increasing
pressure. However, the airspace above the urban area has not yet been
developed. Air transportation becomes a prospective solution for road
congestion. To merge the air and ground traffic, the flying car is proposed as
a class of novel dual-mode vehicles. Figure 6 below illustrates three well-
known flying car prototypes. However, many challenges still remain to be
solved before we can finally see them flying and driving in the cities.

Figure 6: Flying car examples: (a) Pegasus E-Model
(https://bepegasus.com/index.jsp) (b) Aeromobil (https://www.aeromobil.com/)
(c) Terrafugia Transition (https://terrafugia.com/transition/)

At the beginning of the project, we interviewed the project team of 10
engineers that had worked on several flying car design projects regarding the
challenges they came across in their design projects including the present
one. Lightweight design emerges as a major design problem of flying cars and
requires novel solutions. Table 3 lists the corresponding inputs to all three
types of fine-tuned GPT-3 generators (Table 2). For each type, 50 concepts are
generated.

Table 3: Inputs of flying car case study Applications |  Flying car
---|---
Benefits |  Lightweight
Challenge Statement |  A flying car includes a subsystem for flying in the air in addition to a subsystem for driving on the ground. With both flying and driving subsystems in one, the weight might be increased to increase the drive load, demand more propulsion to overcome gravity, and increase fuel consumption. Lightweight design is a challenge for flying cars.

Figure 7 categorizes the biological sources of the GPT-generated bio-inspired
flying car design concepts. Across all three types, the generators are over
50% likely to retrieved birds as an analogy, and then insects. This is
consistent with the cognitive process of humans where people tend to retrieve
information based on attribute similarities (Linsey et al., 2008), i.e.,
retrieve flying animals as analogies for flying cars. However, there are some
unexpected but interesting creatures like reptiles and even plants, which
could result in more novel concepts.

Figure 7: Categories of the retrieved biological systems

The correlation evaluation results are shown in Table 4. Type-1 is excluded
for the evaluation of problem-solution correlation because the problem space
is open. We evaluated the remaining GPT-generated BID concepts passing the
tests and narrow down to a small number of candidates for embodiment design,
detailed design, and physical prototyping. We chose two concepts generated by
each type and asked the flying car design team to evaluate them in two
dimensions of feasibility and novelty.

Table 4: Percentage of generated concepts passing the evaluations |  Type-1 |  Type-2 |  Type-3
---|---|---|---
Problem-Solution |  N/A |  44% |  84%
Nature-Solution |  86% |  90% |  86%
Overall |  86% |  40% |  76%

Table 5 lists the concepts for the team to evaluate. Two concepts from the
Type-1 generator that passes the evaluation are also included. These can
provide an intuitive sense to the readers on the GPT-generated concepts as
well as the performance of our approach. The model is capable of learning how
to retrieve biological information that shares attributional or relational
similarity with the target, and then map it to the target to generate
solutions. For instance, the second concept in Table 5 proposes to mimic the
lightweight skeleton of pterosaurs to design the flying car body. Meanwhile,
both biomimicry and innovation descriptions could be not concise enough and
contain extra information that is irrelevant to the analogy of our interest.

Moreover, we also added two samples of robotics innovations from AskNature as
the benchmarking group for the tam to evlaute. They are: “Safer aerial robot
inspired by pufferfish” (https://asknature.org/innovation/safer-aerial-robot-
inspired-by-pufferfish/) and “Highly maneuverable drone inspired by swifts”
(https://asknature.org/innovation/highly-maneuverable-drone-inspired-by-
swifts/). This could make a better sense of how good our generated concepts
are compared to the existing BID examples.

Table 5: Concepts for team evaluation Type-1 Generation
---
1# |  Biomimicry: Swimming animals must produce thrust to move forward. The thrust is generated by moving the tail fin, which works like a propeller. Innovation: The flying car was inspired by the way a shark swims through water. The shark tail fin moves back and forth to create thrust, just like a propeller. However, the shark fin has shape-shifting skin inside that helps to smooth out the flow of water while swimming. The flying car has a similar shape-shifting skin inside its tail fin to help with thrust, similarly to the shark fin. It has a rectangular profile when at rest. When it moves through the air, the tail fin smoothly changes shape to a wedge-shape profile, similar to a shark fin. The two different shapes affect the way air moves over the tail fin, creating thrust.
2# |  Biomimicry: Pterosaurs are considered the dinosaurs of the Mesozoic era, because they soared as high as 50 m above the ground. They were the largest flying animals ever, with wingspans of approximately 3 m. Their wings were thin and long, and they had a bony breastbone that was shaped like an upside-down ‘V’. Their forelimbs were short, but their claws were long and sharp, helping them to grab on to branches. Innovation: The flying car has a body that is similar in shape to pterodactyls, with a body designed to control drag, lift, and thrust. It also has a lightweight hull and a propeller to generate thrust. The vehicle’s hull is constructed of high-performance carbon fiber, inspired by the lightweight skeletons of pterosaurs. The propeller is mounted on a pivoting arm that is controlled by a joystick. The entire assembly weighs approximately 35 pounds and looks similar to a parasail. The propeller is 16 inches in diameter and is powered by a 930cc marine engine.
Type-2 Generation
3# |  Biomimicry: Hummingbirds are lightweight, small animals that need to stay aloft while carrying food to their nestlings. They do this by using the forces of lift and drag, which are affected by airspeed and wing shape. At low airspeeds, hummingbird wings provide thrust by moving back and forth. At high airspeeds, their curved outer wing surfaces trap air, creating low drag and enabling them to stay aloft at high speeds. Innovation: The FLY Bot is made of carbon fiber and epoxy and weighs only 9.4 ounces (266 grams). It is powered by a brushless DC motor and has three wings that are made of a porous, lightweight material. The wings have a curved profile similar to a hummingbird, which allows it to have high lift at low speeds while also being able to move the robot quickly through the air. The high-aspect-ratio wings have porous materials in the inner cores, including carbon fiber, materials that are similar to the sponges found in sand dollars and the nano structure of sea sponges, which allows them to be lightweight as well as strong and flexible. The wings are also connected to close-fitting anchors that are used to control aerodynamic forces. This allows the robot to fly at high speeds of up to 27 miles per hour. Overall, the FLY-BOT weighs only a pound and its wingspan is 10 feet long.
4# |  Biomimicry: Many birds are able to fly for very long periods of time without tiring. This is because birds are able to keep their heart rates low and oxygen consumption at a minimum. Their lightweight and porous bones allow their bodies to keep warm, which helps to conserve energy. Innovation: The lightweight carbon fiber body of the flying car has many of the same porous and lightweight bone characteristics found in a bird. It has a carbon fiber body with a porous, lightweight lattice of carbon fiber ribs that support the vehicle and minimize drag while in flight. The ribs are attached via carbon fiber spars that are held together by a unique adhesive that was inspired by gecko feet, which allow the vehicle to stay attached when flying through the air.
Type-3 Generation
5# |  Biomimicry: Some insects, such as dragonflies and fruit flies, are known for their high maneuverability and high speed flight. This agility is enabled by a wing geometry that gives the wing a high aspect ratio (length to width ratio) and a thin structure. Innovation: The wing design of the flying car is inspired by the wing shape of dragonflies and fruit flies. It is made of lightweight carbon fiber that has the same stiffness as steel, but is 60% lighter. The wing is able to effectively integrate air and ground characteristics into a single wing design, which could lead to more efficient use of materials, increased maneuverability, and more. The main wing is wedge-shaped and has a very small aspect ratio, similar to fruit flies and dragonflies. The small aspect ratio significantly reduces the total wing surface area, generating greater lift for a given wing area. The high lift generates significantly less drag and less induced drag, which allows the vehicle to achieve greater efficiency and a better aerodynamic performance.
6# |  Biomimicry: The flying fish stores energy for its leap above the water surface. It does this by bending its body in a “U” shape, with its tail lying flat. When it is ready to jump, it springs its tail upwards and propels itself through the air. Innovation: The lightweight flying car has a horizontal propeller fitted with a folding propeller system, similar to a flying fish. The propeller folds when the vehicle is in driving mode, and unfolds when going into flying mode. When folded, the propeller is protected and the vehicle can travel over land and water.

The rubrics for feasibility and novelty evaluation are shown in Table 6. For
feasibility, we generally refer to the idea of workability by Douglas et al.
(2006) which represents the ability to implement the concept. On the other
hand, novelty is the degree that measures how common the concept has already
been seen in the target domain and its related industries. Note that by our
metrics, a concept that had been expressed before but was never successfully
implemented is still considered novel.

Table 6: Feasibility and novelty rating rubrics Feasibility Rank |  Description
---|---
1 |  The concept makes no sense from the engineering perspective.
2 |  The concept makes little sense with today’s technology, but could be possible in the future.
3 |  The concept makes sense, but efforts are needed to work out a practical technical roadmap.
4 |  The concept makes good sense, and a technical roadmap can be easily established to realize it.
5 |  The concept makes perfect sense and there are existing tools, materials, or components to realize it.
Novelty Rank |  Description
1 |  Solution exists and is commonly seen in the target domain (target domain = the target product described in the concept)
2 |  Solution exists but is uncommon in target domain.
3 |  New features are proposed for target domain, but similar approaches or technology can be commonly seen in related industries (related industries = flying car, drone, automobile, robotics, etc.)
4 |  New features are proposed for target domain, and similar approaches or technology can be rarely found in related industries.
5 |  New features are proposed, and no similar approaches or technology can be found nowadays.

After the survey, 9 out of the 10 returned results are useable and are
analyzed in Figure 8 by the generator types. The team gave relatively high
novelty levels but only intermediate feasibility levels to the comparison
group, which is reasonable because a lot of samples from AskNature are not
directly workable with existing materials or components. Comparing to this,
Type-1 and Type-2 concepts got lower feasibility score but higher novelty
score, while the ones generated from Type-3 are considered more feasible but
less novel. This finding is not surprising because Type-1 and Type-2
generations are more open than Type-3, and this is consistent to our finding
from Figure 3 in Section 4.1.

Overall, the results show that feasible and novel BID flying car concepts are
generated by fine-tuned GPT for the engineering team for selecting the one to
move forward for embodiment and detailed design.

Figure 8: Human evaluation results: distribution of (a) feasibility scores;
(b) novelty scores; (c) average score of feasibility and novelty

With the evaluation, we want to select a concept for further development.
Concept 4# received the highest average novelty score among all concepts with
an acceptable feasibility score. It suggests mimicking the porous structure of
the bone of birds for the material design in the flying car and beyond the
expertise of the team. Eventually concept 2# is selected to proceed for
embodiment and detailed design. 2# draws analogy from the pterosaurs that were
the largest known flying animal on earth and yet they can fly fast. The
concept suggests mimicking the body and lightweight skeleton of pterosaurs.
Figure 9 (a) is the sketch drawn by the first author that illustrates the
early embodiment design based on the generated concept. Apart from the body
shape and wing structure mentioned in the GPT-generated concept, we also take
inspiration from the folding mechanism of pterosaurs’ wings, which results in
a combination of the front wheels and the front propellers to further reduce
the weight. Figure 9 (b) depicts the folding process from the aerial mode to
land mode of the flying car compared to that of the pterosaurs.

This interesting example showcases that human-AI collaboration in design
concept generation does not necessarily stop at applying what the algorithm
may offer. Concepts that are represented in natural language can be either too
abstract to understand or too rigid and thus need the human designers to
further take them into embodiment and detailed design. Designers can also draw
further inspiration from the given concept and combine inspiration from other
sources to make it more suitable, add more details and optimize it for their
own project scenario.

((a))

((b))

Figure 9: Design sketch based on concept 2#: (a) folding mechanism (b) air-
land mode transition

##  5 CONCLUDING REMARKS

This paper has introduced a data-driven intelligent methodology for bio-
inspired design concept generation. The core of the methodology is to fine-
tune the generative pre-trained transformers for different concept generation
situations with different data inputs and use them to automatically generate
design concepts in natural language texts. The case study shows reasonable
performance of the methodology in generating good quality BID concepts in
intelligible natural language. Our work responds to the recent calls in the
design research community to adopt pre-trained language models for design
research and applications (Han et al., 2022; Siddharth et al., 2021) To the
best of our knowledge, this study is the first to employ generative pre-
trained transformers for bio-inspired design, while our recent study has
explored the applications of GPT-2 and GPT-3 to generate new design concepts
using a product design data repository (Zhu and Luo, 2021).

Here we aim to contribute to artificial intelligence for BID. Despite being
the first to explore fine-tuning GPT-3 to aid in BID concept generation, our
work is preliminary and limited in a few aspects. First, we only explored the
small AskNature data repository. Future research may explore alternative and
even more fruitful nature or nature-inspired design data sources for applying
our GPT-based BID methodology. Second, we only introduced three model fine-
tuning and concept generation strategies in this paper. There could be more
and more flexible strategies for fine-tuning GPT-3 for concept generation.
Third, we only explored a limited set of metrics for automatically evaluating
the generated concepts in text. Future research may explore more metrics and
develop new metrics for evaluating the quality and novelty of the GPT-
generated concepts in natural language. In addition, the case study is limited
in its scale and scope. The evaluation of 10 engineers from one design project
is insufficient to provide statistical significance. As the next step of this
research, we plan to engage more and more diverse engineers to evaluate the
concepts generated by the fine-tuned GPT for different design problems and
settings. These will allow us to develop more systematic understandings on the
enabling or conditioning factors on the performance of our methodology.

Therefore, these limitations of our work present new research opportunities
and future research directions. In general, we hope our exploratory work
presented here can inspire more research on the use or development of
generative PLM for biologically inspired design, and design-by-analogy in
general.

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