Recursive Function Space
kenrikodaka laboratory (NAGOYA CITY UNIVERSITY / JAPAN)
KENRI KODAKA (VR Programming / Project Management)
KOYO MORI (Computer Graphics Design)

Summary
Recursive Function Space (RFS) is a virtual reality based interactive contents that prominently explores and activates a meta-cognitive function in our human cognition system. RFS consists of four (RFS 1.0) or six (RFS 2.0) clones of the player (HMD wearer), characterized in a distinctive perspective transition among these clones that gives an out-of-body-like experience of seeing oneself, how one looks at oneself, and how one is looked by oneself. This is done with the HMD technology (VIVE), where pulling a lever of the left/right controller enables the player to take the perspective of the left/right hand of HMD wearer. The player can also take the perspective from the right hand of “left hand clone” or that from the “meta-clone” owning the HMD wearer at its left hand clone. Such a recursive exploration of the multiple perspectives is systematically associated with Homunculus’ infinite regress discussed in the context of “hard problem of consciousness”. Thus, RFS embodies a classically universal question concerning the origin of the subject with an up-to-date HMD technology.
Statement
Recursive Function Space (RFS) entices you to find a core of yourself and get an implication that such an attempt has no specific goal after some explorations. Such a universal question concerning the origin of the subject ($\bf{O}$) has been so far dealt with mainly in the mind of a susceptible young person, a story of novels, or notes of prominent philosophers. In RFS, such a thought experiment is unfolded as perspective transitions among multiple self-body images, which is a fairly specific operation in the three-dimensional space implemented in a specific environment of a head-mounted display (HMD) (using VIVE).

RFS is a singular space at an intersection among formal logic (Russell’s paradox), brain science (out-of-body experience), philosophy (Homunculus’ infinite regress), quantum theory (many-worlds interpretation), psychoanalysis (petit objet a), and Buddhism (the palm of Buddha). In RFS, we understand that we ($\bf{O}$, which is assumed to be the core of the subject) are actually dancing on the palm of Buddha and that Buddha is also on the palm of another Buddha. Conversely, we may understand that we are originally something like Buddha for another subject. Thus, RFS provides us with a unique intuition where we (and our reality) are essentially ubiquitous in our phenomenal experience.

Indeed, our cognitive space is accompanied by a recursive structure because the cognition cannot separate “self-image” from “self who image self-image.” As the formal logic demonstrates, completeness of the system allowing recursive mapping is critically denied. Thus, the attempt to find O’s origin ends in failure. As in formal logic, RFS defines the basic elements representing “self-body image” positioning in a three-dimensional space and some basic mappings applied to the element, constructing a universal structure representing the failure to find $\bf{O}$.

Tutorial Movie (3'57)

Description Rule of RFS
The element and the mapping of RFS are defined as follows:
Special Element $\bf{O}$
$\bf{O}$ is a special element exclusively representing a person who wears an HMD. The position and head rotation of $\bf{O}$ follows those of the HMD wearer.
L Map and R# Map
(M1) Position
For a specific element $\bf{P}$, the function that moves $\bf{P}$ to the left hand of $\bf{P}$ is referred to as L Map and denoted as $\bf{PL}$. For a specific element $\bf{P}$, the function that moves $\bf{P}$ to the right hand of $\bf{P}$ is referred to as R# Map and denoted as $\bf{PR}$#.
The head rotation of $\bf{PL}$ (L Map of $\bf{P}$) is in accordance with that of $\bf{P}$.
The head rotation of $\bf{PR}$# (R# Map of $\bf{P}$) is always directing to the head of $\bf{P}$ (Introspective Perspective).
(M3) Scale
L Map and R# Map reduce the size to half; the body size of $\bf{PL}$ and $\bf{PR}$# is half of that of $\bf{P}$.
(M4) Inverse Map
When the L Map of an element $\bf{Q}$ corresponds with $\bf{P}$, $\bf{Q}$ is denoted as $\bf{PL}^{-1}$. Note that $\bf{L}^{-1}$ is the inverse map of L Map, and $\bf{PL}^{-1}\bf{L}$ = $\bf{P}$ is satisfied. Conversely, L Map of $\bf{PL}^{-1}$ is $\bf{P}$; the rule of M1 - M3 is equally applied to the relation between $\bf{PL}^{-1}$ and $\bf{P}$.

RFS Implementation
RFS can freely switch between two modes: the basic space RFS 1.0 and the extended space RFS 2.0.
o RFS 1.0 definition
o Elements
RFS 1.0 includes the following four elements.
size=1size=1/2size=1/4
$\bf{O}$$\bf{OR}# \bf{OL}$$\bf{OLR}$#
o o Map
$\bf{OL}$ is L Map of $\bf{O}$, $\bf{OR}$# is R# Map of $\bf{O}$, and $\bf{OLR}$# is R# Map of $\bf{OL}$.
o o o Position
$\bf{O}$ is at HMD wearer, $\bf{OL}$ is at the left hand of $\bf{O}$, $\bf{OR}$# is at the right hand of $\bf{O}$, and $\bf{OLR}$# is at the right hand of $\bf{OL}$.
o o o o Head Rotation
The head rotation of $\bf{O}$ and $\bf{OL}$ accords with that of the HMD wearer, that of $\bf{OR}$# is directing to $\bf{O}$, and that of $\bf{OLR}$# is directing to $\bf{OL}$.
o o o o o Scale
When the body size of $\bf{O}$ is assumed to be 1, the body size of $\bf{OL}$ and $\bf{OR}$# is 1/2 and that of $\bf{OLR}$# is 1/4.
o o RFS 2.0 definition
o Elements
RFS 2.0 includes the following six elements.
size=2size=1 size=1/2size=1/4
$\bf{OL}^{-1}$$\bf{OL}^{-1}R# \bf{O} \bf{OR}# \bf{OL}$$\bf{OLR}$#
o o Map
$\bf{OL}^{-1}$ is inverse L Map of $\bf{O}$, and $\bf{OL}^{-1}R$# is R# Map of $\bf{OL}^{-1}$.
o o o Position
$\bf{OL}^{-1}$ is an element that has $\bf{O}$ at its left hand, and $\bf{OL}^{-1}R$# is at the right hand of $\bf{OL}^{-1}$.
o o o o Head Rotation
The head rotation of $\bf{OL}^{-1}$ accords with that of the HMD wearer, and that of $\bf{OL}^{-1}R$# is directing to $\bf{OL}^{-1}$.
o o o o o Scale
When the body size of $\bf{O}$ is assumed to be 1, that of $\bf{OL}^{-1}$ is 2 and that of $\bf{OL}^{-1}R$# is 1.

The remaining four elements ($\bf{O}$, $\bf{OR}$#, $\bf{OL}$, and $\bf{OLR}$#) follow RFS 1.0 definition.

o o o Perspective transition
Pulling the lever of HMD’s controller enables perspective transition among multiple elements in RFS. In both RFS 1.0 and RFS 2.0, the initial perspective accords with $\bf{O}$ and pulling the right controller’s lever switches the HMD’s perspective to $\bf{OR}$#, looking over $\bf{O}$ from the right-hand position. In RFS 1.0, pulling the left controller’s lever switches the perspective to $\bf{OL}$# and pulling the levers of both controllers switches the perspective to $\bf{OLR}$#, looking over $\bf{OL}$ from the right-hand position of $\bf{OL}$. In RFS 2.0, pulling the lever of the left controller switches the perspective to $\bf{OL}^{-1}$ and pulling the levers of both controllers switches the perspective to $\bf{OL}^{-1}R$#, looking over $\bf{OL}^{-1}$ from the right-hand position of $\bf{OL}^{-1}$.
Exhibition
2017.1.28-29｜からだは戦場だよ2017 @ やながせ倉庫・ビッカフェ
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