EXPERIMENT IN PERCEIVING CONCEIVING:
HANDEDNESS, BINOCULARITY, & LOGICALITY
On October 10, 2003, Dr. S.B. Hoath wrote: "your most recent construction of the bifocal spirals has a perceptual analogue. If one looks at the logarithmic spiral [inscribed in the golden rectangle], and relaxes one's eyes in the manner required to see 3D 'magic eye' figures, 2 [inscribed] spirals arise. If the head is then tilted by 45 degrees, the resulting figure is remarkably like the structure illustrated in http://dgleahy.com/dgl/p27.html." When the writer performed the experiment he found that the resulting figure was similar to the referenced figure, but significantly different (Figure 4, below). He was led to expand the experiment by viewing the right-handed and left-handed versions of the inscribed logarithmic spiral (see below, "Procedure"), with results as described below under "Summary of Results for Type B."
On October 22, 2003, in response to an inquiry by the writer, Dr. Hoath wrote: "When I view the doubled figure of the right handed spiral and tilt my head to the left (spiral's right) as per the first set of directions [see below under "Procedure"], the [largest components of the] resulting figure (RR) [are not on one linear plane] and [the figure] contains a single red diagonal running from the lower left to the upper right. It is the same as Figure 4 (LL) [see below under "Summary of Results for Type B"], but flipped horizontally. I am right handed and right eye dominant as far as I can tell.
"If my head is not
tilted then the [largest components] of [the] right handed spiral [figure]
remain [on one linear plane] and I see the mirror image of Figure 2 [ibid.].
At first, it was difficult for me to get adequate separation; i.e., to expand
the figures beyond a width of 2. I met some resistance (tension) at first
in trying to [expand the figures beyond a width of 2]—it seems to be a matter
of adjusting my focal length to achieve a truly parallel gaze.
"In summary, whenever my head is tilted I see images [whose largest components
are not on one linear plane] with both right and left handed spirals. Both
spirals yield single diagonals running upwards to the right and left,
respectively. I can expand both types of spirals to attain binocular
images which have a width greater than 2. And, finally, if my head is not
tilted the binocular images remain level."
With this report in hand, the writer was able to form the hypothesis that there are two types of viewers: Type A, right-handed and right eye dominant (exemplified by Dr. Hoath), and Type B, left-handed and left eye dominant (exemplified by the writer). The procedure and results for Type A viewing are as described in Dr. Hoath's October 22 email and incorporated in the procedure described below. The procedure and results for Type B are to be found below under the appropriate headings.
The reader is invited to attempt to
reproduce the results of the experiment. You can communicate your results
to the writer by using the contact button at the bottom of the page. If
you do, please state whether you are right-handed or left-handed and right or
left eye dominant (your dominant eye can be determined by viewing a distant
object through a small opening between two fingers centered at arm's length directly
in front of your face and noting whether the right or the left eye keeps the
object in sight when the right and left eyes are alternately closed one at a
time).
Procedure
Two versions of the inscribed logarithmic spiral are to be used in the experiment (click on the figures to perform the experiment against a plain white background). The Right-handed version:
and the Left-handed version:
1) Click on the Right-handed figure (above) to be able to view it by itself against a plain white background. While viewing the figure relax your eyes until you see the figure doubled.
If you are left-handed: keeping the doubled figure in view tilt your head slowly to the spiral's Right (your left). As you do so you should be able to see a figure taking shape that approximates Figure 1 (below).
If you are right-handed: keeping the doubled figure in view and your head vertical adjust your focus until the width of the figure expands and you see a figure taking shape that approximates the reverse of Figure 2 (below).
2) Click on the Left-handed figure (above) to be able to view it by itself against a plain white background. While viewing the figure relax your eyes until you see the figure doubled.
If you are left-handed: keeping the doubled figure in view tilt your head slowly to the spiral's Right (your left). As you do so you should be able to see a figure taking shape that approximates Figure 2 (below).
If you are right-handed: keeping the doubled figure in view and your head vertical adjust your focus until the width of the figure expands and you see a figure taking shape that approximates Figure 2 (below).
3) Click on the Right-handed figure (above) to be able to view it by itself against a plain white background. While viewing the figure relax your eyes until you see the figure doubled.
If you are left-handed: keeping the doubled figure in view tilt your head slowly to the spiral's Left (your right). As you do so you should be able to see a figure taking shape that approximates Figure 3 (below).
If you are right-handed: keeping the doubled figure in view tilt your head slowly to the spiral's Right (your left). As you do so adjust your focus until the figure expands and you see a figure taking shape that approximates the reverse of Figure 4 (below).
4) Click on the Left-handed figure (above) to be able to view it by itself against a plain white background. While viewing the figure relax your eyes until you see the figure doubled.
If you are left-handed: keeping the doubled figure in view tilt your head slowly to the spiral's Left (your right). As you do so you should be able to see a figure taking shape that approximates Figure 4 (below).
If you are right-handed: keeping the doubled figure in view tilt your head slowly to the spiral's Left (your right). As you do so adjust your focus until the figure expands and you see a figure taking shape that approximates Figure 4 (below).
Summary of Results for Type B
The figures below
are in each case ideal results of the parallel eyes viewing procedure described
above for Type B, where ideal = a completed drawing of the figure that naturally tends to
form in the parallel eyes viewing as the viewer's head is gently tilted in the
direction indicated for each figure. The sequence conforms to the "fundamental
form of all logical statements" (D.G. Leahy, Foundation: Matter the
Body Itself [Albany, 1996], p. 258, n. 3, et passim) pq
+
q + p
+
, and does so at once qua procedure and qua ideal
result.
Where p = total width =
f +
1/f2
= 2 (the ratio of the width of the right-handed figures to that of the
left-handed figures = 2/f2),
and q = figures on one linear plane, and where RR = head tilted to Right-handed
spiral's Right (viewer's left), LR = head tilted to Left-handed spiral's Right
(viewer's left), RL = head tilted to Right-handed spiral's Left (viewer's right)
, and LL = head tilted to Left-handed spiral's Left (viewer's right),
q,
RL = p, and LL =
, as here illustrated:
Figure 2. LR =
q:
Figure 3. RL = p
:
Figure 4. LL =
:
.
Note that only Figure 4 contains a single diagonal, and that it is at
once the
analogue of the virtually left-handed diagonal of the real trinary logic
cornerstone (cf.
http://dgleahy.com/dgl/p19.html; also D.G. Leahy, Faith and
Philosophy: The Historical Impact [Aldershot,
Burlington, and Singapore], Chapter 7) and of the figure
referenced by Dr. Hoath (cf.
http://dgleahy.com/dgl/p27.html). The ratio of the diagonal of
the latter to that of Figure 4 is exactly 2.35114101/3.077683539 = 2/φ2
= the epidermal cell reduction ratio (ibid.).
Next note the increasing distance between the double diagonals in
Figures 1, 2, and 3. The underlying order of the Figures is then the
minimum order: 4, 1, 2, 3 (cf. Leahy, Foundation, Section III.1,
et passim), where the distance between the two diagonals in Figure 1 =
the inverse length of the diagonal in Figure 4.
Where the single diagonal of Figure 4 = δ
= 3.077683539, the distance between diagonals in Figure 1 =
φ0/δ,
in Figure 2 = φ1/δ,
in Figure 3 = φ2/δ,
and 11.70820393 is the diameter of the 'defining bowl' in the "infinite
pentagonal arrangement of 'golden bowls' at the φ-level
of existence itself" (cf.
http://dgleahy.com/dgl/p15.html), the rational product of the
minimum order proportion of the specified lengths is
δ/(φ0/δ)
× (φ1/δ)/(φ2/δ)
= 11.70820393/2.
Comparison of Types A and B
Let a right-handed right eye dominant person be Type A. Let a
left-handed left eye dominant person be Type B. Type A performing
the experiment perceives the right-handed and left-handed spirals symmetrically.
Type B performing the experiment perceives the right-handed and
left-handed spirals asymmetrically.
Type A perceives according to the order (where V = head vertical, and
T = head tilted in the direction opposite that of the spiral) VR =
q,
TR =
,
VL =
q,
TL = .
The symmetry of Type A's perception
is expressed by the non-logical repetition in which VR = VL and TR =
TL. Type A is not
constrained to see the forms pq or p
(although in fact, if it wishes, it appears it might constrain itself
to do so; cf. Hoath's remarks above re 'widening').
The asymmetrical nature of Type B's perception is expressed by the
logical non-repeating distribution
RR = pq, LR =
q,
RL = p, LL =
.
No exact combination of p and q is repeated.
f is the ratio of the asymmetrical
section of the line such that while, in order to provide the elements
of a geometric proportion, all other divisions of the line involve two
cuts, that is, a repetition of the ratio in which the first division
is made, the f ratio does not.
The lack of repetition among the elements of the logical form of all
statements points to a fundamental relationship between the form of
ordinary logic and f
+ 1 = f2. Type B, consistent with the logical
structure of its perception, cannot duplicate
the Type A direct perception of the TR =
repetition of TL = .
Type B perceives the results illustrated above under the following conditions:
Figure 1 (RR = pq) when its head is tilted ~15º to the Right-handed spiral's Right,
Figure 2 (LR =
Figure 3 (RL = p
Figure 4 (LL =
Where movement in accord with RR has the positive sign +, and movement in accord with LL in the opposite direction has the negative sign –, as in matters electric, where, that is, the logical distribution RR + LR + RL + LL = + +, – +, + –, – –,
RL/LL×RR/LR = (~ –31.5º)/(~ –11º)×(~ +15º)/(~ +11º) = RL/LR×RR/LL = (~ –31.5º)/(~ +11º)×(~ +15º)/(~ –11º) = (~ +233.51/4)º.
When Type B's head is tilted ~ +233.5º to the Right-handed spiral's Right it
perceives the inverse of Figure 4, as here illustrated:
Where ~ –11º is the position at which Type B perceives Figure 4 (LL
=
)
and ~ +233.5º the position at which it perceives the latter's inverse,
~ +233.5º + ~ –11º = ~ +222.5º = ~ 360º/φ.
(Note that
RL + LL = ~ –31.5º + ~ –11º = ~ –42.5º = –180º + ~ 360º/φ2.
For the golden angle, 360º/f2 = 137.507764º, and its reflex, 360º/φ = 222.4922359º, cf. http://dgleahy.com/dgl/p27.html).
The constraints on Type A
and Type B are themselves asymmetrical. Type A constraints,
should they exist, are self-constraints. Type B constraints are
f-(not
self-)constraints. Type B perception is logically structured
immediately, i.e., not by choice. Type B can perceive TR =
only in an inverse and reverse form, i.e., only as the inverse of LL =
,
only by means of
tilting the head (indeed, the whole body) to a position
f-defined relative to the
perception LL =
,
that is, to the position at
~ +233.5º = ~ +222.5º – ~ –11º.
The necessarily logical structure of Type B perception is a
consequence of the f-defined constraint
that makes perception of TR =
impossible for Type B.