The

Mirrored Triangles

Illusion

On the perceived distance

between triangles in

mirror image arrangement

W.A. Kreiner

Faculty of Natural Sciences University of Ulm

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1. Illusions on perceived length

There are several illusions where the apparent length of a line depends on the size, the

shape or the type of context elements. Examples are the Oppel-Kundt illusion (Oppel, 1854;

Kundt, 1863; Spiegel, 1937; Surkys, 2007; Surkys, Bertulis, Bulatov, and Mickiene, 2008), the

Müller-Lyer (1889) or the Baldwin (1895) illusion (Brigell, Uhlarik, and Goldhorn, 1977;

Wilson and Pressey, 1988; Kreiner, 2011). In addition, there are variants of these illusions,

where the context elements are not attached to the target line, but leaving a gap in between

(Pressey, Di Lollo, and Tate, 1977; Kreiner, 2012).

In the illusion discussed here, not the length of a line serves as a target, but an empty space,

ie, the distance between the tips of two isosceles triangles.

In case of the Baldwin illusion, the apparent length of the target line decreases with

increasing size of the adjacent boxes. In case of the mirrored triangles, one finds that the

apparent distance of the dips is a function of the radius of the outer circle. It decreases with

increasing radius, and vice versa. This can be interpreted as due to a size constancy effect.

Originally, the term size constancy referred to the observation that the perceived size of an

object as a function of its distance does not follow the laws of geometrical optics (Schur,

1925). With increasing distance, it appears rather larger than one would expect it from the

size of the retinal image. While the retinal image decreases with distance d according to a

power function d-1, the apparent size does not decrease as rapidly. It rather follows a

function dn-1.

Schur´s observation has been confirmed by Gilinsky (1955). In addition, she has found a

corresponding effect on objects of different size presented at the same distance of

observation. In comparison with larger objects, smaller targets appear rather larger than one

would expect it from their true dimension. This can be described by a similar function as in

the case of observation at varying distances. Both variants of size constancy are explained

such that in case of a smaller retinal image the visual angle is reduced, which, in turn, causes

subjective magnification (Kreiner, 2004).

It should be mentioned that, in case the field of sight is somewhat restricted by artificial

frames, a significant influence of the restriction on the apparent length of vertical or

horizontal lines has been observed (Gavilán et al, 2017).

2. The experiment

2.1 Stimuli: 14 pairs of isosceles black triangles (Fig 1) above bright background were

presented in the top half of transparencies. The triangles were arranged such that their dips

faced each other. The distance of the dips was held constant. It served as the target. In the

original drawing on a DIN A4 sheet of paper the distance was 60 mm. The area of the

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triangles was kept nearly constant to 756(10) units square (mm2), while the ratio between

the trangles´ height (horizontal dimension) and their base was varied between 0.15 and

5.94. A comparison scale of seven horizontal red lines was presented in the lower right half

of the transparency. The lengths of the comparison lines decreased from top to bottom by a

factor of 1.3. On different transparencies, different response scales were presented, where

the absolute length of the standards varied up 29% (Fig 1).

15x100/1.=3

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2

3

4

5

6

7

1

22x68/1.=2

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2

3

4

5

6

7

11

56x27/1.=5

1

2

3

4

5

6

7

10

80x19/1.=7

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5

6

7

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Fig 1 Examples of pairs of triangles, where the separation of the dips is kept constant. Their apparent distance

was determined from comparison with the reference lines (Kreiner, 2012).

2.2 Subjects: 11 healthy volunteers took part (among them the author), all of them age

above 54. Vision was corrected to normal.

2.3 Experimental procedure: The transparencies were projected with a beamer. First, the

triangles were shown for 4 seconds, then, for another 6 seconds, the standard lines were

added. An empty transparency followed for 2 seconds. 6 participants were seated at a

distance of 3 meters. The target subtended an angle of 0.081rad. 5 participants observed

from a distance of 4.5 meters, the target subtending 0.054rad.

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Fig. 2a Apparent distance of

the equilateral triangles´

apices as a function of their

height (=horizontal extension).

The horizontal line at 60 mm

marks the true separation of

the tips.

Fig 2b The red curve gives the

radius R of the outer circle, the

blue one its inverse, multiplied

by some arbitrary number. The

intensity of the illusion seems

to be correlated with the

inverse of the radius. The

vertical line indicates the

minimum radius which

coincides with the maximum

of the illusion, as obtained

from the fit (Fig 6).

3. Results

Fig 2a gives the result of the experiment. Plotting the apparent separation of the triangles as

a function of their horizontal extension (their height), one finds that it first increases and

then, after a maximum, steeply decreases towards the triangles becoming more and more

elongated horizontally. Depending on the triangles´ shape, the apparent distance was found

to be larger or smaller than their true distance. The perceived length oft he target appears to

be negatively correlated with the radius R of the outer circle (Fig 2a,b). R is determined after

the theorem of Pythagoras, as shown in Fig 3. The solid line at x= 22 indicates the maximum

of the function given by Eq (1). See Fig 6.

0 20 40 60 80 100

45

50

55

60

65

70

75y A

ppa

rent

dis

tance

/ m

m

x Horizontal dimension of a triangle [mm]

0 20 40 60 80 100

60

70

80

90

100

110

120

130

R(o

ute

r circle

= r

ed d

ots

) [m

m]

Horizontal extension [mm]

[1/R]*8000

R

5

Fig 3

The radius R of the outer circle is

found after the theorem of Pythagoras.

T means the target length.

x

The smallest apparent distance corresponds to the largest circle, and vice versa. In the

following, from a conceptual model, an algebraic function is derived.

The mathematical function

The conceptual model is based on the idea of size constancy. Originally, size constancy

meant that, with increasing distance, the decrease in apparent size is less pronounced than

one would expect it from the size of the retinal image [Schur (1925)]. While the retinal image

follows the function 1/d (d= distance) which can be written as d-1, the apparent size

follows a function dn-1, where n is a so called size constancy parameter. A similar law

applies to the case where not the distance, but the target´s size r (ie, any linear dimension of

the target) is varied. Small objects appear enlarged, and vice versa. This has been shown by

Gilinsky (1955) on triangles. With respect to the retinal image, the d and the r are inversely

proportional to each other: Both, a large distance d as well as a small dimension r of an

object result in a small retinal image. Therefore, at constant distance, the apparent size is

proportional to (1/r)k-1. k stands for the size constancy parameter in case the object´s linear

dimension is varied instead of its distance. This is shown in Fig 4. There, R means the

radius of the outer circle of the triangles, its radius being varied between R =1 and R =3:

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Fig 4

At the left, a small circle is seen (R=1). Another

circle, three times as large, appears to be

reduced in size (red dashed circular line). So does

the apparent distance of the triangles,

surrounded by the circle. Drawing not exactly to

scale.

The perceived length of the radius is Rperc= (1/R)(k-1). If there were no size constancy effect

(k=0), one had Rperc= (1/R)(0-1) =R. Division yields

Rperc/R = (1/R)k or R-k.

The expression R-k gives the factor by which a larger circle appears smaller than one would

expect it from the size of the retinal image. Here, R=1 means the reference circle (Table 1).

This reference is an aribitrary choice. One could take the larger circle as the reference as

well. In that case, the smaller circle would appear enlarged.

Function: R(perceived) = (1/R)(k-1)

Table 1 Perceived size of circles due to the size constancy effect. The column on the the right-hand side gives the amount of shrinking. For example 1.275/1.5 equals 0.85. See Fig 5.

R R(perc) [k= 0.4] R/R(perceived)

1 1 1

1.5 1.275 0.850

2 1.516 0.758

2.5 1.733 0.693

3 1.933 0.644

Fig. 5

Reduction of the apparent size of circles

with radii between R=1 and R=3 (arbitrary

units) due to the size constancy effect. A

target of linear extent within the circle

(eg, the separation of the traingles´ dips)

appears to shrink by the same amount.

1,0 1,5 2,0 2,5 3,0

0,6

0,7

0,8

0,9

1,0

1,1

d/d

0

Radius r

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Assuming the apparent distance of the dips to be in proportion to the apparent radius of the

outer circle R (Fig 3), the function to be fitted to the experimental results is

y (apparent distance) = 60*A*[Rperceived/R0] = 60*A*[((30 + x)2 + (756/x)2)/62.332] -k/2

Eq (1)

60 means the true distance in millimeters. k and A are the parameters to be fitted. The

expression in square brackets is the square of the radius R of the outer circle, divided by the

square of the smallest radius which can occur (62.3mm). It corresponds to a horizontal

extent of one triangle of 22.2 mm. (30 + x) means the sum of half the target´s length plus the

horizontal dimension x of one triangle. A can be called the illusion factor.

0 20 40 60 80 100

45

50

55

60

65

70

75

Appare

nt dis

tance [

mm

]

Horizontal extension of triangel [mm]

Fig 6 Fitting Eq 1 to the results of the experiment. The values obtained for A and k are given in Table 2. The

maximum oft he curve is around x=22mm. Underneath, there are examples of stimuli: A pair of the tallest

triangles (left), of the longest ones (right), and a pair of the ones which are nearly rectangular.

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756/x means half of the triangle´s base (its vertical dimension), as 756 units (mm2) is the

triangle´s area. The exponent contains a square root (1/2), which means that the apparent

distance is assumed to be a function of the linear extent of the outer circle.

In Fig 6, and in Table 2, the results of the fitting procedure are shown. The value obtained for

the size constancy parameter k= 0.477(33) can be compared with the values derived from

the results reported by Gilinsky (1955), which are between 0.359(23) and 0,500(42) (Kreiner,

2004), depending on the distance of observation, which was larger in her experiment by

more than two orders of magnitude.

A / Illusion factor k /size constancy parameter 2red Table 2 Result of fitting Eq 1 1.139(12) 0.477(33) 0.239

4. Discussion

The mirrored triangles illusion is interpreted as due to a size constancy effect. The result

shows that the intensity of the illusion correlates inversly with the diameter of the outer

circle. From this, it is concluded that the size of the visual angle is in proportion to the outer

circle of the stimulus. This can be compared with the Baldwin illusion, where the intensity of

the illusion appears to be correlated with the size of the outer circle as well [Kreiner (2011)].

In contrast to the latter, in case of the triangles illusion it is not the area of the context

elements which is altered, but their shape.

5. Size constancy as a consequence of limited data processing capacity

The apparent size is not in proportion to the size of the retinal image (Lühr, 1898; Cornish,

1937; Schur, 1925; Gilinsky, 1955; Kreiner, 2004). Comparing two retinal images of different

size, the smaller one appears somewhat enlarged, or the larger one reduced in size.

Concerning the illusion, it does not matter whether it is the image of a small object nearby or

of a large object at far distance [Schur, 1935; Gilinsky, 1955).

Size constancy can be interpreted as due to the limited information processing capacity of

the visual system (Kreiner, 2004). Figuratively speaking, there is only a certain number of

pixels which can be processed by the brain per unit time, finally producing the perceived

image. Choosing a wide visual angle and collecting the pixels from a large area will result in

an overview of the scenery, but at low resolution, while a narrow visual angle will improve

resolution on the expense of overview. A conceptional model of the size constancy effect is

based on the following assumptions:

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- The information processed by the visual system is retrieved from only part of the

retinal area.

- The size of this section increases or decreases with the size of the object one gives a

close inspection.

- The information processing capacity of the visual system is limited. Usually it makes

use of its full capacity, independent of the object´s size and the visual angle chosen.

- The information collected, processed and transformed to give finally the perceived

image, is projected onto kind of an internal visual memory screen which always

exhibits constant size (Kreiner, 2004). This leads to enlargement, in case a small visual

angle had been chosen. Small or large have to be understood as relative to a

standard size.

The „internal visual memory screen“ is just an expressive comparison. It means that the

visual system always uses its full data handling and storage capacity, regardless of the size of

the visual angle.

In the following, the consequence of a limited channel capacity on the perceived image is

illustrated. It is a trade off between the size of the image and the resolution achieved. Fig 7

shows a photo taken in the French town of Pontrieux (Pontrev)/ Côtes-d´Armor. The image

consists of 481.000 pixels. Lets assume that, within a certain time given, the visual system

can manage only 19.200 pixels (=481.000/25). Now, if each 25 pixels (squares of 5 times 5

=25 pixels) are replaced by one large pixel, one gets the situation to be seen in Fig 8: One

cannot recognize details. However, it is possible to achieve high resolution again from

sacrifizing overview: Concentrating the 19.200 pixels on a smaller section where each side

has been reduced by a factor of 5, one can achieve the resolution of the original photo. For

this purpose, the visual angle has to be reduced by a factor of 5 too, horizontally as well as

vertically (Fig 9).

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Fig 7 Scenery in the town of Pontrieux (France). Picture taken with a total of 481.000 pixels. The frame

indicates the area one will soon give a close inspection.

Legends to the figures on next page:

Fig 8 (At the top oft he following page)

In case the capacity of the system amounts to only 19.200 pixels (per time given), the image appears blurred.

High resolution can be achieved only by sacrifizing overview, ie, retrieving all the pixels from a smaller area

(white rectangle).

Fig 9

All the 19.200 pixels have been concentrated on an area smaller by a factor of 25 compared to the overview (a

factor of 5 on each side). Resolution has been improved considerably. it matches the resolution presented in

Fig 7. However, the improvement is achieved on the expense of gaining knowledge on what´s going on around.

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Fig 8

Fig 9

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