As Column Taper Increase, Greek Column Illusion Increases and Then Becomes Multistable

John H. Krantz and Roger L. Terry

Hanover College

This research was presented as a poster at the sixth annual convention of the American Psychological Society.

Outline
Abstract...Introduction...Method...Results...Conclusions...References/Acknowledgments

Abstract

Ancient Greeks leaned Doric columns inward to make them appear vertical countering an illusion. The relationship between the size of illusion and column tapering was determined using a staircase procedure. Initially, the illusion is linearly related to column taper; however, as taper increases, the perception becomes multistable. (01)(21)(57)

Introduction

The ancient Greeks modified Doric temples to countermany visual illusions (Coren and Girgus, 1978; Dinsmoor,1950). For example, on many temples the columns lean towards the middle of the row of columns. Apparently, thearchitects were trying to prevent the columns from appearing to lean outward as can be seen when the temples were builtwith truly vertical columns (Photograph of the Temple of Apollo at Corinth below).

Previous research (Krantz, 1993) investigated whether column taper (the narrowing of the column diameter as the column rises) or the height of the columns were factors in theillusion. The results of two experiments found strong evidence that column taper was a factor in the illusion, but no evidence that the height of the columns were a factor.

In the present experiment, column taper wassystematically manipulated over a wide range. It was hoped to determine the functional relationship between column taper and the size of the illusion. At first the range was over values that encompassed and exceeded those used in actual Greek temples, then the range was extended to see if the obtained linear function broke down. In addition, one of the hypotheses for the illusion was that of misapplied size-constancy. To test this possibility, columns were tested in both a vertical and horizontal orientation.
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Method

Subjects

. Twenty-eight colleagues and students (ages 7 ª 55) participated in the experiment. All subjects wore corrective lenses if prescribed.

Stimuli and Apparatus

. The stimuli were various sets ofthree columns. One set of columns was linearly tapered, called tapered, and the other sets were not, called straight. The basic dimensions of the columns are shown in Figure 1.
The column sets were matched for surface area and total area between the columns. Column taper was measured as the diameter of the top of the column divided by the diameter of the bottom of the column. The first range of column tapers were .5, .75 (used in Krantz, 1993), .875, 1,1.14, 1.33, 2. This range was chosen because it exceeded the range used in actual columns. To further test the relationship between column taper and the illusion a second set of column tapers were developed: .125, .25, .5, 1, 2, 4, 8.

The columns were presented on a three-dimensional mock temple. The center column served as the vertical reference. The outer columns could be attached at the top tomany different positions causing the columns to tilt varying degrees. The middle position was vertical. The two outercolumns were always attached in symmetrical positions. The entire apparatus could be placed either horizontally or vertically. Viewing distance was 2m.

Procedure

. Column taper was a within subject variable for each column taper range and column orientation was a between subject variable. A forced-choice staircase procedure was used to determine the perceived vertical forthe outer columns as a pair (Cornsweet, 1962, Figure 2).
Two staircases were run in each condition, one starting ateach of the extreme positions. The staircases were run for 10 turnarounds with the perceived vertical for the staircase wasthe mean of the 10 turnarounds. The perceived vertical for the condition was the mean of both staircases. The design ofthe experiment is shown in Figure 3 and 4 for the small and large taper ranges, respectively.

Conditions were counterbalanced across subjects.
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Results

Small Column Taper Range (.5 - 2)(n=14)

The means perceived vertical as a function of log columntaper are shown in Figure 5.
No column taper is a log taper value of 0. The solid line represent the data from vertical columns and the dashed line shows the data from horizontal columns. As can be seen both data sets show that the perceived vertical is a linear functions of columntaper. Columns narrower at the top than bottom lean in and columns wider at the top than bottomlean out when perceived vertical. These findingsare consistent with previous research (Krantz, 1993)and the Greek refinement (Dinsmoor, 1950). The slopes and r2 values for linear regressions run separately on each column orientation are also shown in Figure 5. The r2 are very high suggesting that perceived vertical is a linear function of column taper. This tight linear relationship is also seen in the individual subjects data where only three subjects had an r2 below .8. Surprisingly, the slope for the regression line fittingcolumn taper to perceived vertical for horizontalcolumns is nearly twice the slope for vertical columns.

Large Column Taper Range (.125 - 8)(n=14 new subjects)

The subjects who were run on the large range of column tapers had their results combined with the subjects who were run on the narrow range of column tapers. Figure 6 show the mean perceived vertical as a functionof log column taper across the entire range of columntaper tested.
For the horizontal columns, the tilt of the perceived vertical increase as log column taper increases away from 0 up to a point and then an asymptote appears. For the vertical columns, the results are more perplexing. When the top is larger than the bottom, the results seem similar to the horizontal columnsonly not as extreme. When the top is smaller than the bottom, thetilt of the perceived vertical first increases and then decreases and even, perhaps, reverses. Two subjects, to date, have been run under all conditions, and their results are similar to the above description. Some subjects from the large column range reported that the tilt of the columns appeared to vary from one moment to the next. This suggested that the extremevalues of column taper may lead to a multistableperception like the Necker Cube. The data across subjects were reanalyzed separately for staircases beginning at the inside and outside positions. The results are shown in Figures 7 and 8 for vertical and horizontal columns, respectively. The data in Figure 7 shows that for extreme valuesof column taper, the staircases beginning at the two positions lead to very different and not overlapping values for mean perceived vertical. As expected from the greater regularity from the horizontal column data, the two staircase starting positions do not lead to significantly different values for perceived vertical.
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Conclusions


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Refences

Coren, S. & Girgus, J. S. (1978). Seeing is deceiving: The psychology of visual illusions. Hillsdale, NJ: LawrenceErlbaum.

Cornsweet, T. (1962). The staircase-method in psychophysics. American Journal of Psychology, 75,485-491.

Dinsmoor, W.R. (1950). The architecture of the ancient Greeks. New York: Batsford.

Krantz, J. (1993). The taper of Greek doric columns cause perception of column tilt. Poster presented at the Fifthannual convention of the American PsychologicalSociety. Manuscript submitted to Perception.

Acknowledgments

The authors would like to heartily thank Stephanie Schwartzkopf and, especially, Jody Ballard for their assistance in collecting the data.
Outline
Abstract...Introduction...Method...Results...Conclusions...References/Acknowledgments