Figure 1.
A version of the Delboeuf illusion.
Circles c and s are the same size, but c appears smaller when surrounded by a larger circle, r. If one adjusts the size of c to appear the same as that of s, one makes it larger than it should be.
Figure 2.
Data from one of the participants in our study to conventional displays of the Delboeuf illusion.
The inner circle (c) had a fixed diameter of 10 mm. The outer, inducing circle had a range of values from 20 mm (relative inducer size of 2) to 120 mm (relative inducer size of 12) in 10-mm steps. The participant adjusted the size of a circle (s; not shown) in the lower left of a computer-monitor screen to match the apparent size of c. Three such stimuli are illustrated, spatially to scale, in grey–one in which the relative inducer size is 2, one in which the relative inducer size is 7, and one in which the relative inducer size is 12. The small plot circles are the mean differences between the adjusted size of s and its true size (10 mm) for each value of relative inducer size from three trials each (the participant received the trials in a completely random order); the vertical bars are standard errors. The red line is the statistically significant, best-fitting, positively sloping, linear function.
Figure 3.
Schematic representation of a cross-section of a single-rooted tooth, such as a canine.
The gum on the left is on the side of the lips; the gum on the right is on the side of the tongue.
Figure 4.
Schematic representation of the same tooth from Figure 3, illustrating the three main stages in root-canal therapy.
A. The tooth with infection in the apical part of the root following death of the pulp. The dentist begins by making a cavity in the crown, entering the pulp space. B. The dentist cleans the root canal, and some of the surface of the canal, using fine, tapered files. The dentist also treats the infection with medication, leaving it there for one week. C. The dentist fills the canal with gutta percha and the crown with a composite resin.
Figure 5.
Schematic representation of the same tooth from Figure 4C, of the three main stages in apicectomy.
A. The tooth with persistent infection in the apical part of the root. The specialist dentist, endodontist, begins by resecting (reflecting) the gum, exposing the bone of the lower jaw. B. The endodontist removes bone to expose the apical part of the root, creating the bony crypt. Then the endodontist removes and discards the apical 3 mm of the root, leaving the cut face. Then the endodontist uses a very fine, ultrasonic cutting tip to prepare a cavity in the root canal. C. The endodontist fills the cavity and sutures the gum. Eventually the surrounding bone will grow to fill the bony crypt.
Figure 6.
A prepared tooth, with a resected root end and a canal filled with gutta percha, mounted in a tube for work by an endodontist.
On the left is the side view. On the right is a magnified plan view of the resected root end on its painted, red background of resin.
Figure 7.
Illustration of the simulated bony crypt.
Left: The simulated bony crypt. Right: The simulated bony crypt in place over a prepared resected root end.
Figure 8.
Cut faces of four root ends prior to the endodontists’ operating on them (root tips removed and pink gutta percha visible), showing some of the measures we took.
For the upper left tooth we show the length of the long axis of the root acoss the root face (rl) and the length of the canal on the same axis (cl). For the upper right tooth, we show the width of the short axis of the root acoss the root face (rw), and the width of the canal on the same axis (cw). We repeated all measures after the teeth had been operated on by the endodontists, except that the central lengths and widths across the root face were of the cavities.
Figure 9.
Relation between the potency of the Delboeuf illusion (relative inducer size) in teeth supplied to each endodontist to how much he or she cut into the tooth along the long axis of the root (adjusted length – canal length).
Each graph shows the regression equation, the correlation coefficient for the relationship, and whether the relationship is statistically significant (*p<.05; ****p<.0001). In general, endodontists increased the length of cavities more in teeth showing a strong Delboeuf illusion than in teeth showing a weak Delboeuf illusion.
Figure 10.
Relation between the potency of the Delboeuf illusion (relative inducer size) in teeth supplied to each endodontist to how much he or she cut into the tooth along the short axis of the root (adjusted width – canal width).
Each graph shows the regression equation, the correlation coefficient for the relationship, and whether the relationship is statistically significant (*p<.05; ****p<.0001). In general, endodontists increased the width of cavities more in teeth showing a strong Delboeuf illusion than in teeth showing a weak Delboeuf illusion.
Figure 11.
Scale drawing of the average shape of the roots across the root face (in blue), the average shape of the cavities endodontists made (in red), and the average shape of the canals of the teeth (in green).
The canals were fatter ellipses than the cavities. The cavities resembled the shape of the roots more than of the canals.