Rotating dual-layered checkerboard illusion

Abstract

We report a novel visual illusion, where a 2 × 2 two-colored checkerboard square rotating against an identical background appears to morph into a circle with a size change. This illusion can be categorized as a subtype of the breathing illusion (BI) based on its phenomenological characteristics. However, it also exhibits intriguing features that may offer new insights into BI's underlying mechanisms, not fully captured by existing displays.

How to Cite this Article

Usui, K., Ishikawa, M., Taya, S., & Kitaoka, A. (2026). Rotating dual-layered checkerboard illusion. i-Perception, 17(1), 1–5. https://doi.org/10.1177/20416695251409283

In everyday visual environments, physical continuity of objects is frequently obscured by occlusion or low contrast with their surroundings, resulting in physically continuous contours appearing as fragmented segments in retinal images. However, through motion of the observer or the objects, the visual system can perceive unified contours and shapes not detectable when static. This perceptual completion across space and time is called spatiotemporal boundary formation (SBF; Shipley & Kellman, 1994). In this paper, we introduce a novel visual illusion related to SBF.

When a 2 × 2 two-colored checkerboard square is superimposed on an identical background and rotated, its perceived shape undergoes dynamic changes, transitioning sequentially from a square to a rounded square, then to a circle, and back again (Movie 1 left). The perceived size also varies with shape, appearing largest with the square, smaller with the rounded square, and smallest with the circle (Figure 1). We refer to this phenomenon as the Rotating Dual-layered Checkerboard Illusion (RDCI).

Figure 1.

Static version of RDCI.

Several visual illusions show changes in perceived shape and size linked to SBF in rotational motion. Among them, those creating a “breathing” impression are classified as “breathing illusions” (BI; Bruno, 2001). Under this definition, RDCI can be considered a new BI variant, though its stimulus configuration has distinctive features: in conventional BI displays, changes in visible edge length are determined by the aperture size (the slit between the occluders; see Movie 1 right), whereas in RDCI they are determined by the zero-contrast border between foreground and background. This gives RDCI a far larger visible-edge variation range than BI: in BI, it runs only from the aperture size $a$ to $\sqrt{2}a$ , which covers only part of the full side length of the rotating square, s, whereas in RDCI, it runs from 0 to s. Here, “0” indicates the moment when the square becomes fully embedded in the background and disappears (Figure 2A, rightmost). This brief loss usually goes unnoticed, demonstrating successful SBF, whereas the changes in shape and size reflect its failure. Thus, RDCI compellingly demonstrates both the success and failure of SBF in a single dynamic display.

Figure 2.

(A) RDCI static frames in several rotation angles (SR in parentheses). (B) Participants’ ratings vs. SR/rotation angle, and (C) ordinal probit model fit. Solid line and ribbon indicate the mean/fit with 95% CI. Dots indicate individual data points.

We considered whether visible edge proportion might determine the apparent shape of the RDCI. The ratio of visible to total (visible + illusory) edge length, called the support ratio (SR), is known to affect the strength of contour completion (Shipley & Kellman, 1992). Yet, this definition is not applicable to BI-family displays, where illusory contour length changes over time. Hence, here we define SR as the proportion of visible edge relative to the perimeter of the distal stimulus (rotating square) defined when generating the stimulus. To test its effect, 10 participants viewed 18 static RDCI frames (0°–85°, 5° steps, Figure 2A), each presented once in random order, and rated perceived shape (1 = square, 5 = circle) on Google Forms. The results first revealed a strong impact of visible corners: square percepts dominated when corners were visible (before 45°, i.e., SR > 0.5), whereas “circle” responses increased gradually after corners disappeared (Figure 2B). We analyzed this response pattern using an ordinal probit model assuming a slope change at SR = 0.5 (Figure 2C), which supports the view that, after corners disappeared, reductions in SR were linked to an increase in “circle” responses (Nagelkerke's pseudo-R² = .78).

Two main accounts have been proposed for the BI. The first attributes it to motion perception per se, where limited edge visibility leads to mislocalization of the perceived rotation center, resulting in the apparent size/shape change (Shiffrar & Pavel, 1991). The second explanation focuses on contour completion (Bruno, 2001). From this perspective, the illusory effect in the BI arises from the difference in size/shape of the completed contours formed by visible edges in each static frame. The fact that the RDCI effect can also be observed in static images (Figure 2) would support the latter account.

Interestingly, when the square element of the RDCI is translated, the size/shape changes similar to those seen during rotation can still be observed (Movie 2 left). This also might support the explanation based on the contour completion in BI, as it predicts that the illusion occurs regardless of motion type. However, it has been suggested that the BI weakens or disappears without rotational motion. For example, a translating square behind occluders typically appears as a nonrigid blob (Movie 2 right, “eccentric square” in Gerbino, 2017) or could appear as a rigid square only under certain conditions (Bruno, 2001, p. 545), neither showing the characteristic “breathing” impression.

Nevertheless, phenomenological differences in these translational versions may also support contour completion as the main mechanism of BI-family phenomena. In translational BI, visible edges appear asymmetrically (Figure 3B), and completion laws predict irregular contours (e.g., Gerbino, 2024). Although such asymmetric visible edges are also shown in translational RDCI (Figure 3A), the cross-shaped border formed by the white and black regions on the moving square remains constant in both size and shape, which may constrain the degrees of freedom of contour completion; for example, this may keep completed contours symmetric, and the perceived size may not fall below the size of the cross. In sum, while stimulus structure shapes how completion occurs, both phenomena can be understood as arising from the contour completion process.

Figure 3.

Static frames of translational RDCI (A) and BI (B).

Although the present discussion has treated BI and RDCI as products of a common mechanism, their stimulus differences complicate direct comparison, leaving their precise relationship open for future study.