Purpose of Research

The intent is to use mathematics to investigate visual illusions, and to reveal their mechanisms.

There are two main benefits of using mathematics. The first is that as the phenomenon can be described in the form of computational procedures, it is possible to predict what will happen when conditions are changed. The second is that as the strength of an illusion can be expressed using the numerical value, it becomes possible to control illusory effect by raising and lowering that value.

These benefits are useful for developing fields of application. By minimizing the strength of visual illusion it is possible to create a readily identifiable environment and improve safety. By maximizing the strength of visual illusion, it is possible to provide new ways of expressing information and improve cultural richness.

These research activities enable the building of flexible and robust mathematical modeling methods to support elucidation of senses and awareness and of computational theory for analysis of those models, and contribute to the progress of mathematics itsel.

Outline of the Project


One important outstanding research topic in the natural world is elucidation of human intelligence. One aspect that epitomizes intelligence is the sensory functions for awareness of environment. Illusions are a phenomenon that confuses the sensory functions, but rather than being an undesirable pathological phenomenon, they are something that manifests in extreme form in circumstances where they serve a function required by the senses, and it is becoming generally accepted that that function is usually useful. Consequently, research into illusions is nothing other than direct research into sensory functions. In addition, because the functions manifest in extreme form, it can be hoped they will make it easier to get to the true nature of the senses.

Research into illusion has historically been undertaken primarily in the field of sensory psychology, and there is already a large body of academic opinion.
Research to date has, however, typically been about observing illusory phenomena and discussing them qualitatively, so that more quantitative analysis is hoped for.

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Objective and focus of research

The foregoing is the context for research from a mathematical science and mathematical engineering perspective, using mathematical modeling techniques together with a focus on visual illusory phenomena. In particular,
(1)  the mechanism of visual illusion will be understood by building and analyzing mathematical models of visual illusions;
(2)  methods will be developed to express visual illusionary effects quantitatively; and
(3)  methods will be explored to control visual illusory effects by optimizing the strength of visual illusion.

Methods will next be created to adapt the deliverables from the above work to the two objectives of minimizing and maximizing the strength of visual illusion, to contribute widely to society.

In the first instance, minimizing the strength of visual illusion provides pointers to the environment needed to enable accurate apprehension of surrounding circumstances. Specifically, it provides mathematical pointers to avoiding accidents that occur as a result of misapprehended circumstances, or traffic congestion that arises as a result of misapprehended incline in a road, and to regulating the over-blown advertising that invites such misapprehensions.

In the second instance, through amplification of the visual illusory effect, maximizing the strength of visual illusion develops many new visual methods of expression, providing ways to make life richer. Specifically, it is useful in preventing important signage being overlooked by making it more striking, providing new means of entertainment using visual illusion, and creating new forms of artistic expression using visual illusory effects.

In addition, the aforementioned understanding of visual illusion through mathematical modeling and repeated return to society of its deliverables will build flexible, robust mathematical modeling methods capable of supporting elucidation of senses and awareness, and computational theories for their analysis.


The first concept to cite in our research is analysis of visual illusions through mathematical models. When it is possible to represent a visual illusion in a mathematical model, the visual illusion phenomenon can be replicated and explained using computation, and it becomes possible to quantitatively express the strength of a visual illusion with numerical values. It also becomes possible, using the computational process, to forecast what will happen if conditions are changed. We name the concept that together describes these methods and their effect as "computational illusion." We hope to develop new directions in research into visual illusion in association with researchers in Japan and abroad.

Second, optimization of degree of visual illusion is also a concept fundamental to our research. It is made possible precisely because it enables quantification of the effects of visual illusion using mathematical models, and opens up many possibilities for practical applications. In other words, by minimizing the strength of visual illusion we contribute to creating a safer society, while conversely, by maximizing the strength of visual illusion we are able to provide new media expression.

The third fundamental concept is establishment of a robust computational method to enable describing and operation of flexible human sensory functions in the modeling of visual illusory phenomena and the computational processes that support that modeling. The behavior of humans, including their senses, is not so clear or not capable of being described in simple mathematics; it is an unsophisticated thing, rife with uncertainties and exceptions. Hence we need to develop new mathematics for modeling. Succeeding in developing mathematics for that purpose then becomes a breakthrough to new possibilities in both mathematics and the field under study. Trusting in these real experiences, in our research we are also engaged in developing robust computational methods that confront reality.

Fourth, the Illusion Museum is another concept fundamental to our research, as a locus for research activity relating to illusions. In the museum we exhibit the latest illusory works of research participants, free to the public, and measure visual illusory effect through surveys of museum visitors, while at the same time the museum serves as a hub for exchange between researchers into visual illusions from all over Japan. The Illusion Museum functions to publicize our research in an easy-to-understand format, and to fulfill our obligation to explain to taxpayers how research funds are used. In addition, the wisdom of illusion researchers not directly involved in our research can be gathered through the museum's function as a forum for exchange among researchers.

Future vision

In our research we focus on the visual illusion phenomenon, but for the future, our aim is an "illusion science" that looks cross-sectorally and explores the principles common to other illusions relating to the five human senses, including touch, taste, smell, and hearing, and to illogical human behavior in social activities, deeming it too as illusion. Through computational theory-based consideration of visual illusion by the research director to date, at least in high-order visual illusions, a basic principle is emerging that illusions appear when there has been failure, in a seeking to supplement insufficient information, and it is a principle that is expected to be capable of application to such illusions as misunderstanding in communication. Based on this expectation, we are trying to propose "cross-sectoral illusion science" as a new field of the Grant-in-Aid for Scientific Research on Innovative Areas of the Ministry of Education, Culture, Sports, Science and Technology. This adds to the illusions of the five senses a broad range of human activity, including illogical choices by humans in economic or voting behavior, misunderstanding in communication, and misinterpretation of media expression, deems them all illusion, and seeks to explore their common principles. As a locus for research exchange, the Illusion Museum is a first step in the ultimate goal of establishing a wide-ranging cross-sectoral illusion science, and positions our research therein.

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