Read this first
This page is a working theory, not settled science. Where a statement rests on published research, we cite it. Where it is our own idea, analogy or speculation, it is marked “Working theory”, “Metaphor” or “Speculation”. None of the Bindu Tessera exercises has been tested in a study of insight, creativity or brain networks.
Insight as a rhythm
Insight rarely arrives on command. The research suggests it comes less from one special state of mind than from a rhythm of several. Here is that rhythm as we understand it:
1. Focus: real, effortful engagement with a problem or an object of attention.
2. Notice and switch: catching the moment the mind drifts, or the image flips, and choosing where attention goes next.
3. Release and incubate: setting the problem aside, ideally while doing something easy.
4. Sleep onset: the drowsy edge of sleep, and then a proper night's sleep.
5. Aha: a solution or a new way of seeing arrives, often suddenly.
6. Check: testing whether the idea actually works.
Several parts of this are well supported. A meta-analysis of incubation studies found that setting a problem aside does help, especially after longer preparation and when the break is filled with an undemanding activity (Sio & Ormerod 2009). In one experiment, people who slept after practising a number task were roughly twice as likely to discover a hidden shortcut in it as people who stayed awake (Wagner et al. 2004). In another, spending just 15 seconds or more in the lightest stage of sleep, at sleep onset, raised the rate of discovering a hidden rule from 30% to 83%, and the advantage disappeared for people who drifted on into deeper sleep (Lacaux et al. 2021).
The “notice and switch” step has a brain-network story too. During focused-attention meditation, the cycle of mind-wandering, noticing, shifting attention and holding it maps in turn onto the default mode, salience and executive networks (Hasenkamp et al. 2012). The salience network, centred on the anterior insula, is thought to pick out what matters and switch the brain between inward-facing default-mode activity and outward-facing executive control (Menon & Uddin 2010).
Working theory
The visual exercises rehearse steps 1 to 3 in miniature: steady attention, noticing the flip, and letting go. Our hunch is that practising the moves between these states, rather than any one of them, is what matters for insight.
Bringing the default mode network into line
The default mode network (DMN) is a set of brain regions that is most active when we are not focused on the outside world: mind-wandering, remembering, imagining and thinking about ourselves. Meditation is often said to “quiet” it. Newer research points somewhere more interesting: toward cooperation.
Highly creative people show stronger connections between the default, salience and executive networks, systems that usually work in opposition, and this pattern predicted creative ability in independent groups of people (Beaty et al. 2018). Meditation research points the same way. Experienced meditators showed lower activity in the main DMN regions while meditating, along with stronger coupling between the DMN's central hub and regions involved in attention and control (Brewer et al. 2011). In a randomised trial, a three-day mindfulness retreat, compared with relaxation training, increased connectivity between the posterior cingulate (a DMN hub) and the left dorsolateral prefrontal cortex (Creswell et al. 2016). And in a 31-day training study of 46 people with an active control group, mindfulness training increased connectivity between the default mode and salience networks, and between the salience and executive networks (Bremer et al. 2022).
Two honest caveats. First, simply staring at a fixed point does not quiet the DMN. Looking at a fixation point is the standard resting baseline in brain imaging, and it is a condition in which the DMN is active (Raichle et al. 2001). Second, nobody has tested whether our illusion exercises change brain networks at all. “Bringing the DMN into line” is our metaphor for the coordination findings above, not a measured effect of this practice.
Working theory
The aim is integration, not shutdown: a mind whose wandering, noticing and focusing systems work together and hand over smoothly. The exercises are meant as a small, repeatable workout for those handovers.
The mind groks in its sleep
In 2022, a group of machine-learning researchers described a curious effect in small neural networks trained on simple algorithmic puzzles. The networks first memorised their training examples while doing no better than chance on new ones. Then, with continued training long after that point, their performance on new examples climbed from chance to perfect. The authors called this “grokking” (Power et al. 2022).
Sleep seems to do something loosely similar for people. In the Wagner study above, participants practised a number task that contained a hidden rule they had not been told about. Sleep did not just strengthen what they had practised; it made them more likely to find the structure underneath it (Wagner et al. 2004).
Working theory
Practice loads the examples; incubation and sleep give the mind time to find the rule beneath them. This is an analogy between two very different systems, not a claim that brains and neural networks learn in the same way.
Metaphor
Mathematics has its own pictures of this. Galois theory grew from Évariste Galois's work on when equations can be solved by radicals, and it studies an equation through a hidden group of symmetries among its roots (Galois 1832). In topos theory, Olivia Caramello's “bridges” approach shows how two mathematical theories that look entirely different can express the same underlying content, so that results can be carried from one to the other (Caramello 2017). We use these only as metaphors for what insight feels like: seeing the hidden structure that turns many separate facts into one.
Ramanujan, the extreme case
Srinivasa Ramanujan (1887–1920) is often described as a mystic who received mathematics from a goddess. The record is more interesting than that. The earliest account comes from two people who knew him in Madras, his former teacher P. V. Seshu Aiyar and his patron R. Ramachandra Rao, writing just after his death:
“Ramanujan used to say that the goddess of Namakkal inspired him with the formulae in dreams. It is a remarkable fact that frequently, on rising from bed, he would note down results and rapidly verify them, though he was not always able to supply a rigorous proof.”
P. V. Seshu Aiyar and R. Ramachandra Rao, memoir in the Journal of the Indian Mathematical Society (1920), reprinted in the Collected Papers of Srinivasa Ramanujan (1927) and quoted by G. H. Hardy (1940).
G. H. Hardy, his collaborator at Cambridge, quoted that passage and then rejected the mystical reading: “I am sure that Ramanujan was no mystic.” Hardy called his religion “a matter of observance and not of intellectual conviction”, and recalled Ramanujan telling him “that all religions seemed to him more or less equally true” (Hardy 1940). Decades later his widow, Janaki, gave the mathematician Bruce Berndt a similar picture: “No he was not religious at all. He never went to the temple, he was too busy doing his sums.” She added: “Often I had to feed him and put food in his hand so he could continue doing his sums and not have interruptions.” Witnesses disagreed about how devout he was, and none of Ramanujan's own surviving letters mention Namagiri, dreams or God (Berndt & Rankin 1995).
One famous line is often quoted as his: “An equation for me has no meaning unless it expresses a thought of God.” It is an attribution, not his writing. It comes from S. R. Ranganathan's 1967 biography, which reports it as something Ramanujan said to a friend, nearly fifty years after his death (Ranganathan 1967; Alladi 2016).
Working theory
Ramanujan was not a mystic channelling formulas, but a mind doing the structure-finding described above at an extraordinary level: intense absorption by day, and results that often surfaced on waking. Whatever he believed about where they came from, he checked them. This is speculation about one remarkable person, and it cannot be tested directly.
A short history of insight
Many famous discoveries are told as sudden flashes. Some of those stories are well documented; others are legend. Here is where the best-known ones stand.
Documented
William Rowan Hamilton, 16 October 1843. Walking along the Royal Canal in Dublin, Hamilton saw the rule for quaternions, i² = j² = k² = ijk = −1, and carved it into the stone of Brougham Bridge. He described the moment in an 1865 letter to his son. The carving is gone, but a plaque marks the spot, and the flash came after some fifteen years of work on the problem (Graves 1885).
Henri Poincaré. In his essay “Mathematical Creation”, Poincaré described how, after setting aside a problem about Fuchsian functions, the key idea came to him just as he stepped onto an omnibus at Coutances. He argued that this kind of unconscious work follows, and depends on, a period of intense conscious effort (Poincaré 1910).
Isaac Newton. Humphrey Newton, his assistant during the years Newton wrote the Principia, recalled that Newton “very rarely went to Bed, till 2 or 3 of the clock, sometimes not till 5 or 6”, and that “oftimes he has forget to eat at all” (letters to John Conduitt, 1727/8). This was written about forty years later by an eyewitness, and it describes absorption, not a method. Newton suffered a serious breakdown in 1693.
Legend or disputed
Archimedes and “Eureka”. The bath story comes only from Vitruvius, writing some two centuries later, and Galileo argued that the method it describes would not have worked in practice. Treat it as legend.
Évariste Galois's last night. On the night before his fatal duel in 1832, Galois did write a long letter summarising his results, including the line “je n'ai pas le temps” (“I have no time”). But he did not invent his theory overnight. The work dates from 1829–31, and the letter itself says the ideas had been in his head for nearly a year (Rothman 1982; Galois 1832).
August Kekulé's snake. In an 1890 speech, Kekulé said the ring structure of benzene came to him in a half-sleep reverie of snakes biting their tails. He told the story 25 years after the fact. Some historians have argued he invented it; others defend it while noting that it cannot be verified and that the theory grew from years of work (Wotiz & Rudofsky 1983; Rocke 1985). Disputed.
A note on sleep. Some of these stories involve long nights, but that is not the lesson. Losing sleep reliably harms attention and working memory (Lim & Dinges 2010), and one night without sleep impaired innovative thinking and flexible decision-making (Harrison & Horne 1999). What the research links to insight is absorption followed by rest and sleep, not deprivation.
The fill-in mind: the hollow face
Look into the back of a hollow mask and you will very likely see a normal face pointing toward you. Move, and it seems to turn and follow you. This is the hollow-face illusion. It holds even though your two eyes report that the surface curves away: the brain's expectation that faces stick out overrides the depth cues (Hill & Johnston 2007; Schneider et al. 2002).
The hollow face: watch the mask turn.
Watch for the moment the hollow back of the mask turns toward you. It still looks like a face pointing out at you, and it seems to turn the wrong way.
Face mask derived from “Infinite, 3D Head Scan” by Lee Perry-Smith (Infinite-Realities), CC BY 3.0. Modified: cropped to a hollow mask and re-rendered.
Disney's Imagineers stumbled on the same effect. According to Jason Surrell's history of the Haunted Mansion, Rolly Crump and Yale Gracey walked past the back of a mould of Abraham Lincoln's face and saw its eyes seem to follow them. That “happy accident” became the Haunted Mansion's busts that appear to turn and watch visitors (D23, quoting Surrell).
The psychologist Richard Gregory argued that perceptions work like hypotheses: the brain's best guesses, built from sensory data plus knowledge from the past (Gregory 1997). The hollow face makes that guessing visible. In one experiment by Gregory and colleagues, people saw the illusory outward face, yet their quick finger movements still went to the real, hollow positions of targets on it (Króliczak et al. 2006).
How strongly we fill in also varies. People with schizophrenia are less likely to see the illusion and judge the mask's real shape more accurately; in one study the illusion returned toward normal as symptoms improved with treatment (Schneider et al. 2002), and another found fewer illusions across faces, scenes and viewing conditions (Keane et al. 2013). A brain-imaging study suggested that in patients, bottom-up signals were strengthened and top-down influences weakened while viewing hollow faces (Dima et al. 2009). A reduced illusion has also been reported in regular cannabis users (Semple et al. 2003), and infants of 7 to 8 months appear to see a screen-based rotating hollow face as hollow, unlike adults (Tsuruhara et al. 2011).
Working theory
This fill-in capacity may be a gift shaped by evolution: a mind built to find faces and structure in incomplete data. It may also be part of how human minds differ from today's AI systems, and one way people could complement them. The exercises let you watch that filling-in happen, and notice the awareness that sees it. This is speculation; we know of no study testing it.
Dreams and Illusions: Two Windows on the Same Engine
A leading idea in neuroscience, often called predictive processing, holds that the brain is constantly generating a model of the world and uses the senses mainly to check and correct it. The neuroscientist Rodolfo Llinás argued that waking and dreaming are fundamentally similar brain states that differ mainly in how much weight the senses get: wakefulness, in his words, is an intrinsic state “modulated by sensory parameters” (Llinás & Paré 1991), and in dreams we are “released from the tyranny of our sensory input” (Llinás 2001). J. Allan Hobson and Karl Friston (2012) framed sleep and dreaming in similar terms: the brain's model of the world keeps running with the senses largely offline, and is tuned up while it does.
The exercises on this site show the same engine from the waking side. The hollow face shows prediction overriding the senses: your eyes report a hollow, but the model insists on a face. The Necker cube shows the model switching between two equally good interpretations while the lines on the screen stay exactly the same. Dreams show the generator running with much less error correction.
“dreams run the same rendering engine as waking perception with half the error correction switched off, which makes them the only place most people ever see the raw output of their own world-generator.”
Working theory
Our working theory is that the meditative skill lies in noticing the moment of generation itself: catching the world as it is being built, so that it feels freshly painted, while the awareness watching it stays steady. This has not been tested. It is something to try in the exercises, not a finding.
The Interface: Perception as a Desktop
The cognitive scientist Donald Hoffman goes a step further. His interface theory of perception proposes that evolution shaped perception not to show us the world as it is, but to guide useful action, much as a computer desktop does (Hoffman, Singh & Prakash 2015). A file's icon is not blue inside the machine. The desktop hides the transistors and voltages so you can act quickly. In evolutionary simulations, and in a formal “fitness beats truth” theorem, perceptual strategies tuned to fitness routinely outcompeted strategies tuned to the true state of the world (Mark, Marion & Hoffman 2010; Prakash et al. 2021). His book The Case Against Reality (2019) makes the full case.
This is a serious, peer-reviewed hypothesis, but a contested one. Critics argue that perception which must serve many goals at once ends up carrying much of the world's real structure (Berke et al. 2022). Hoffman's further claims, that spacetime is not fundamental and that reality is made of conscious agents, go well beyond what the models show (Reynolds 2019).
The exercises on this site fit the metaphor. In interface terms, the hollow face is an icon the brain draws because “face” is the useful answer, even when the evidence says hollow. The Necker cube is one set of lines with two icons on offer.
Working theory
Our view is very similar to Hoffman's: at some point, evolution may have found a desktop, a way to compress an overwhelming flood of input into icons you can act on fast. The practice, as we see it, is learning to see the desktop as a desktop without losing the ability to use it. This is a way of looking, not a finding.
Owen Barfield made a related argument in 1957, starting from the Necker cube; see the Necker cube page.
Mind changes, awareness stays
Every exercise on this site shows the same thing from a different angle. The Necker cube flips while the drawing stays still. The magic eye image reveals a shape that was in the dots all along. The mandala turns around a still centre. The hollow face looks solid even when you know it is hollow. In each case the mind's construction changes, and you can notice that the awareness seeing it does not.
Working theory
That contemplative point is also the right starting posture for insight: holding your current picture lightly enough to let a better one appear.
How this could be tested
A theory like this should be tested, not just believed. Here is a simple study design we would welcome, whoever runs it:
Who: adults new to meditation, randomly assigned to two groups.
What: four weeks of short daily sessions, either with the Bindu Tessera exercises or with an active control of the same length and screen time, such as a simple visual attention task.
Measured before and after: insight problems, the Alternate Uses Task for divergent thinking, voluntary control over Necker cube reversals (holding one view and switching on request), and resting-state connectivity between the default mode, salience and executive networks.
How: hypotheses and analysis plan preregistered in advance, with the people scoring the tests unaware of who was in which group.
If the exercises did no better than the control, that would be worth knowing too.
Sources
Lacaux, C., et al. (2021). Sleep onset is a creative sweet spot. Science Advances, 7(50), eabj5866.
Raichle, M. E., et al. (2001). A default mode of brain function. PNAS, 98(2), 676–682.
Hardy, G. H., Seshu Aiyar, P. V., & Wilson, B. M. (eds.) (1927). Collected Papers of Srinivasa Ramanujan. Cambridge University Press (memoir, pp. xi–xix). (Book.)
Berndt, B. C. (2020). Living with Ramanujan for 40 years. Phil. Trans. R. Soc. A, 378, 20180437.
Berndt, B. C., & Rankin, R. A. (1995). Ramanujan: Letters and Commentary. American Mathematical Society. (Book.)
Ranganathan, S. R. (1967). Ramanujan: The Man and the Mathematician. Asia Publishing House. (Book.)
Alladi, K. (2016). Touched by the Goddess. Inference, 2(2).
Graves, R. P. (1885). Life of Sir William Rowan Hamilton, vol. 2 (includes Hamilton's 1865 letter to his son). (Book.)
Poincaré, H. (1910). Mathematical creation. The Monist, 20, 321–335.
Humphrey Newton to John Conduitt, 17 January and 14 February 1727/8. The Newton Project.
Wotiz, J. H., & Rudofsky, S. (1983). Kekulé or Kekule? Ambix, 30, 133–136.
Vitruvius, De Architectura IX, preface; Galileo, La Bilancetta (1586). (Classical texts.)
Gregory, R. L. (1997). Knowledge in perception and illusion. Phil. Trans. R. Soc. B, 352, 1121–1127.
Llinás, R. R., & Paré, D. (1991). Of dreaming and wakefulness. Neuroscience, 44(3), 521–535.
Llinás, R. R. (2001). I of the Vortex: From Neurons to Self. MIT Press.
Reynolds, H. D. (2019, 8 August). Reality check [review of The Case Against Reality]. Physics World.
Barfield, O. (1957). Saving the Appearances: A Study in Idolatry. Faber & Faber.