Childhood is when we learn how to piece the world together. We have to find our way through a labyrinth of riddles and conundrums, which can only be solved by the accident-prone inductive method. Few of us progress through this wonderland with its mad games and bizarre rules as confidently as Lewis Carroll’s Alice; hard work is needed to decipher it, and the mechanical tasks to which we graduate when we grow up are a doddle in comparison to all that desperate infantile cogitation.

In brainy Manhattan – where acculturation starts when embryos listen to Mozart in the womb, and admission to the best kindergartens depends on a series of quizzes and interviews to establish mental aptitude – a new museum recognises that childhood is a time spent thinking, not idly playing. Occupying two floors of a building just off Fifth Avenue on East 26^th Street, the Museum of Mathematics was founded last year by Glen Whitney, a Harvard-educated specialist in logical puzzles like Gödel’s infamous theorems, whose mathematical career included a spell as a quantitative analyst for a hedge fund. Whitney believes that “kids are explorers”, and here he invites them to explore a gymnasium in which the climbing frames and slippery slides are purely conceptual.

The museum’s abbreviated name is the first of its tricky intellectual tests. It is known as MoMath, which mimics MOMA, the acronym of the grandiose Museum of Modern Art, two miles away uptown. Its logo is π, the symbol for the Greek letter pi, with its thin legs and elegant twirly crossbar, which stands for a number that is irrational; the icon that identifies the place is a ring of pis with intertwined limbs that jump up and down like rambunctious infants venting excess energy, and a Logo Generator in the museum allows you to choreograph any number of pis and make them jiggle in ever more frenetic rings.

Cleverly, the exhibits placed nearest the entrance are rides for both brain and body, substitutes for the thrills that less highbrow children enjoy at amusement parks. Instead of screaming as you are tormented by g-forces on a roller coaster, you can sedately trundle across a pit of acorns in MoMath’s Coaster Roller, enjoying a smooth passage because the balls have a constant diameter.

The pis above the front door at MoMath seem to threaten a numerical orgy, which gave me a headache in advance. In fact, apart from a calculator made from a parabola of string that hangs in the stairwell between the two floors, arithmetic is mercifully absent. MoMath is really about geometry, which is the study of magnitudes and the way they cohere in space. A Structure Studio coaxes you to build in three dimensions, with creations that curve instead of growing at stiff, gawky right angles, so you can make geodesic domes and transparent footballs rather than the boxy contraptions I used to fit together when playing with my Lego or Meccano sets.

These are cosmological exercises, attempts at what the great physicist Arthur Eddington called “world building”. Eddington’s theorising had its origins in the shaky assemblages of bric-a-brac he played with as a boy; not being God, he confessed “I cannot make the world out of nothing”, but he said that if he possessed the right ingredients – to be exact, 256 numerical coefficients – he could design what he called “the palace of the mind”. The frowning, preoccupied children I watched were doing exactly the same.

The games played at MoMath have a more practical application. They also function as a training for American life – for its cut-throat competitiveness, its relentless work ethic, and its insane invention of numbers that are, in the jargon of the mortgage market, “sub-prime”. In a seven-sided temple called the Mathenaeum, children standing at what looks like the tiller of a ship at sea customise polyhedrons and submit their creations to be voted on; the smartest design is printed in 3-D at the end of the day. In the Tile Factory, shapes can be bent and stretched into tessellating patterns. The challenge here is cerebral, but the video screen accompanying the display shows an actual factory with industriously fuming chimneys – a relic from the days before America outsourced manufacturing to China and India. An exhibit entitled Edge FX is a better apprenticeship for the current American economy, which depends on a financial abracadabra that juggles ever-growing rows of zeroes. Here balls representing business transactions tumble down chutes and are classified as good or bad, compelling you to adjust the bias of the ball you’re aiming. You can then sit back and watch your profits pile up in a teetering skyscraper of gains, or see yourself wiped out. What better introduction could there be to the chancy fatalism of Wall Street trading? I suspect Edge FX exudes bad karma, and on the day I visited no one but me went near it.

The exhibits that pleased me most are about art, not wealth. Kids are hyperactive, so MoMath sits them in a hyperboloid, where a spinning chair turns straight lines into a curved surface. Another exhibit is a reminder that painting, as Marcel Duchamp said, is more than a merely “retinal” activity. The implements provided are notional, abstract: the brush has no fibres, the pots of paint are empty, the canvas a slick digital screen on which your hand can’t make a mark. But the motions of the dry brush conjure up webs of glorious intricacy, and as they materialised I began to understand why William Blake found symmetry “fearful”: we are part of a hidden order that is invisible to us, and every random movement we make obeys inflexible laws.

I ended, bemused by the dual ingenuity of nature and technology, by learning about fractals while being changed into a Human Tree. My body appeared on a wall, multiplied by successively smaller copies of its already puny self. As these fractional twins sprouted from my limbs, they turned me into vegetation: my legs were tree trunks, my arms branches, my fingers twigs, and there were probably birds’ nests in my hair. Beside me, some children who looked like animated shrubs pranced about, waving their arms to make themselves grow into tall fractal trees. I wasn’t vigorous enough to manage this endless self-duplication, and stood there as if waiting to wilt and lose my leaves. Childhood requires physical dynamism as well as mental force; it is probably best left to the young.