When Jason Padgett pours cream into his morning coffee, this is what he sees:

“I watch the cream stirred into the brew. The perfect spiral is an important shape to me. It’s a fractal. Suddenly, it’s not just my morning cup of joe, it’s geometry speaking to me.”

Padgett’s world is bursting with mathematical patterns. He is one of a few people in the world who can draw approximations of fractals, the repeating geometric patterns that are building blocks of everything in the known universe, by hand. Tree leaves outside his window are evidence of Pythagoras’ theorem. The arc that light makes when it bounces off his car proves the power of pi.

He sees the parts that make up the whole. And his world is never boring, never without amazement. Even his dreams are made up of geometry.

“I can barely remember a time,” the 43-year-old says, “when I saw the world the way most everyone else does.”

Flash back 12 years: Padgett had dropped out of Tacoma (Wash.) Community College, and was a self-described “goof” with zero interest in academics, let alone math. The only time he dealt in numbers was to track the hours until his shift ended at his father’s furniture store, tally up his bar tab, or count bicep curls at the gym.

With his mullet, leather vest open to a bare chest, and skintight pants, he was more like a high-school student stuck in the 1980s — even though it was 2002, and he was a 31-year-old with a daughter.

He would race his buddies in a freshly painted red Camaro. His life was one adrenaline rush after another: cliff-jumping, sky-diving, bar-hopping. He was the “life of the party.” The guy who would funnel a beer before going out and would slip a bottle of Southern Comfort in his jacket pocket to avoid paying $6 for mixed drinks.

“I thought it would go on that way forever,” Padgett says.

Party time came to end the night of Friday, Sept. 13, 2002, at a karaoke bar near his home. There, two men attacked him from behind, punching him in the back of the head, knocking him unconscious.

He fell to the ground as the two men punched and kicked him, stopping only when he handed over his worthless jacket.

He was rushed to the hospital, where a CT scan revealed a bruised kidney. He was released that same night.

The next morning, while running the water in the bathroom, he noticed “lines emanating out perpendicularly from the flow. At first, I was startled, and worried for myself, but it was so beautiful that I just stood in my slippers and stared.”

When he extended his hand out in front of him, it was like “watching a slow-motion film,” as if ­every slight movement was in stop-motion animation.

Days went by, but the visuals remained.

Padgett, who had scored relatively high on IQ tests in elementary school but reached only pre-algebra in high school, soon became “obsessed with every shape in my house, from rectangles of the windows to the curvature of a spoon.”

When he looked at numbers, colorful shapes superimposed over them.

He stopped going to work and began to read anything he could get his hands on about math and physics. He developed a fascination with fractals and pi.

The doctors called what happened to him a “profound concussion.” Little did they know just how profound it was.

Padgett is one of only 40 people in the world with “acquired ­savant syndrome,” a condition in which prodigious talents in math, art or music emerge in previously normal individuals following a brain injury or disease.

Acquired savants like Padgett raise remarkable questions about the rest of us average folk: Do we all have inner Einsteins just waiting for the right bop on the head to be set free? Do we possess inner greatness?

“I believe I am living proof that these powers lie dormant in all of us,” Padgett writes in his memoir, “Struck by Genius: How a Brain Injury Made Me a Mathematical Marvel” (Houghton Mifflin Harcourt), out Tuesday.

“If it could happen to me, it could happen to anyone.”

After Padgett’s brain was shaken up, so was his perception of the world.

“I noticed the light bouncing off a car window in the form of an arc, and the concept came to life,” he writes. “It clicked for me ­because the circle I saw was subdivided by light rays, and I realized each ray was really a representation of pi.”

Overcome by his realization, he began to draw out the images. Although he never had an aptitude for art before, now it was as if “someone else were clutching my fist and guiding my hand.” Drawings had to be perfect. Sometimes they took days; a few took weeks.

During one of his meditations, he came to the conclusion that “circles don’t exist.”

“It was like a bomb went off in my mind. In a matter of minutes, I was no longer just a receiver of geometric imagery or a researcher; I was a theorist,” he writes.

The Post showed some of Padgett’s drawings to Tim Chartier, a math and computer-science professor at Davidson College in North Carolina and author of “Math Bytes” (Princeton University Press).

“It’s remarkable that he sees the world this way without any real training,” says Chartier. “Is that genius? I think you have to be careful when you use that word, but, yes, to be able to see that. That’s just wild.”

Padgett reminds Chartier of Srinivasa Ramanujan, an early-20th-century mathematician who significantly contributed to number theory despite never receiving formal training.

But Padgett is not the next Stephen Hawking. This ability allows him to see the world in a unique way — but it’s highly ­unlikely that his ability will land him a Fields Medal.

“He needs the help of a trained mathematician,” says Chartier.

There were downsides that came along with the new Padgett. Once gregarious and outgoing, he now refused to leave the house. He nailed blankets to the window and refused visitors. He became obsessed with germs and washed his hands until they were red and raw.

He couldn’t even hug his own daughter until she washed her hands.

He began to fear that this wasn’t a gift at all — that it all was a sign of mental illness.

Reassurance came from a BBC documentary that featured Daniel Tammet, who could recite pi to the 22,514th digit. Tammet is an autistic savant, as well as a synesthetic one, which means that multiple senses are evoked — such as “smelling” colors or, in Padgett’s case, matching numbers and colors.

“That’s it! That’s what’s going on with me. Oh, my God! Someone else can see what I see!” Padgett recalls thinking.

He began to Google and found that there were others — people, unlike Tammet, who had ­acquired their “gifts.”

There was Orlando Serrell, who, after being struck by a baseball at age 10, could suddenly tell you the day of the week of any given date; Dr. Anthony Cicoria, who began expertly playing the piano after he was struck by lightning; and Alonzo Clemens, who was a child with an IQ of 40, yet, after falling on his head, could sculpt any animal out of clay down to the most minute detail after seeing it only briefly.

Padgett reached out to Wisconsin psychiatrist Dr. Darold Treffert, the world-recognized expert on savantism who had studied Kim Peek, the inspiration for “Rain Man,” and championed use of the word “savant syndrome” instead of “idiot savant” in 1980.

Via e-mail — and later in person — Treffert diagnosed Padgett with acquired savant syndrome, one of only 30 people identified at the time. (The number has since risen to 40, Treffert tells The Post.)

Padgett wasn’t alone, and this comforted him. He tore the blankets off the windows and enrolled in a local community college.

Further reassurance arrived in 2011, when Dr. Berit Brogaard, now director of the Brogaard Lab for Multisensory Research at the University of Miami, invited him to Finland for a three-day ­research work-up.

She used fMRI machines and transcranial magnetic stimulation (TMS) to find that the right side of Padgett’s brain had been compromised and that there was greater activation on the left side. She noted significant increased activity in the left parietal lobe — which is where neuroscientists say “math lives.”

The parietal lobe is involved with many complex computations used in our daily lives. Reach out for a cup of coffee while reading, and your brain is making seriously complex calculations (charting the distance, the weight, the velocity of movement, etc.) — all of this without our realizing it.

“One could speculate that [Padgett] has better access to these areas than the rest of us,” Brogaard says.

This supports emerging research that shows that bilateral involvement in the parietal lobe (meaning both sides are activated) actually is correlated with worse math abilities. The brain likes to be specialized — and Padgett is ­hyper-specialized.

But how did it get this way? How did the brain know to specialize after an injury?

Theories involving neuroplasticity, or the ability of the brain to make new connections, abound. Brogaard hypothesizes that the trauma of the event flooded the brain with neurotransmitters, which ultimately changed its structure.

Treffert believes that the structural changes allow Padgett to tap into his “genetic memory” — the same kind of instinctual memory that guides birds to fly in a “V” formation — freeing up areas that are inhibited in healthy brains.

“It shows us that ordinary people have untapped abilities,” says Brogaard. This sentiment is one that every researcher interviewed by The Post repeated.

In a series of studies at the University of Sydney in Australia, people wearing a “thinking cap,” a device that immobilizes parts of the brain, were able to draw in greater detail and complexity, find mistakes in written language, solve complex puzzles and more accurately guess the number of objects in a large sample size. But these advances happen only while wearing the cap and fade an hour later.

It’s enough for author Allan Snyder to conclude, “Savant skills are latent in all of us.”

The truth is we know very little about our 3-pound organs, says Dr. V.S. Ramachandran, director of the Center of Brain and Cognition at the University of California, San Diego.

“All the progress and advances we’ve made in neuroscience over the years, yet we know precious little of higher brain functions. These anomalies, as scientists call them, show the depth of our ignorance,” he says.

But do we even want to know? Would we be happier as savants?

Asked whether Padgett would go back to his old life if he could, he responded:

“No.”

Then, after a pause, he added, “though sometimes I do miss the blissful ignorance of life before.”