A shape with finite volume but infinite surface area has been known since the 1600s

3:18

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A shape with finite volume but infinite surface area has been known since the 1600s

the gist

Gabriel's horn is made by taking the curve y = 1/x for x of 1 or more and spinning it around the x-axis, giving a trumpet-shaped solid that gets ever narrower and runs on forever. Its volume adds up to a finite amount, exactly π, while its surface area is infinite. The paradox rests on the fact that a sum of ever-thinner circles can converge while the sum of the outside surfaces diverges. Evangelista Torricelli studied it in 1643, and the name comes from the Christian idea of the archangel Gabriel blowing a trumpet on Judgment Day.

how it works

The solid is formed by rotating a curve whose height shrinks as you move outward. The circular cross-sections get thin fast enough that their total volume adds up to a finite number, but the strip of surface along the length shrinks more slowly, so its total area keeps growing without limit.

so what

It is a clean example of how infinite processes behave differently from everyday intuition, and it helped build the early calculus of infinite sums and integrals. Mathematicians still use it to show that finite and infinite quantities can separate in surprising ways.

the bigger pattern

Similar infinite-yet-finite results appear in the Koch snowflake and other fractals, which enclose finite area while their edges are infinitely long. The same tension between finite and infinite shows up in Zeno's paradoxes of motion.

fun fact

Torricelli is also known for the barometer, which he developed in 1643 after studying air pressure.

From Wikipedia, the free encyclopedia · text CC BY-SA 4.0

source: Wikipedia ↗

historyWikipediaclassic1336–1565 · Karnataka, India

Hampi was once one of the world's largest cities, until its Vijayanagara capital was destroyed in 1565

the gist

Hampi is a UNESCO World Heritage Site in Karnataka, India, with more than 1,600 surviving remains spread over 4,100 hectares. It was the capital of the Vijayanagara Empire from 1336 to 1565. Travellers from Persia and Portugal described it as a prosperous, wealthy and grand city with temples, markets and farms along the Tungabhadra River. By 1500 it was estimated to be the world's second-largest city after Beijing, and probably India's richest at the time. In 1565 a coalition of Muslim sultanates defeated the Vijayanagara Empire, its capital was conquered and destroyed, and the city was left in ruins. Hampi remains a religious centre, with the active Virupaksha Temple at its heart.

how it works

The city was built around a fortified core with temples, markets and water structures along the river, supplied by an extensive network of roads and waterworks. When the empire fell, the city was not rebuilt, so its stone monuments were left standing.

so what

Hampi shows how quickly a great city can vanish after a military defeat, and how a ruined capital can remain sacred. Its ruins give historians a view of medieval South Asia that few other sites can match.

fun fact

The name Hampi comes from Pampa, an old name for the Tungabhadra River.

From Wikipedia, the free encyclopedia · text CC BY-SA 4.0

Image: Wikimedia Commons

source: Wikipedia ↗

2 min

politicsshortCNA2:01

False allegations about Singapore’s hand in Indonesia’s markets and a made-up meeting

from CNA

CNA Verify has found close to 400 posts pushing falsehoods tying Singapore to Indonesia’s stock market rout and to a secret meeting that never took place. Together, they drew nearly 12 million views, and some were circulated by accounts that appeared to be working in concert, across multiple social media platforms.

Here’s how the disinformation campaign targeting Singapore was uncovered.

on YouTube: CNA ↗