New Scientist - USA (2021-02-13)

13 February 2021 | New Scientist | 19

SOME Chinese noblemen were

using cosmetic face cream

2700 years ago. Archaeologists

have found an ornate bronze jar

containing the remains of a face

cream, which was made from a

mixture of animal fat and a rare

substance called moonmilk that

is found in caves.

The discovery is the earliest

evidence of a Chinese man using

cosmetics, although Chinese

women did so earlier than this.

In 2017 and 2018, Yimin Yang

at the University of the Chinese

Academy of Sciences in Beijing

and his colleagues excavated a site

called Liujiawa in northern China.

It dates from the Spring and Autumn

period (771 to 476 BC) of Chinese

history, centuries before the country

was first unified by the Qin dynasty.

During this period, Liujiawa was the

capital city of a state called Rui.

In the 2700-year-old tomb of a

nobleman, the researchers found a

bronze jar. Inside were lumps of a

soft, yellow-white material. They

immediately suspected that this

was cosmetic cream. Chemical

analyses later confirmed this and

revealed two main ingredients.

The first was animal fat,

which came from a ruminant

animal that had been fed lots

of grass-like plants.

The second ingredient was a

form of watery calcium carbonate

called moonmilk. It is a soft, white,

creamy substance that forms inside

caves. The cream would have made

the man’s face white, says Yang

(Archaeometry, doi.org/ftvg).

The face cream is the earliest

example associated with a Chinese

man. However, evidence of Chinese

women using cosmetics goes back

further. In 2016, Yang’s team

studied red cosmetic sticks from

1980 to 1450 BC, which were

buried with women in China.  ❚

UNRAVELLING knots just got

easier. One of the biggest problems

in the mathematical study of

knots is recognising the difference

between an actual knot and a piece

of string that can be untangled

into a single loop. A new algorithm

can find this “unknot” far faster

than any previous one can.

Mathematically, the definition

of a knot is a closed curve – like a

piece of string with the ends tied

together – that can’t be untangled

into a simple loop. Anything that

can be untangled into a simple

loop, no matter how complicated

or tangled it appears at first, is

called the unknot. “Just like zero

isn’t a number, the unknot isn’t

a knot,” says Mark Dennis at the

University of Birmingham, UK.

Mathematicians have been

working on algorithms to tell

whether a given knot is actually

the unknot for about 100 years,

and pioneering mathematician

and computer scientist Alan

Turing even wrote about it in his

final published paper in 1954.

Now, Marc Lackenby at the

University of Oxford has come up

with an algorithm that can make

this distinction far faster, which he

presented at a recent seminar at

the University of California, Davis.

“Although finding the unknot

seems quite intuitive because

throughout our lives we are

untangling wires and pieces

of string and headphone cords

and things, it turns out that

mathematically it touches on

much more abstract areas of

maths, questions to do with

geometry in higher-dimensional

spaces,” says Dennis. “There are

knot diagrams that you need to

make more complicated before

you can simplify them down,” he

says, and computers aren’t great

at recognising when to do so.

The level of complexity of

a given knot is defined by the

number of crossings it contains.

A crossing is the spot at which one

part of the string passes over or

under another part, and any tangle

that can be manipulated so that

it has no crossings is the unknot.

“You might expect it not to be a

difficult problem, and the issue is

that when you start to think about

how a computer would actually

decide such a question, you realise

that you don’t have the right tools

to even come to a decisive answer

about whether a thing is or isn’t

knotted,” says Lackenby.

Other mathematicians have

designed algorithms that can find

whether a given tangle is knotted

or not, but every added crossing

doubles the time needed to solve

the problem. Lackenby’s algorithm

can figure it out faster than that.

His work relies on defining each

knot as representing the edge

of a three-dimensional shape.

“You can imagine a round

unknot just lying in the plane,

well that’s the boundary of a disc,”

says Lackenby. “Or you can imagine

taking a strip of paper and gluing it

together in a loop with some little

twists in it, and the boundary of

that strip of paper will be a knot.”

If the shape corresponding

to a knot can be manipulated

and simplified into a disc, that

knot is actually the unknot.

Determining whether a knot

is the unknot has far-ranging

applications, from studying

how DNA is tangled up within

cells to understanding the loops

of plasma that make up stars,

so a faster algorithm could be

enormously helpful. ❚

Maths

Leah Crane

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News

“ There are knot diagrams

that you need to make

more complicated before

you can simplify them”

Archaeology

Michael Marshall

Breakthrough for 100-year-old

knotty maths problem

The 2700-year-old bronze

jar (left) and the face cream it

contained (right) were found

in northern China

1 cm 1 cm

Ancient face cream

was made from cave

‘milk’ and animal fat