Yes, very impressive Richard. I would suggest that it all depends upon how close you are on each side and perhaps it uses the parallax effect… similar to looking at the different ends of a cone?

I think the four limbs are (roughly) quadrants of a circle on the surface of a pyramid. When the eye is placed at the apex of the pyramid, all quadrants are seen edge on, hence the square – just as the pyramid itself appears square in projection. If the eye is at the centre of the “base” of the pyramid, the quadrants are seen to be curved, and make the circle. Wonderfully imaginative!

This installation is called “Squaring the Circle” and is/was at at the Michael Kohn Gallery in Los Angeles

It is not what it appears: the suspended wire sculpture looks to be wrought into a perfectly drawn black circle. View it from another angle, however, and you’ll find that it’s also a square. This defiance of geometry (a sculptural sleight of hand) involves a hybrid shape of right-angled points and 180-degree curves, plus light reference to satirist Edwin Abbott’s 1884 short story Flatland, in which the denizens of a two-dimensional world reject the possibility of life in three dimensions.

The artists are collectively known as Troika. Eva Rucki (b. 1976, Germany), Conny Freyer (b. 1976, Germany) and Sebastien Noel (b. 1977,France) have worked together as an artist trio since 2003.

With a particular interest in perception and the spatial experience, their collective multi-media
works challenge the experience of seeing and knowing. They manipulate our perception of the
world and ask the question why we know what we know, and whether this knowledge is certain.

This installation is called “Squaring the Circle” and is/was at at the Michael Kohn Gallery in Los Angeles

It is not what it appears: the suspended wire sculpture looks to be wrought into a perfectly drawn black circle. View it from another angle, however, and you’ll find that it’s also a square. This defiance of geometry (a sculptural sleight of hand) involves a hybrid shape of right-angled points and 180-degree curves, plus light reference to satirist Edwin Abbott’s 1884 short story Flatland, in which the denizens of a two-dimensional world reject the possibility of life in three dimensions.

The artists are collectively known as Troika. Eva Rucki (b. 1976, Germany), Conny Freyer (b. 1976, Germany) and Sebastien Noel (b. 1977,France) have worked together as an artist trio since 2003.

With a particular interest in perception and the spatial experience, their collective multi-media works challenge the experience of seeing and knowing. They manipulate our perception of the world and ask the question why we know what we know, and whether this knowledge is certain.

The following is what seems to be a mathematical proof that ten equals 9.999999….
What’s wrong with it?
a = 9.999999…
10a = 99.999999…
10a – a = 90
9a = 90
a = 10

Eddie, the “proof” is wrong, because 9.99999… is not a number upon which we can meaningful perform arithmetic. it’s more like ∑(9)*10^(-n), and you can’t multiply or subtract an infinite series from itself (recall the “proof” that 1+2+3+… = -1/12?).

This is an example of what mathematicians call a simple similtaneous equation.

These sorts of equations are treacherous and should not really be engaged with by people who have better things to do with their time.

Almost all problems presented as similtaneous equations can be solved in more intuitive ways using simple direct methods.

And indeed the methods of the heart always supersede the methods of the so called rational equations.

It is not thus so surprising that even a simple pair of equations like yours give rise to a conundrum that is then set as a puzzle. It is an excellent example of how misleading and irrelevant they are.

Even the most complex maths I encounter in my daily life – geocentric orbital mechanics for star charts – is better based in reality (and gives evidence-based results) than the sort of maths presented in “puzzles”.

DAve – if 9.9recurring and 10 are different numbers then there must other numbers between them (ie at least one number which is less than 10 but more than 9.9recurring). But, in fact they are the same number and it is impossible for you to name a number in-between.

Anyway, the proof that 0.9recurring = 1 is discussed in many places on the web, so it’s probably not worth wasting more pixels on it here.

The object is simply what you get if you intersect a cone with a four-sided square pyramid, . Viewed from the vertex of the cone it’s a circle. From the vertex of the pyramid it’s a sqare. I decided to construct this using Sketchup (a free 3-d CAD program).

Can I figure it out? Yes.

Yes, very impressive Richard. I would suggest that it all depends upon how close you are on each side and perhaps it uses the parallax effect… similar to looking at the different ends of a cone?

I think the four limbs are (roughly) quadrants of a circle on the surface of a pyramid. When the eye is placed at the apex of the pyramid, all quadrants are seen edge on, hence the square – just as the pyramid itself appears square in projection. If the eye is at the centre of the “base” of the pyramid, the quadrants are seen to be curved, and make the circle. Wonderfully imaginative!

This installation is called “Squaring the Circle” and is/was at at the Michael Kohn Gallery in Los Angeles

It is not what it appears: the suspended wire sculpture looks to be wrought into a perfectly drawn black circle. View it from another angle, however, and you’ll find that it’s also a square. This defiance of geometry (a sculptural sleight of hand) involves a hybrid shape of right-angled points and 180-degree curves, plus light reference to satirist Edwin Abbott’s 1884 short story Flatland, in which the denizens of a two-dimensional world reject the possibility of life in three dimensions.

The artists are collectively known as Troika. Eva Rucki (b. 1976, Germany), Conny Freyer (b. 1976, Germany) and Sebastien Noel (b. 1977,France) have worked together as an artist trio since 2003.

With a particular interest in perception and the spatial experience, their collective multi-media

works challenge the experience of seeing and knowing. They manipulate our perception of the

world and ask the question why we know what we know, and whether this knowledge is certain.

Thought you should know this

Slight copy/paste fail there, DAve, but nice try.

Sorry Mum.

This installation is called “Squaring the Circle” and is/was at at the Michael Kohn Gallery in Los Angeles

It is not what it appears: the suspended wire sculpture looks to be wrought into a perfectly drawn black circle. View it from another angle, however, and you’ll find that it’s also a square. This defiance of geometry (a sculptural sleight of hand) involves a hybrid shape of right-angled points and 180-degree curves, plus light reference to satirist Edwin Abbott’s 1884 short story Flatland, in which the denizens of a two-dimensional world reject the possibility of life in three dimensions.

The artists are collectively known as Troika. Eva Rucki (b. 1976, Germany), Conny Freyer (b. 1976, Germany) and Sebastien Noel (b. 1977,France) have worked together as an artist trio since 2003.

With a particular interest in perception and the spatial experience, their collective multi-media works challenge the experience of seeing and knowing. They manipulate our perception of the world and ask the question why we know what we know, and whether this knowledge is certain.

Thought you should know this

The following is what seems to be a mathematical proof that ten equals 9.999999….

What’s wrong with it?

a = 9.999999…

10a = 99.999999…

10a – a = 90

9a = 90

a = 10

I’ll let you have an answer when I manage to find my Rubik’s Cube

Eddie

I’m not sure of the relevance of this to Richard’s post. Could you not find some other outlet for your quizzings?

I’m just trying to keep the old Friday Puzzle going. It was the best thing on this blog.

there’s nothing wrong with it. 10 does = 9.9 recurring

Nothing is wrong! 9.99999… does equal 10! Another way to say it is limit(9 + .9 + .09 + .009 + …etc.) = 10.

Eddie, the “proof” is wrong, because 9.99999… is not a number upon which we can meaningful perform arithmetic. it’s more like ∑(9)*10^(-n), and you can’t multiply or subtract an infinite series from itself (recall the “proof” that 1+2+3+… = -1/12?).

MathMiles has the better proof below.

@Edie

You are saying that a=9.999999 and a=10

This is an example of what mathematicians call a simple similtaneous equation.

These sorts of equations are treacherous and should not really be engaged with by people who have better things to do with their time.

Almost all problems presented as similtaneous equations can be solved in more intuitive ways using simple direct methods.

And indeed the methods of the heart always supersede the methods of the so called rational equations.

It is not thus so surprising that even a simple pair of equations like yours give rise to a conundrum that is then set as a puzzle. It is an excellent example of how misleading and irrelevant they are.

Even the most complex maths I encounter in my daily life – geocentric orbital mechanics for star charts – is better based in reality (and gives evidence-based results) than the sort of maths presented in “puzzles”.

Nothing wrong. 10 does equal 9.9 recurring.

No it isn’t. To get from 9.9 recurring to 10, you do so by rounding it up. Which implies that 9.9 recurring is and always will be less than 10

DAve – if 9.9recurring and 10 are different numbers then there must other numbers between them (ie at least one number which is less than 10 but more than 9.9recurring). But, in fact they are the same number and it is impossible for you to name a number in-between.

Anyway, the proof that 0.9recurring = 1 is discussed in many places on the web, so it’s probably not worth wasting more pixels on it here.

Yes there are differing numbers in between. 9.999 recurring will NEVER equal 10. I refer you to the Kluivert Axiom

The object is simply what you get if you intersect a cone with a four-sided square pyramid, . Viewed from the vertex of the cone it’s a circle. From the vertex of the pyramid it’s a sqare. I decided to construct this using Sketchup (a free 3-d CAD program).