Construction a Golden Rectangle

This Challenge allegedly has deep connections to art, geometry and our psyche.

The Golden Ratio is calculated to be 1.618::1.
It describes a pattern that is found in sunflowers, playing cards, architecture and web design layout.

Here are two presentations of a Golden Rectangle where the sides are in Golden Ratio proportion:

Attachment 122576

Now it can be shown that the Golden Ratio is integral to the pentagon, where it is Diagonal Length::Edge Length:

Attachment 122575
It is therefore possible to construct the blue Golden Rectangle from this knowledge, as shown.

In this Challenge, I wish you to uncover a simpler construction of the Golden Rectangle.

Acorn

P.S. Phi (ɸ) is (√5 + 1) / 2 ≃ 1.6180339887498948482045868343656

first attachement not showing here... unless you have a deleted image broken link...

well at least you spared us the fibonacci sequence

how accurate do you want it - 1.6 is usually ok for this lowly cartoonist, and since that is one:three-fifths makes it all a whole load easier... but I tend not to use it that much, I go by what looks right..

Quote:

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Originally Posted by handrawn

well at least you spared us the fibonacci sequence
how accurate do you want it - 1.6 is usually ok for this lowly cartoonist, and since that is one:three-fifths makes it all a whole load easier... but I tend not to use it that much, I go by what looks right..

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I actually slipped in the Fibonacci Sequence through the sunflowers.

The attachment got deleted instead of the one wrongly enumerated in the second diagram.

The diagram should have been:

Attachment 122582

Acorn

This one had two typos:

Quote:

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Originally Posted by Acorn

I actually slipped in the Fibonacci Sequence through the sunflowers.

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:D
don't tell me... next up is the golden spiral ;)

Quote:

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Originally Posted by handrawn

:D
don't tell me... next up is the golden spiral ;)

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Nope, only pink elephants:

Attachment 122583

Acorn

Quote:

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Originally Posted by Acorn

Nope, only pink elephants:

Attachment 122583

Acorn

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Ha ha
H

Quote:

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Originally Posted by Acorn

This Challenge allegedly has deep connections to art, geometry and our psyche.

The Golden Ratio is calculated to be 1.618::1.
It describes a pattern that is found in sunflowers, playing cards, architecture and web design layout.

Here are two presentations of a Golden Rectangle where the sides are in Golden Ratio proportion:

Attachment 122576

Now it can be shown that the Golden Ratio is integral to the pentagon, where it is Diagonal Length::Edge Length:

Attachment 122575
It is therefore possible to construct the blue Golden Rectangle from this knowledge, as shown.

In this Challenge, I wish you to uncover a simpler construction of the Golden Rectangle.

Acorn

P.S. Phi (ɸ) is (√5 + 1) / 2 ≃ 1.6180339887498948482045868343656

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Acorn , I used to be very aware of the "Golden rectangle" but now a days like handrawn I tend to go by what looks right. I tend to think of the value of Pi as 22/7 or rounded to 3.14159.

:)


Attachment 122590

Quote:

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Originally Posted by handrawn

:)


Attachment 122590

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Cheers, your eye is half of the solution. If the elephant has a left eye squint as well, you have the complete answer!

Acorn

Quote:

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Originally Posted by wizard509

Acorn , I used to be very aware of the "Golden rectangle" but now a days like handrawn I tend to go by what looks right. I tend to think of the value of Pi as 22/7 or rounded to 3.14159.

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For pi, use 355/113 (3.14159292), c.f. pi at 3.14159265; 355/113 is 2,000 times more accurate.

Agreed, people do forget that 5/3 (1.6) is very close to Phi (ɸ), around 99% accurate.

Acorn

Here is a quick demonstration of the crossed-eyed elephant:

Attachment 122604

1. Create a Red circle.
2. Clone, colour Green and align centres vertically to edges.
3. Clone again, colour Yellow and double size.
4. Clone this, colour Cyan.
5. Align bottom Yellow and Green, top Red and Cyan.

The (red) length of the Red/Green intersections is taken as unity.
Extend the right end to the the right-side intersection fo Yellow/Cyan. This (blue) length is Phi (ɸ).

Note this is the Golden Ratio and a little more would be needed to get a Golden Rectangle but the eyes have it.

Acorn

ah now I see what you mean... very good

[btw the two circles in my eye were in ratio 1:0.618]

Quote:

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Originally Posted by handrawn

ah now I see what you mean... very good
[btw the two circles in my eye were in ratio 1:0.618]

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handrawn, 'eye' saw that straight away - same as 1.618::1 as Phi (ɸ)'s inverse is ɸ - 1.

Acorn

OK, perhaps the concept is a bit too exotic but keep it in mind for future reference.

Here is a construction that delivers a Golden Rectangle from an initial pentagon.

1. Create a Red pentagon, with rotation 180°.
2. Clone it, colour it Green and change to a circle.
3. Place its centre on the pentagon left base corner.
4. Shift-drag until the circle's edge touches the pentagon's opposite sides. [I used a mixture of Guides and Zooms as there is no snap point unless you add set the circle's rotation as 36° (or rotate all the construction by the same)!]
5. Clone the pentagon, colour it Blue and change to a square.
6. Resize and position its base to match the pentagon.
7. Drag up the top to touch the green circle.

Attachment 122631

I have left a simpler approach outstanding should anyone still want to have a go.

Acorn

Here goes:

Attachment 122666

1. Create a Red square.
2. Clone it, change to a circle, colour it Green and place behind.
3. Place its centre on the midpoint of bottom edge of the Red circle.
4. Shift-drag the circle's edge to touch the top right corner of the Red square.
5. Clone the Red square, colour it Blue and place immediately under the Red square.
6. Drag its right edge so its top right corner touches the edge of the Green circle.

This construction does rely on a steady hand as there is no snap for the circle radius.

Other suggestions are very welcome.

Acorn

Quote:

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Originally Posted by Acorn

... This construction does rely on a steady hand as there is no snap for the circle radius.

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If you draw the original square on a grid, rather than clone and change to circle, you can just draw a circle radius from the centre of the bottom to the top corner and snap to the grid points though ...

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Originally Posted by ss-kalm

If you draw the original square on a grid, rather than clone and change to circle, you can just draw a circle radius from the centre of the bottom to the top corner and snap to the grid points though ...

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Cheers Keith, that is a good remedy. I will although still look for one that doesn't have a dimension constraint.

Found one!

Attachment 122672

1. Create a Red square.
2. Half its width.
3. Clone it, colour it Green.
4. Change the Red square to editable Shapes and delete its top left corner.
5. Position the red triangle to the top right of the Green rectangle.
6. Rotate it about this point so its longest edge it horizontal.
7. Clone the Green rectangle, colour it Blue.
8. Drag its right edge to the right end of the Red triangle.

Acorn

Quote:

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Originally Posted by Acorn

[*]Rotate it about this point so its longest edge it horizontal.

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How do you ensure that it's horizontal?

Quote:

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Originally Posted by ss-kalm

How do you ensure that it's horizontal?

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Oops! Thanks Keith. Just hold down the Ctrl key when rotating (the shape, not yourself).

Acorn

Great challenge Acorn. I learned from this one. Here is a comparison of my eyeballed one to your correct one.

Attachment 122692

Quote:

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Originally Posted by wizard509

Great challenge Acorn. I learned from this one. Here is a comparison of my eyeballed one to your correct one.

Attachment 122692

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Larry, thank you for your approach.
I have been trying to avoid math / maths and approximations but as an engineer myself, I appreciate there is always another way.
Unfortunately, your sums have gone awry as the final rectangle has a side ratio of 1.79::1. We need 1.618::1.

Earlier, handrawn suggested 1.6::1 was good enough.

A similar construction to your could map out a ratio of 1.6.
Thereafter just increase the width by 1% to achieve 1.616::1, accurate to 0.1%.

Attachment 122694

My Yellow/Cyan blobs are a 3/5th scaled ruler.

Acorn

Here there is another way...

Attachment 122695

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Originally Posted by Marco D

Here there is another way...

Attachment 122695

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Marco, it is that.

Same construction really as Attachment 122666 where I dropped the Blue GR down out of the way and I used one Red square to your two ((1) & (2)). Both have the same underlying math / maths.

The point of these challenges is to discover Xara constructs that are useful to you.

Acorn

Quote:

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Originally Posted by Acorn

Marco, it is that.

Same construction really as Attachment 122666 where I dropped the Blue GR down out of the way and I used one Red square to your two ((1) & (2)). Both have the same underlying math / maths.

The point of these challenges is to discover Xara constructs that are useful to you.

Acorn

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Yes the math is the same but this was an attempt to adress the snapping problem, you mentioned there.

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Originally Posted by Marco D

Yes the math is the same but this was an attempt to adress the snapping problem, you mentioned there.

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Macro, thank you for the clarification.

I don't, however, see any snapping that stops on the top right corner.

Acorn

Quote:

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Originally Posted by Acorn

... I don't, however, see any snapping that stops on the top right corner ...

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You don't need to snap that corner, you can use magnetic alignment. As far as I'm aware, the only snapping issue originally, was the centre of the bottom of your red square, which Marco has solved.

While drawing the circle (radius creation option) you can snap from the corner of the first to the corner of the second rectangle.
I thought that the problem in your construction was finding the center of the base of the square, but I now seen that you can easely snap there too, so it was already ok that way... ;)

Marco.

Quote:

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Originally Posted by Marco D

... but I now seen that you can easily snap there too ...

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Maybe I'm missing something, but I don't see any way to snap to the centre of the base of the square - at least not accurately.

If you draw the circle and then you move it, you can snap the center of the circle with the center of the square base. After that (selecting the circle tool and holding the shift key) you can resize the circle to hit the square corner. That done, the only thing to do is to resize the square clicking on the right side handle until you reach the snap point at the circle.

Marco.

Quote:

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Originally Posted by Marco D

...you can resize the circle to hit the square corner...

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Sorry Marco, this is the step that doesn't have an exact snap point.

Acorn

Quote:

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Originally Posted by Acorn

Sorry Marco, this is the step that doesn't have an exact snap point.

Acorn

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If you resize the circle while in the circle tool, you can select a control point on the circonference and drag it to the corner of the square pressing shift to preserve the aspect ratio.

Quote:

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Originally Posted by Marco D

If you resize the circle while in the circle tool, you can select a control point on the circonference and drag it to the corner of the square pressing shift to preserve the aspect ratio.

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Thank you Marco, you have made me very happy.

I had never thought of resizing with the Circle tool and a control point this way before.

Acorn

Quote:

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Originally Posted by Acorn

Thank you Marco, you have made me very happy.

I had never thought of resizing with the Circle tool and a control point this way before.

Acorn

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You are very welcome Acorn. ;)