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flat assembler > Heap > A simple mathemagical trick for interested ones.

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YONG



Joined: 16 Mar 2005
Posts: 8000
Location: 22° 15' N | 114° 10' E
Fantastic! I know it is difficult to make a smooth transition as the tube is intersecting with itself (from 3 to 4) but you managed to do it.

A little suggestion: It may be a good idea to use two different colors to represent the inner and outer surfaces of the finished tube.

Wink
Post 05 Jan 2016, 05:15
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MHajduk



Joined: 30 Mar 2006
Posts: 6023
Location: Poland
How to calculate the volume of a flat-bottomed, corked bottle, partially filled with a fluid having only the ruler as a measuring tool?
The 3D graphics made by me and presented below gives an answer to this question.

Image
Post 08 Jan 2016, 20:45
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YONG



Joined: 16 Mar 2005
Posts: 8000
Location: 22° 15' N | 114° 10' E
MHajduk wrote:
How to calculate the volume of a flat-bottomed, corked bottle, partially filled with a fluid having only the ruler as a measuring tool?
How about the case when there is only a very small amount of fluid? When the fluid can't even completely fill up the non-cylindrical portion of the bottle, what should we do? Rolling Eyes Wink
Post 09 Jan 2016, 04:00
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sleepsleep



Joined: 05 Oct 2006
Posts: 8364
Location: ˛                             ⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣Posts: 334455
hi MHajduk,
what software you use to made that 3D bottles?
Post 09 Jan 2016, 04:53
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MHajduk



Joined: 30 Mar 2006
Posts: 6023
Location: Poland
YONG wrote:
How about the case when there is only a very small amount of fluid? When the fluid can't even completely fill up the non-cylindrical portion of the bottle, what should we do? Rolling Eyes Wink
We can do as Archimedes once did:
take a can with precisely known volume, fill it completely with a fluid with far less density than water (because almost empty, corked bottle may not sink in water) - it can be ethanol, kerosene or something like that - put carefully the bottle inside - it will sink and displace amount of the fluid that will be equal to the outer volume of the bottle. Take the bottle off and measure amount of the fluid that left in the can. Difference between the initial and final amounts will give us the outer volume of the bottle. It wouldn't be precise but always... Wink
Post 09 Jan 2016, 11:16
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MHajduk



Joined: 30 Mar 2006
Posts: 6023
Location: Poland
sleepsleep wrote:
hi MHajduk,
what software you use to made that 3D bottles?
Blender 2.76b and a handful of other graphic editors (but Blender is the main tool). Smile
Post 09 Jan 2016, 11:22
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YONG



Joined: 16 Mar 2005
Posts: 8000
Location: 22° 15' N | 114° 10' E
MHajduk wrote:
YONG wrote:
How about the case when there is only a very small amount of fluid? When the fluid can't even completely fill up the non-cylindrical portion of the bottle, what should we do? Rolling Eyes Wink
We can do as Archimedes once did:
take a can with precisely known volume, fill it completely with a fluid with far less density than water (because almost empty, corked bottle may not sink in water) - it can be ethanol, kerosene or something like that - put carefully the bottle inside - it will sink and displace amount of the fluid that will be equal to the outer volume of the bottle. Take the bottle off and measure amount of the fluid that left in the can. Difference between the initial and final amounts will give us the outer volume of the bottle. It wouldn't be precise but always... Wink
Genius! Razz
Post 10 Jan 2016, 03:01
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idle



Joined: 06 Jan 2011
Posts: 359
Location: ukraina
Just so it in a social network: _2016 = 666+666+666+(6+6+6)
Post 10 Jan 2016, 18:54
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MHajduk



Joined: 30 Mar 2006
Posts: 6023
Location: Poland
idle wrote:
Just so it in a social network: _2016 = 666+666+666+(6+6+6)
Yeah, I saw it a few days ago. People are very creative in finding this kind of relations. Smile
Post 10 Jan 2016, 20:19
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MHajduk



Joined: 30 Mar 2006
Posts: 6023
Location: Poland
In case you are interested in drawing circles with a huge diameter (by "huge" I mean here really big ones, measured in kilometers or so) there is a tricky idea needed because the rope-and-the-stick method won't work for circles with a diameter bigger than several dozen meters. Here you can use a method based on the well known fact from geometry stating that all angles inscribed in a circle and subtended by the same chord (lying on the same side of the chord) are equal.

Image

Image

Image
Post 14 Jan 2016, 23:54
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MHajduk



Joined: 30 Mar 2006
Posts: 6023
Location: Poland
"Still life with vases", 3D graphics.

Image
Post 17 Jan 2016, 22:56
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MHajduk



Joined: 30 Mar 2006
Posts: 6023
Location: Poland
Prove the following inequality:

Image
Post 19 Jan 2016, 08:49
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MHajduk



Joined: 30 Mar 2006
Posts: 6023
Location: Poland
Image
Post 19 Jan 2016, 17:58
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shoorick



Joined: 25 Feb 2005
Posts: 1605
Location: Ukraine
impressive vase! Smile
Post 20 Jan 2016, 06:15
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sleepsleep



Joined: 05 Oct 2006
Posts: 8364
Location: ˛                             ⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣Posts: 334455
like the vase too!

cycles engine? MHajduk?
Post 20 Jan 2016, 19:47
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MHajduk



Joined: 30 Mar 2006
Posts: 6023
Location: Poland
shoorick wrote:
impressive vase! Smile
Thanks. Smile
sleepsleep wrote:
like the vase too!

cycles engine? MHajduk?
Yes, that's Cycles Render - a bit slow but gives a truly good rendering results. Smile
Post 20 Jan 2016, 21:42
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tthsqe



Joined: 20 May 2009
Posts: 721
the most impressive math done by MHajduk is not in his formulas.
Post 21 Jan 2016, 01:13
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MHajduk



Joined: 30 Mar 2006
Posts: 6023
Location: Poland
Ptolemy's theorem: A convex quadrilateral can be inscribed in a circle if and only if the product of the lengths of one pair of opposite sides added to the product of the lengths of the other pair is equal to the product of the lengths of the diagonals. Thus, in a cyclic quadrilateral ABCD we have

AB*DC + AD*BC = AC*BD

Image
Post 21 Jan 2016, 01:35
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YONG



Joined: 16 Mar 2005
Posts: 8000
Location: 22° 15' N | 114° 10' E
tthsqe wrote:
the most impressive math done by MHajduk is not in his formulas.
Agree. The most impressive part is how he presents it. I would say MHajduk has created a new art form by combining "classic" math with "contemporary" computer graphics. What an interesting blend! Wink
Post 21 Jan 2016, 09:51
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shoorick



Joined: 25 Feb 2005
Posts: 1605
Location: Ukraine
perfect blender!
Post 21 Jan 2016, 18:57
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