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Author
YONG

Joined: 16 Mar 2005
Posts: 8000
Location: 22° 15' N | 114° 10' E
YONG
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.

05 Jan 2016, 05:15
MHajduk

Joined: 30 Mar 2006
Posts: 6034
Location: Poland
MHajduk
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.

08 Jan 2016, 20:45
YONG

Joined: 16 Mar 2005
Posts: 8000
Location: 22° 15' N | 114° 10' E
YONG
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?
09 Jan 2016, 04:00
sleepsleep

Joined: 05 Oct 2006
Posts: 8898
Location: ˛　　　　　　　　　　　　　　　　　　　　　　　　　　　　　⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣Posts: 334455
sleepsleep
hi MHajduk,
what software you use to made that 3D bottles?
09 Jan 2016, 04:53
MHajduk

Joined: 30 Mar 2006
Posts: 6034
Location: Poland
MHajduk
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?
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...
09 Jan 2016, 11:16
MHajduk

Joined: 30 Mar 2006
Posts: 6034
Location: Poland
MHajduk
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).
09 Jan 2016, 11:22
YONG

Joined: 16 Mar 2005
Posts: 8000
Location: 22° 15' N | 114° 10' E
YONG
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?
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...
Genius!
10 Jan 2016, 03:01
idle

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

Joined: 30 Mar 2006
Posts: 6034
Location: Poland
MHajduk
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.
10 Jan 2016, 20:19
MHajduk

Joined: 30 Mar 2006
Posts: 6034
Location: Poland
MHajduk
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.

14 Jan 2016, 23:54
MHajduk

Joined: 30 Mar 2006
Posts: 6034
Location: Poland
MHajduk
"Still life with vases", 3D graphics.

17 Jan 2016, 22:56
MHajduk

Joined: 30 Mar 2006
Posts: 6034
Location: Poland
MHajduk
Prove the following inequality:

19 Jan 2016, 08:49
MHajduk

Joined: 30 Mar 2006
Posts: 6034
Location: Poland
MHajduk
19 Jan 2016, 17:58
shoorick

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

Joined: 05 Oct 2006
Posts: 8898
Location: ˛　　　　　　　　　　　　　　　　　　　　　　　　　　　　　⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣⁣Posts: 334455
sleepsleep
like the vase too!

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

Joined: 30 Mar 2006
Posts: 6034
Location: Poland
MHajduk
shoorick wrote:
impressive vase!
Thanks.
sleepsleep wrote:
like the vase too!

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

Joined: 20 May 2009
Posts: 724
tthsqe
the most impressive math done by MHajduk is not in his formulas.
21 Jan 2016, 01:13
MHajduk

Joined: 30 Mar 2006
Posts: 6034
Location: Poland
MHajduk
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

21 Jan 2016, 01:35
YONG

Joined: 16 Mar 2005
Posts: 8000
Location: 22° 15' N | 114° 10' E
YONG
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!
21 Jan 2016, 09:51
shoorick

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