Think TwiceDragon Curve is one of many self-similar fractal curves. It is also an example of a space-filling curve. The curve never crosses itself and does not meet at the ends. The same pattern is scaled by square root of two and twisted by 45 degree angle.
You can build your own dragon curve by folding paper in half many times, and then unfolding it by 90 degrees. Learn more: en.wikipedia.org/wiki/Dragon_curve
Thanks for watching :) _________________________________________________________________
Unfolding The Dragon | Fractal Curve |Think Twice2017-10-22 | Dragon Curve is one of many self-similar fractal curves. It is also an example of a space-filling curve. The curve never crosses itself and does not meet at the ends. The same pattern is scaled by square root of two and twisted by 45 degree angle.
You can build your own dragon curve by folding paper in half many times, and then unfolding it by 90 degrees. Learn more: en.wikipedia.org/wiki/Dragon_curve
Thanks for watching :) _________________________________________________________________
youtube.com/channel/UCTfl24SP1qRn00D_AItf0TAGenerating Conic Sections with Circles | Part 3. The HyperbolaThink Twice2021-07-16 | Learn mathematics in a fun and interactive way at: brilliant.org/ThinkTwice ----------------------------------------------------------------------------------------------------------- Help me create more videos by supporting Think Twice on:
----------------------------------------------------------------------------------------------------------- Contact me: ► thinktwiceask@gmail.comGenerating Conic Sections with Circles | Part 2. The ParabolaThink Twice2021-01-14 | Strengthen your problem-solving skills at: brilliant.org/ThinkTwice ----------------------------------------------------------------------------------------------------------- Let C be a circle centered at F and let L denote a line on the same plane as C which doesn't intersect C. Then construct a variable circle tangent to L and C and denote its center as X. A collection of all such possible centers X is a parabola. ----------------------------------------------------------------------------------------------------------- Help me create more videos by supporting the channel on:
----------------------------------------------------------------------------------------------------------- Contact me: ► thinktwiceask@gmail.comGenerating Conic Sections with Circles | Part 1. The EllipseThink Twice2020-08-29 | Learn key problem-solving techniques at: brilliant.org/ThinkTwice ----------------------------------------------------------------------------------------------------------- Take any circle and pick any one of its interior points. Then the collection of the centers of circles passing through that point and tangent to the initial circle is an ellipse. ----------------------------------------------------------------------------------------------------------- Help me create more high-quality videos by supporting Think Twice on:
----------------------------------------------------------------------------------------------------------- Contact me: ► thinktwiceask@gmail.comEulers Formula V - E + F = 2 | ProofThink Twice2020-06-13 | Explore the world of 3-dimensional geometry by signing up for free at: brilliant.org/ThinkTwice ----------------------------------------------------------------------------------------------------------- Proofs for two theorems used in this video: ► Polygon triangulation: youtube.com/watch?v=2x4ioToqe_c ► Area of a spherical triangle: youtube.com/watch?v=Y8VgvoEx7HY
Euler's polyhedron formula is one of the simplest and beautiful theorems in topology. In this video we first derive the formula for the area of a spherical polygon using two theorems proven in the previous two videos which are linked above. This result is then used to prove the fact that V-E+F = 2 is true for all convex polyhedra by projecting the polyhedron on the surface of the sphere and doing some algebraic manipulation. -----------------------------------------------------------------------------------------------------------
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----------------------------------------------------------------------------------------------------------- #mathematics #geometry #EulerEvery Polygon can be Triangulated Into Exactly n-2 Triangles | Proof by InductionThink Twice2020-05-13 | Learn more about propositional logic and dive into the world of beautiful geometry at: brilliant.org/ThinkTwice ----------------------------------------------------------------------------------------------------------- Help me create more content by supporting Think Twice on:
----------------------------------------------------------------------------------------------------------- 🎵 Music by : Jonkyoto - fiverr.com/jonkyotoSpherical Geometry: Deriving The Formula For The Area Of A Spherical TriangleThink Twice2020-03-21 | For more fun and challenging 3D geometry problems head to: brilliant.org/ThinkTwice ----------------------------------------------------------------------------------------------------------- Please consider supporting Think Twice on:
----------------------------------------------------------------------------------------------------------- Summary: ► A spherical triangle is a surface area of a sphere bounded by 3 arcs of great circles. ► Any spherical triangle has it's antipodal duplicate ► Each of these spherical triangles are intersections of 3 different spherical lunes. ► Adding up the area of all 6 lunes results in the surface area of a sphere with 4 additional spherical triangles.
►Girard's Theorem: the area (T) of a spherical triangle, with interior angles a,b and c, is given by T = r^2 (a + b + c - pi). The
----------------------------------------------------------------------------------------------------------- 🎵 Music: chill. by sakura Hz soundcloud.com/sakurahertz Creative Commons — Attribution 3.0 Unported — CC BY 3.0 Free Download / Stream: http://bit.ly/chill-sakuraHz Music promoted by Audio Library youtu.be/pF2tXC1pXNoThe Fermat Point of a Triangle | Geometric construction + Proof |Think Twice2020-02-09 | Learn more theorems in Euclidean geometry and their applications at: brilliant.org/ThinkTwice ----------------------------------------------------------------------------------------------------------- Please consider supporting Think Twice on:
The Fermat point of a triangle ABC is a point P such that the sum of distances PA+PB+PC is a minimum.
To find the Fermat point of a triangle ABC: 1. Construct equilateral triangles on each side of ABC 2. Connect vertices A,B and C to the opposite and outermost vertex of equilateral triangle. 3. The point at which the three lines intersect is a Fermat point of triangle ABC.
In the case where one of the angles of triangle ABC is greater than 120 degrees the Fermat point will be located at the obtuse-angled vertex of ABC.
About the video: 1. Pick any polygon 2. Split it up into triangles (While it is trivial to triangulate any convex polygon it is has been proven that any polygon, convex or concave, can be decomposed into triangles and there are many triangulation techniques). 3. Construct a rectangle of equal area for each triangle. 4. Construct a square of equal area for each rectangle. 5. Construct a larger square equal to the sum of each smaller squares via Pythagorean Theorem.
Music by: instagram.com/miras2hotVisual Calculus: Derivative of sin(θ) is cos(θ)Think Twice2019-04-09 | Build an understanding behind different concepts of calculus that will help you tackle challenging problems at: brilliant.org/ThinkTwice _________________________________________________________________
Proof: Derivative of sin(θ) is cos(θ) _________________________________________________________________
youtube.com/watch?v=vy9jeNFCuSUWhat is the area under an arc of a cycloid curve?Think Twice2019-01-27 | Build an understanding behind different concepts of geometry that will help you tackle challenging problems at: brilliant.org/ThinkTwice _________________________________________________________________
Why does the area under a cycloid curve equal to 3 times the area of the circle used to trace out that curve? _________________________________________________________________
Napoleon's theorem: "On each side of a triangle, erect an equilateral triangle, lying exterior to the original triangle. Then the segments connecting the centroids of the three equilateral triangles themselves form an equilateral triangle."
Finding an equation for the volume of a sphere using Cavalieri's Principle ( assuming we already know the equation for the volume of a cone) _________________________________________________________________
" Gone For Now "- youtube.com/watch?v=bkJ-5Pb-514Pythagorean theorem | 3 Visual Proofs |Think Twice2018-07-11 | Some of my favorite proofs of Pythagorean theorem. Enjoy:)
"FollowHer" - youtube.com/watch?v=Nh_oj5hsAbsApproximating Pi ( Monte Carlo integration ) | animationThink Twice2018-06-10 | Pi approximation using Monte Carlo integration.
_________________________________________________________________Finding the general formula for nth octagonal number | Visual proof |Think Twice2018-04-26 | Sometimes creative approach leads to simple and elegant solutions...
_________________________________________________________________Arithmetic mean vs Geometric mean | inequality among means | visual proofThink Twice2018-04-05 | hi~
It's quite hard to upload videos from china because of YouTube being banned, but I'll be back home in two/three months and will be uploading more frequently (i will try at least).
"so far away" by outlaxInfinite Sums | Geometric Series | Explained VisuallyThink Twice2018-02-01 | Geometric series are probably one of the first infinite sums that most of us encountered in high-school. When I first heard of an infinite sum(two or three years ago), I was really amazed that some of them would equal to a finite number. It seemed very strange that even if I keep adding numbers forever I would get a finite answer. At school I was just taught to plug the numbers into the formula, without fully understanding why or how it works.
In this video I go over some examples of geometric series and how we can get some insight on why it works by using visuals.
Thanks for watching~
P.S. I will be moving to China for 5 months tomorrow, so I'm not sure if I will have an access to YouTube there, due to many sites being blocked by the government.
youtube.com/watch?v=27w-GP6ZGF0Geometry of Binomial Theorem | Visual Representation | 2 examplesThink Twice2018-01-19 | A visual representation of binomial theorem. In this video I used only two examples where the exponent is equal to 2 and 3. However the same analogy can be carried on to higher exponents. It would just be a lot harder to animate hypercubes.
I Hope everyone is doing fine, and as always let me know what do you guys think about the animation. Thanks a lot~
Another visualization and a proof of some simple euclidean geometry. I think it's pretty neat :)
Two chords of a circle intersecting at 90 degrees make up 4 segments. The squares of these 4 segments always add up to the diameter squared. Hope the video is clear and understandable.
Quick update:
We made it over 2k subs! Thanks everyone for all the comments and support. I wouldn't be here without your help and other bigger YouTubers that help me out along the way.
youtube.com/watch?v=PffzsCjnnRQGeometry: Vivianis theorem | Visualization + Proof |Think Twice2017-12-13 | Viviani's theorem basically states that the sum off lengths of 3 lines, drawn at 90 degrees from the sides of an equilateral triangle to any inner point is always equal to the height.
saw this theorem online and thought that I would program a nice and simple visualization for it. What do you think?
Click the link below to interact with the sketch that I programmed:
Don't really know what to say. Just enjoy the video and let me know what you think of it:) _________________________________________________________________
youtube.com/watch?v=ewBOcdz29SwChaos Game | Fractals emerging from chaos | Computer simulation |Think Twice2017-11-07 | I wanted to make a video about this topic for a long time, however without knowing how to code it would be really inefficient to animate every single dot by hand. So for the past few weeks I've been learning how to program animations and so I managed to simulate a chaos game. Thanks for watching :) _________________________________________________________________
Thank you everyone for 1000 subs. It's really cool that over a thousand people are actually interested in my content. I'm thankful for all the comments and support that I've been getting from you. I decided to create a Patreon page. If you feel like you would like to see more of my content as well as get to know more about me consider becoming a patron at :
________________________________________________________________Double pendulum | Chaos | Butterfly effect | Computer simulationThink Twice2017-09-30 | A system is considered chaotic if it is highly sensitive on the initial conditions.
If a system is chaotic it doesn't mean that it is random. A chaotic system is completely deterministic. Given enough time and precise initial conditions of the system it would be possible to calculate precisely, how it will evolve.
Given enough time, two identical setups, set to initial conditions that are as identical as possible, will look entirely different. _________________________________________________________________
soundcloud.com/hvetter-1Cutting a Möbius strip in half (and more) | Animated Topology |Think Twice2017-09-11 | ________________________________________________________________
About the video: Exploring the properties and other unexpected shapes that we get by cutting up some Mobius strips.
This video is a lot more text heavy compared to most of my previous ones, however it's quite difficult to explain something in more depth without using a lot of words.
Thanks for watching:) _________________________________________________________________
________________________________________________________________Area of dodecagon | Beautiful geometry | Visual mathematicsThink Twice2017-08-23 | Just a short and simple animation of how to find the area of dodecagon by knowing its radius. Obviously there are many different ways to do it, but I thought that this one was the most visually appealing.
We've reached 600subs faster than I thought so just wanted to thank everyone for the support:) (really)
Your comments, thoughts and ideas are always appreciated.
_________________________________________________________________Sum of first n odd numbers | Visual mathematics |Think Twice2017-07-19 | Short animation about the sum of first n odd numbers.
here is an algebraic proof:
The first n odd natural numbers are 1, 3, 5, ... 2n-1. This forms an arithmetic progression with first term 1, common difference 2 and last term (2n-1).
The sum is given by S = 1+3+5+...+(2n-1). Using S = n (First term + Last term)/2, we get S = n (1+2n-1)/2, which simplifies to S = n^2.
Subscribe for more:) _________________________________________________________________
CLAUDE DEBUSSY: CLAIR DE LUNEBeautiful visualization | Sum of first n Hex numbers = n^3 | animationThink Twice2017-07-09 | In this animation I'll show why the sum of first n Hex numbers is equal to nxnxn. Hex number (or centered hexagonal number) is just a number of dots that surround the center dot in a hexagonal lattice.
Hope you like this video. _________________________________________________________________
Nocturne op. 9 no. 2Sum of n squares | explained visually |Think Twice2017-06-20 | There is a simple algebraic proof for why 1^2 + 2^2 + 3^2 +...+ n^2 = (n(n+1)(2n+1))/6 , and it's not that interesting. However I think that the visual explanation is a lot more beautiful and so I made a simple animation about it.
I know that I used only one example where n=4, but the same will work with any integer n. It's not a proof, but it will make you understand the logic behind the formula. _________________________________________________________________
Bach - Prelude in C MajorNicomachuss theorem | Visualisation | 3-D animation |Think Twice2017-05-28 | A visual proof of Nicomachus's theorem. It states that the sum of the first n cubes is the square of the nth triangular number. That is,
1^3 + 2^3 + 3^3 +...+ n^3=(1 + 2 + 3 +...+ n)^2.
I thought it was one of the most visually appealing proofs so I decided to make a short 3D animation.