Spherical Geometry: Deriving The Formula For The Area Of A Spherical Triangle Think Twice 2020-03-21 | For more fun and challenging 3D geometry problems head to:brilliant.org/ThinkTwice-----------------------------------------------------------------------------------------------------------Please consider supporting Think Twice on: ► Patreon: patreon.com/Think_twice ► Twitter: twitter.com/thinktwice2580 (@thinktwice2580) ► Instagram: instagram.com/thinktwice_ltu (@thinktwice_ltu)-----------------------------------------------------------------------------------------------------------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 -----------------------------------------------------------------------------------------------------------Contact me:► thinktwiceask@gmail.com-----------------------------------------------------------------------------------------------------------🎵 Music:chill. by sakura Hz soundcloud.com/sakurahertzCreative Commons — Attribution 3.0 Unported — CC BY 3.0 Free Download / Stream: http://bit.ly/chill-sakuraHzMusic promoted by Audio Library youtu.be/pF2tXC1pXNo
Generating Conic Sections with Circles | Part 3. The Hyperbola Think Twice 2021-07-16 | Learn mathematics in a fun and interactive way at:brilliant.org/ThinkTwice-----------------------------------------------------------------------------------------------------------Help me create more videos by supporting Think Twice on: ► Patreon: patreon.com/Think_twice ► Twitter: twitter.com/thinktwice2580 ► Instagram: instagram.com/thinktwice_ltu (@thinktwice_ltu)-----------------------------------------------------------------------------------------------------------Contact me:► thinktwiceask@gmail.com
Generating Conic Sections with Circles | Part 2. The Parabola Think Twice 2021-01-14 | 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: ► Patreon: patreon.com/Think_twice ► Twitter: twitter.com/thinktwice2580 (@thinktwice2580) ► Instagram: instagram.com/thinktwice_ltu (@thinktwice_ltu)-----------------------------------------------------------------------------------------------------------Contact me:► thinktwiceask@gmail.com
Generating Conic Sections with Circles | Part 1. The Ellipse Think Twice 2020-08-29 | 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: ► Patreon: patreon.com/Think_twice ► Twitter: twitter.com/thinktwice2580 (@thinktwice2580) ► Instagram: instagram.com/thinktwice_ltu (@thinktwice_ltu)-----------------------------------------------------------------------------------------------------------Contact me:► thinktwiceask@gmail.com
Eulers Formula V - E + F = 2 | Proof Think Twice 2020-06-13 | 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=Y8VgvoEx7HYEuler'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.-----------------------------------------------------------------------------------------------------------Help me create more content by supporting Think Twice on: ► Patreon: patreon.com/Think_twice ► Twitter: twitter.com/thinktwice2580 (@thinktwice2580) ► Instagram: instagram.com/thinktwice_ltu (@thinktwice_ltu)-----------------------------------------------------------------------------------------------------------Contact me:► thinktwiceask@gmail.com-----------------------------------------------------------------------------------------------------------🎵 Music by : Jonkyoto - fiverr.com/jonkyoto-----------------------------------------------------------------------------------------------------------#mathematics #geometry #Euler
Every Polygon can be Triangulated Into Exactly n-2 Triangles | Proof by Induction Think Twice 2020-05-13 | 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: ► Patreon: patreon.com/Think_twice ► Twitter: twitter.com/thinktwice2580 (@thinktwice2580) ► Instagram: instagram.com/thinktwice_ltu (@thinktwice_ltu)-----------------------------------------------------------------------------------------------------------Contact me:► thinktwiceask@gmail.com-----------------------------------------------------------------------------------------------------------🎵 Music by : Jonkyoto - fiverr.com/jonkyoto
The Fermat Point of a Triangle | Geometric construction + Proof | Think Twice 2020-02-09 | Learn more theorems in Euclidean geometry and their applications at:brilliant.org/ThinkTwice-----------------------------------------------------------------------------------------------------------Please consider supporting Think Twice on: ► Patreon: patreon.com/Think_twice ► Twitter: twitter.com/thinktwice2580 (@thinktwice2580) ► Instagram: instagram.com/thinktwice_ltu (@thinktwice_ltu)-----------------------------------------------------------------------------------------------------------Summary: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 ABC2. 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.-----------------------------------------------------------------------------------------------------------Contact me: ► thinktwiceask@gmail.com-----------------------------------------------------------------------------------------------------------Music by: ► Lee: youtube.com/watch?v=Epi2rsr04Os
A fun probability puzzle with a neat geometric solution. Think Twice 2019-12-06 | Spread the love of math by gifting your friends a Brilliant Premium subscription : brilliant.org/ThinkTwice -------------------------------------------------------------------------------------------------- Please consider supporting Think Twice on: ► Patreon: patreon.com/Think_twice ► Twitter: twitter.com/thinktwice2580 (@thinktwice2580) ► Instagram: instagram.com/thinktwice_ltu (@thinktwice_ltu) --------------------------------------------------------------------------------------------------Any further questions: ► thinktwiceask@gmail.com Programs used: ► Cinema 4D ► Processing Music by ► Lee: youtube.com/channel/UCaH9AQy86DBIg-BpWzj-csQ
Constructing a Square of Equal Area to a given Polygon Think Twice 2019-08-19 | Explore the world of Euclidean geometry by solving geometry puzzles at:brilliant.org/ThinkTwice--------------------------------------------------------------------------------------------------Please consider supporting Think Twice on: ► Patreon: patreon.com/Think_twice ► Twitter: twitter.com/thinktwice2580 (@thinktwice2580) ► Instagram: instagram.com/thinktwice_ltu (@thinktwice_ltu)--------------------------------------------------------------------------------------------------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.--------------------------------------------------------------------------------------------------Any further questions: ► thinktwiceask@gmail.comPrograms used: ► Cinema 4D ► ProcessingMusic by ► Lee: youtube.com/watch?v=q7oBUl7AM_k
Alternating series #2 | Visual solution | Think Twice 2019-05-27 | You can learn more about CuriosityStream at curiositystream.com/thinktwice_________________________________________________________________Support my animations on:patreon.com/Think_twice_________________________________________________________________Any further questions:Email: thinktwiceask@gmail.comTwitter: twitter.com/thinktwice2580Instagram: @thinktwice_ltu https://www.instagram.com/thinktwice_..._________________________________________________________________Programs used:- Cinema 4D - Processing_________________________________________________________________Music by: instagram.com/miras2hot
Visual Calculus: Derivative of sin(θ) is cos(θ) Think Twice 2019-04-09 | 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)_________________________________________________________________Support my animations on:patreon.com/Think_twice_________________________________________________________________Any further questions:Email: thinktwiceask@gmail.comTwitter: twitter.com/thinktwice2580Instagram: @thinktwice_ltu https://www.instagram.com/thinktwice_..._________________________________________________________________Programs used:- Cinema 4D - Processing_________________________________________________________________Music by Zeeky:"Saw you"youtube.com/watch?v=vy9jeNFCuSU
What is the area under an arc of a cycloid curve? Think Twice 2019-01-27 | 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? _________________________________________________________________Support my animations on:patreon.com/Think_twice_________________________________________________________________Any further questions:Email: thinktwiceask@gmail.comTwitter: twitter.com/thinktwice2580Instagram: @thinktwice_ltu https://www.instagram.com/thinktwice_..._________________________________________________________________Programs used:- Cinema 4D - Processing_________________________________________________________________Music by Yusei:"Do you ever miss the old you"youtube.com/watch?v=dln2ji0x5tw
Napoleons theorem | Proof | Think Twice 2018-12-28 | Explore the world of geometry at:brilliant.org/ThinkTwice_________________________________________________________________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."_________________________________________________________________Support my animations on:patreon.com/Think_twice_________________________________________________________________Any further questions or ideas:Email: thinktwiceask@gmail.comTwitter: twitter.com/thinktwice2580Instagram: @thinktwice_ltu instagram.com/thinktwice_ltu_________________________________________________________________Programs used:- Cinema 4D - Processing_________________________________________________________________Music by Zeeky Beats:" Rain Drops "youtube.com/watch?v=Hh98NWHtoNY
Cavalieris Principle in 3D | Volume of a sphere | Think Twice 2018-10-02 | To improve your problem solving skills, go to: brilliant.org/ThinkTwice_________________________________________________________________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)_________________________________________________________________Support my animations on:patreon.com/Think_twice_________________________________________________________________Any further questions or ideas:Email - thinktwiceask@gmail.comTwitter - twitter.com/thinktwice2580_________________________________________________________________Programs used:- Cinema 4D _________________________________________________________________Music:" [[FREE]] LoFi Type Beat -- "4 am" "-youtube.com/watch?v=5Xrju_O2RyI
Alternating series #1 | Visual solution | Think Twice 2018-08-07 | A short animation about a visual solution to an alternating series:)_________________________________________________________________Support my animations on:patreon.com/Think_twice_________________________________________________________________Any further questions or ideas:Email - thinktwiceask@gmail.comTwitter - twitter.com/thinktwice2580_________________________________________________________________Programs used:- Cinema 4D _________________________________________________________________Music:" Gone For Now "- youtube.com/watch?v=bkJ-5Pb-514
Pythagorean theorem | 3 Visual Proofs | Think Twice 2018-07-11 | Some of my favorite proofs of Pythagorean theorem. Enjoy:)_________________________________________________________________Support my animations on:patreon.com/Think_twice_________________________________________________________________Any further questions or ideas:Email - thinktwiceask@gmail.comTwitter - twitter.com/thinktwice2580_________________________________________________________________Programs used:- Cinema 4D _________________________________________________________________Music:"FollowHer" - youtube.com/watch?v=Nh_oj5hsAbs
Approximating Pi ( Monte Carlo integration ) | animation Think Twice 2018-06-10 | Pi approximation using Monte Carlo integration. _________________________________________________________________Support my animations on:patreon.com/Think_twice_________________________________________________________________Any further questions or ideas:Email - thinktwiceask@gmail.comTwitter - twitter.com/thinktwice2580_________________________________________________________________Programs used:- Cinema 4D - Processing- Adobe Premiere Pro_________________________________________________________________
Finding the general formula for nth octagonal number | Visual proof | Think Twice 2018-04-26 | Sometimes creative approach leads to simple and elegant solutions...Enjoy~_________________________________________________________________Support my animations on:patreon.com/Think_twice_________________________________________________________________Any further questions or ideas:Email - thinktwiceask@gmail.comTwitter - twitter.com/thinktwice2580_________________________________________________________________Programs used:- Cinema 4D - Adobe Premiere Pro_________________________________________________________________
Arithmetic mean vs Geometric mean | inequality among means | visual proof Think Twice 2018-04-05 | 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). Same thing with Patreon.._________________________________________________________________Support my animations on:patreon.com/Think_twice_________________________________________________________________Any further questions or ideas:Email - thinktwiceask@gmail.comTwitter - twitter.com/thinktwice2580_________________________________________________________________Programs used:- Cinema 4D - Adobe Premiere Pro- Processing_________________________________________________________________Music: "so far away" by outlax
Infinite Sums | Geometric Series | Explained Visually Think Twice 2018-02-01 | 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. _________________________________________________________________Support my animations on:patreon.com/Think_twice_________________________________________________________________Any further questions or ideas:Email - thinktwiceask@gmail.comTwitter - twitter.com/thinktwice2580_________________________________________________________________Programs used:- Cinema 4D - Adobe Premiere Pro_________________________________________________________________MUSIC: A time of wonderyoutube.com/watch?v=27w-GP6ZGF0
Geometry of Binomial Theorem | Visual Representation | 2 examples Think Twice 2018-01-19 | 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~ _________________________________________________________________Support my animations on:patreon.com/Think_twice_________________________________________________________________Any further questions or ideas:Email - thinktwiceask@gmail.comTwitter - twitter.com/thinktwice2580_________________________________________________________________Programs used:- Cinema 4D - Adobe Premiere Pro_________________________________________________________________MUSIC: Petite Partieyoutube.com/watch?v=5ruW_Vo0zlc
Four squares with constant area | Visual Proof | Squaring the segments | Think Twice 2017-12-27 | ==Hello everyone==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. _________________________________________________________________You can also support me on:patreon.com/Think_twice_________________________________________________________________Any further questions or ideas:Email - thinktwiceask@gmail.comTwitter - twitter.com/thinktwice2580_________________________________________________________________Programs used:- Processing - Adobe Premiere Pro_________________________________________________________________MUSIC: Cellophane Roses - Reachingyoutube.com/watch?v=PffzsCjnnRQ
Geometry: Vivianis theorem | Visualization + Proof | Think Twice 2017-12-13 | 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:openprocessing.org/sketch/483179_________________________________________________________________Support me on:patreon.com/Think_twice_________________________________________________________________Any further questions or ideas:Email - thinktwiceask@gmail.comTwitter - twitter.com/thinktwice2580_________________________________________________________________Programs used:- Processing - Adobe Premiere Pro_________________________________________________________________MUSIC:youtube.com/watch?v=NOiFtoFc8yE&list=RDMMNOiFtoFc8yE
Fitting a Cube Through a Copy of Itself | Ruperts Cube | Think Twice 2017-12-03 | I'm back :)Don't really know what to say. Just enjoy the video and let me know what you think of it:)_________________________________________________________________Support me at:patreon.com/Think_twice_________________________________________________________________Any further questions or ideas:Email - thinktwiceask@gmail.comTwitter - twitter.com/thinktwice2580_________________________________________________________________Overall render time: ~ 8 hoursPrograms used:- Cinema 4D - Adobe Premiere Pro_________________________________________________________________Music from Anime: Death Paradeyoutube.com/watch?v=ewBOcdz29Sw
Chaos Game | Fractals emerging from chaos | Computer simulation | Think Twice 2017-11-07 | 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 :patreon.com/Think_twice_________________________________________________________________Any further questions or ideas:Email - thinktwiceask@gmail.comTwitter - twitter.com/thinktwice2580_________________________________________________________________Overall render time: ~ 4 hoursPrograms used:- Processing (Game simulation)- Cinema 4D (shape outline)- Adobe Premiere Pro (Editing)_________________________________________________________________Music by: Tom DayHiatus - youtube.com/watch?v=bXQbdWlkS4U________________________________________________________________
Unfolding The Dragon | Fractal Curve | Think Twice 2017-10-22 | 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_curveThanks for watching :)_________________________________________________________________Any further questions or ideas:Email - thinktwiceask@gmail.comTwitter - twitter.com/thinktwice2580_________________________________________________________________Render time: ~ 70 hoursPrograms used:- Cinema 4D (3D animation)- Adobe Premiere Pro (Editing)_________________________________________________________________Music by: AlanKey86Time Passes - youtube.com/watch?v=0B97RGF03UMyoutube.com/channel/UCTfl24SP1qRn00D_AItf0TA
Double pendulum | Chaos | Butterfly effect | Computer simulation Think Twice 2017-09-30 | 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._________________________________________________________________Support my animations on:patreon.com/Think_twice_________________________________________________________________Any further questions or ideas:Email - thinktwiceask@gmail.comTwitter - twitter.com/thinktwice2580_________________________________________________________________Overall render time: ~ 45 hoursPrograms used:- Cinema 4D (3D animation)- Adobe Premiere Pro (Editing)- Adobe Animate (Thumbnail)_________________________________________________________________Music:Hvetter:youtube.com/watch?v=fDx2FbqKjFISound Cloud:soundcloud.com/hvetter-1
Cutting a Möbius strip in half (and more) | Animated Topology | Think Twice 2017-09-11 | ________________________________________________________________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:)_________________________________________________________________Support my animations on:patreon.com/Think_twice_________________________________________________________________Any further questions or ideas:Email - thinktwiceask@gmail.comTwitter - twitter.com/thinktwice2580_________________________________________________________________More in depth info:Tadashi Tokieda lecture:youtube.com/watch?v=SXHHvoaSctcen.wikipedia.org/wiki/M%C3%B6bius_strip_________________________________________________________________Programs used:- Cinema 4D _________________________________________________________________Music:Tom Day - Foreword: youtube.com/watch?v=p13ihTQpKwESound Cloud:soundcloud.com/tomday________________________________________________________________
Area of dodecagon | Beautiful geometry | Visual mathematics Think Twice 2017-08-23 | 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.See you soon, TT_________________________________________________________________Support my animations on:patreon.com/Think_twice_________________________________________________________________Any further questions or ideas:Email - thinktwiceask@gmail.comTwitter - twitter.com/thinktwice2580_________________________________________________________________Programs used:- Cinema 4D _________________________________________________________________
Sum of first n odd numbers | Visual mathematics | Think Twice 2017-07-19 | 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:)_________________________________________________________________Support my animations on:patreon.com/Think_twice_________________________________________________________________Any further questions or ideas:Email - thinktwiceask@gmail.comTwitter - twitter.com/thinktwice2580_________________________________________________________________Overall render time: ~ 8 hoursPrograms used:- Cinema 4D (3D animation)_________________________________________________________________Music:CLAUDE DEBUSSY: CLAIR DE LUNE
Beautiful visualization | Sum of first n Hex numbers = n^3 | animation Think Twice 2017-07-09 | 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._________________________________________________________________Support my animations on:patreon.com/Think_twice_________________________________________________________________Any further questions or ideas:Email - thinktwiceask@gmail.comTwitter - twitter.com/thinktwice2580_________________________________________________________________Overall render time: ~ 15 hoursPrograms used:- Cinema 4D _________________________________________________________________Music:Nocturne op. 9 no. 2
Sum of n squares | explained visually | Think Twice 2017-06-20 | 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. _________________________________________________________________Support my animations on:patreon.com/Think_twice_________________________________________________________________Any further questions or ideas:Email - thinktwiceask@gmail.comTwitter - twitter.com/thinktwice2580_________________________________________________________________Overall render time: ~ 25 hoursPrograms used:- Cinema 4D _________________________________________________________________Music:Bach - Prelude in C Major
Nicomachuss theorem | Visualisation | 3-D animation | Think Twice 2017-05-28 | 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. _________________________________________________________________Support my animations on:patreon.com/Think_twice_________________________________________________________________Any further questions or ideas:Email - thinktwiceask@gmail.comTwitter - twitter.com/thinktwice2580_________________________________________________________________Programs used:- Cinema 4D _________________________________________________________________
