"The precision with which these pyramids align to the cardinal points of the compass is just mind-boggling."
"Quite often when you see people solve hard geometry problems, it comes down to adding something to the picture. The most beautiful geometry involves adding something that seems out of left field and it just illuminates everything."
"Tensors are the key to understanding the geometry of curved spaces."
"The seeds [of sunflowers] spiral out in two directions so that they can pack in tightly without overlapping."
"The angle between leaves...minimizes how much one leaf blocks out the next, allowing every leaf optimal exposure to sunlight and rainwater."
"So we worked together and we came up with these results, this whole structure...It was very nice geometry."
"Minkowski showed that the space of special relativity is really four dimensional and it's this kind of geometry in which you can have distances which is zero although the points are a sort of long way away from each other."
"Several ratios of significance have been discovered that are based on mathematical constants like pi and phi, the golden proportion."
"In our problem, we've been asked to find the area of the largest rectangle, what they're asking us for is to maximize the area."
"The area of the largest rectangle that can be inscribed in a semicircle that has radius r is r squared."
"If it is true that there are these geometric patterns in the galaxy, I argue that the mind also operates geometrically."
"When you create a thought, you create a vibration, and that vibration, as it turns out, creates geometries."
"The cubic Bézier is by far, the most common curve."
"The shortest distance between two points on the surface of a sphere is called a geodesic."
"Geometrically, an eigenvector... points in a direction which is stretched by the transformation."
"Curvature is a local phenomenon. If you want to talk about global phenomena... you're talking about topology, which is a whole other discipline of mathematics."
"Euclidean geometry is sometimes literally called tabletop geometry."
"In similar triangles, the ratio of medians is equal to the ratio of heights is equal to the ratio of circumradius is equal to the ratio of angular bisector is equal to the ratio of side."
"Geometry was important...a link between the Earth and the sky. So there's that consistency."
"Geometry, the plasmoids have an affinity for certain geometries and they are self-organizing."
"So let's start with something much simpler; the platonic solids."
"I want to make something that looks like this, right? So this is a hypercube, otherwise known as a tesseract."
"I'm gonna call this, for 4D, this is rotation X, Y."
"Lines 1 and 3 are parallel with a gradient of 3."
"So we can think about other angle rules while we're doing this."
"That's finding side lengths using sohcahtoa."
"Pie is the mathematical number used to calculate the ratio of a circle's circumference to its diameter."
"Life is created by the geometry; the geometry itself is alive, it is conscious, it thinks, it makes biological life."
"The ancient mystery of the dodecahedron persists."
"You can balance them with this triangle here."
"The sacred geometry starts to become very powerful when you look at it from this aspect."
"The constraints on our physical space seem to have constrained our intuitions about geometry."
"Just because you can't visualize something doesn't mean you can't still think about it visually."
"Plato believed that geometry and mathematics exist in their own ideal world."
"These are the five platonic solids, the only shapes that meet the criteria we discussed earlier."
"Geometry exists in every part of nature; its order underlies the structure of all things."
"He truly appreciated the beauty of circular forms."
"At the heart of the design of Washington DC is geometry."
"It's just unusual to see such complex angled geometry and high-quality animated textures like this."
"It's honestly a marvel, the kind of geometry that's out of bounds in some of these areas."
"It usually gives you nice answers like geometric answers."
"We all know what a trapezoid is, you might be the only one."
"The Pythagorean theorem will always get the exact distance between two points."
"Thanks to our friend Pythagoras and geometry."
"The Great Pyramid is eight-sided, locked into the Giza Plateau, and likely part of an energetic system."
"Robocompass can help demonstrate constructions easily."
"The ultimate version has seen a significant improvement in geometry and texture detail."
"The universe speaks geometry, and if you can harmonize with it, more becomes available."
"Complex numbers provide a visceral connection between geometry, trigonometry, and algebra."
"Cosine of 30 degrees: the long side of that, from everything we talked through in the last lecture, is square root of 3 over 2, square root of 3 over 2."
"I'm using the principles of geometry to solve this aerial equation, look at that game we're getting on him right now."
"An oval and ellipse, if you draw a circle and deform that circle in perspective, that is going to be too thick."
"We're going to look at how we can take a collection of vertices in three-dimensional space and form a surface of a solid object out of triangles."
"Dot products actually represent projection because we're going to be using them a lot."
"If you give me three points in a 3D space, I can draw a plane through it - there is only one.. single.. answer."
"A rectangle is quite a simple but really powerful thing, and, well, it literally is just the shape of a rectangle, but this shape of a rectangle can do quite a few different things."
"The total number of lattice points inside a big circle with radius R should be about pi*R^2."
"The precise layout of the three pyramids at Giza illustrate the Egyptian obsession with geometry."
"Everything in the universe is geometric whether it's people trees cats planets solar systems stars you name it."
"What physics is telling us is that everything comes down to geometry and the interactions between elementary particles."
"Geometric shapes: drawing various shapes on the chart."
"Wow, what a change! The lines are now completely straight, no warping in sight. Trigonometry saves the day, that's beautiful."
"Find the complement of 30 degrees: 60 degrees."
"Find the supplement of 40 degrees 35 minutes: 139 degrees 25 minutes."
"Opposite angles add up to 180... We've got the 70 over there... So opposite angles have to add up to 180."
"The mathematics we use is based around spherical geometry."
"What this immediately piqued my interest and got me thinking is it possible for a perfect cube to exist in nature?"
"Geodesics are if you like the straightest possible curves in the space they have the property that the covariant derivative of the tangent vector is zero."
"Let's make that shape that is the two circles and the rectangle."
"Geometry and proportions are key to understanding the human head's construction."
"Geometry was considered in the classical world as essential."
"When I beheld the divine geometry of creation."
"What are the interior angles of a triangle always sum up to? 180 degrees."
"So you see now, the connection with area and with thickness, by no means trivial."
"The golden ratio... it can be defined by taking a line and breaking it into two separate pieces."
"Now we have created an amazing thing, a rectangle that has these proportions is something that we call a golden rectangle."
"You probably get the 2D plane of x y, and if you add the third dimension Z, you go for depth. I have a degree in engineering, I feel."
"In real-life applications, the Z coordinate is depth."
"Creating rectangles, triangles, and even a smiley face - canvas artistry at its finest."
"This isn't magic, it's just using the fact of reference angle."
"What quadrant your angle is in, that's telling you what's positive and what's not."
"It's an awesome visualization of geometries."
"That's right, yes. For this from the Greek meaning beside another plane surface, what term denotes a geometrical solid with six phases each in the form of a parallelogram? Trilogy, freeland, ho, another plate ed."
"You can find the most precise angles and repeat it over and over."
"Analyzing and defining the 2D shape is essential for generating geometry along a spline."
"Lines have infinite points, but specifically, you need at least two."
"What's the two things you need for sure to make the equation of a line? Slope and what?"
"If you know the angle, well, you can find the slope."
"Could you show me how it goes? What is the XY plane now?"
"And similar-- you probably are use to the word in just everyday language-- but similar has a very specific meaning in geometry."
"We are allowed to rotate it, flip it, slide it. That's all allowed."
"The quickest point between two points is a straight line."
"Corresponding angles and sides must be equal in ratio for triangles to be similar."
"Let's ask the all-important question, the one we're all thinking all the time, the one that keeps us up at night, the one that makes us wonder whether it really all even makes any sense—is does this count as an inverted hexagonal pyramid?"
"Hexagons are common shapes that show up everywhere."
"Regular polyhedra is simply a solid shape where each of the faces is identical, that's all there is to it."
"Fermat primes and regular polygons: a remarkable connection."
"What we just proved is that the geometric mean of two positive numbers is always less or equal to their arithmetic mean."
"Your body has geometry, yes, bruh, six feet. Take one cell out your body and pull the DNA out, I'll be as tall as you. Just one cell."
"The cool thing with this technique is that I've solved my geometry only with quads."
"We can retrieve the center position of these four points by adding all those positions and divided by four."
"The height of the pyramid can be determined mathematically with the angle and the base length."
"You can't take a three-dimensional object such as a globe and perfectly project it onto a two-dimensional surface."
"The Flower of Life: the oldest symbol ever."
"Mandelbrot's lifelong Quest was to find a simple mathematical basis for the rough and irregular shapes of the real world."
"Underlying nearly all the shapes in the natural world is a mathematical principle known as self-similarity."
"If you've never made a circle out of a square, this is a really cool trick."
"You have a perfect circle that you made out of a square, which for a woodworker, especially a beginner, is really satisfying."
"The angle he found, 180 degrees, within observational error."
"The purpose of this tutorial is just to show some easy process to create this kind of a quad remesh geometry."
"Having neighboring four quads is the best situation as much as possible."
"That's the kind of angle you really want for the quads."
"Let me show how you can implement this from scratch for any kind of geometry."
"First thing first, what you going to have is this kind of any kind of face that you want to remesh."
"The 5-pointed star is the only constructable geometry that can get us to Golden Mean, the Divine Proportion."
"There’s something unique and important about the five major aspects because they’re situated not just as geometric configurations."
"So if we say hypercube graph, yeah, exactly, this is just coordinates practically."
"Multiplying two numbers on the circle will give us another number on the circle."
"We've generated eight equidistant points, in other words, a regular octagon."
"To construct a regular N-gon, we need to start with an angle of 360 over N."
"These are the only prime-sided constructible polygons."
"Most engineers could tell you that structurally speaking, the triangle is the strongest of all shapes."
"Parallel lines have a lot in common, but they never meet. Every other line meets once and then drifts apart forever."
"This triangle construction is actually the first proposition proved in Euclid's Elements."
"Doubling the cube is the problem of given a unit cube, construct a cube of double the volume."
"The District of Columbia is in the shape of a diamond."
"Can you explain the formula of finding the area of a circle?"
"A polygon with four verts is known as a quad, the holy grail of topology."
"We know only need to go from theta zero to PI because the line is extended so to speak."
"In any right angle triangle, the square which is described upon the side subtending the right angle is equal to the squares described upon the sides which contain the right angle."
"Once I have the two new circles, I just need to look at all of the new triplets of three mutually tangential circles and get the next new circles, and so on and so forth."
"The largest angle always opens up to the largest side."
"The smallest angle opens up to the smallest side."
"Why is a pizza box square if the pizza is a circle and a slice is a triangle?"
"It's kind of an iterative process; you add some geometry or loops, and then you sculpt it a bit to get things in place."
"The universe could only be understood in the language of mathematics whose characters are triangles, circles, and other geometric figures."
"Straight geometry frustrates common problem."
"This triangulation algorithm actually has a name, and it's called the ear clipping algorithm."
"The wedge product will distinguish if it's to the right or to the left."
"You can only understand things because of geometry."
"The golden rectangle is frequently used to frame the composition of paintings and architecture."
"The circle and the sphere are really the first forms in sacred geometry."
"The Delaunay triangulation is such that no other points live inside of any of the circumcircles of any of the triangles."
"The idea of Lloyd's algorithm is to look for what's called the centroid of each one of these polygons."
"Geometry expands your perceptions."
"...geometry is so powerful; it's like a QR code for the subconscious mind to awaken this electrical canopy around your pituitary gland and your pineal gland."
"The ratio of the small to the large is exactly the same as the large to the whole."
"Geometrodynamics is sort of the theoretical geometry of theoretical physics and the idea that natural phenomenon and the universe itself can be described and quantitated through geometry."
"The coolest thing about Archimedean solids is that they're physical objects that can be made in real life, and that just doesn't apply with self-intersecting faces."
"...so you have lines and and the points that are collected together and if you look at any of those points any of those vertices there are so many lines that coming in together and meeting at that point."
"Take this one: geometry fundamentals. It's one thing to just remember a triangle's angles add to 180°, but when you stretch and squash and see the results, it just makes it make sense."
"Possibly extending down to Planck scale geometry."
"One of the things that really makes a model look like high-definition or realistic is the gaps between geometry."
"Most people get a bit afraid of it because they don't understand, but all you have to do is go to more shapes."
"The pyramid shape is itself sacred because it mimics the sun's rays which cover the earth."
"The designs that he had in mind and the geometry that he wanted to build into this project we've definitely retained."
"Vectors are quite important because not only do they describe a point in space but they also describe a direction."
"Why stop with two-dimensional bivectors? Let's make an oriented volume! We call this a trivector."
"The ancient stone circle was designed and constructed using highly Advanced geometry, in fact, Anthony claims the Britain living on the island during the Stone Age had such powerful knowledge of geometry it could have rivaled Pythagoras."
"Euclidean geometry is a specific example of a way that geometry could be. But there's other options where there is curvature."
"It's just Pythagoras theorem souped up to three dimensions."
"A generalization of this kind of idea at every point in space would tell you everything you need to know about the geometry."
"I think the geometry of the space is what's fundamental."
"There's a deep difference between our approaches: your Spade gets turned at the level of geometry and mine does at the level of dynamics."
"If you go back to what Minkowski said, he was at heart a geometer."
"The universe as far as we know is like that. It's three-dimensional but no matter what its global geometry is, there's no reason to think it has an edge."
"Symmetry means that there's some operation... that brings the object into self-coincidence."
"Penrose suggests that certain characteristics of the universe, especially its geometry, may be transferred through these cycles."
"A triangle is the sturdiest shape known. It absorbs any external force and evenly distributes it across all sides."
"The origin of the word 'geometry' may come from the measurement of the curvature of the Earth."
"With Leonardo, it's impossible to determine what is sacred geometry or practical geometry used for art, engineering, and architecture."
"The secretive tradition of sacred geometry centers on number, harmony, geometry, and cosmology."
"Let me tell you something about Cheez-Its that you probably don't know. They're not square."
"Leonardo used geometry both conceptually as a thought exercise and applied it practically in the creation of machines and inventions."
"You have to look for triangles in the universe so you can measure triangles on earth you can actually it's hard to show that the earth is curved but you can show the earth is curved using triangles."
"That's the difference of two squares."
"You get an emergent notion of geometry of distances and times out of the quantum mechanical properties of entanglement."
"May the geometry always be in your favor."
"Any point on the Lie algebra can be thought of as a vector from the identity. And because the Lie algebra is tangent to the Lie group, this vector is just a tangent vector of the Lie group."
"A rectangle is also considered a closed vector."
"I wonder what that intersection looks like. It doesn't look like anything because I can't even see an intersection between that flat and that radius. Okay, so... it's just been around, you can see the radius runs right way around the outside, beyond the 180, and into a point."
What is the minimum number of stones you have to take with you so you know for certain at some point you can say, "I am definitely at the exact opposite vertex of where I started"?
"Triangles are one of the greatest shapes to build with."
"The great idea is that if you have any point on the plane and you draw the vector from your favorite point... to any other point on that plane, that vector is in the plane and so it's perpendicular to their normal vector."
"Controllers are just curves that we use and the reason why we use curves is because they're easy to select, they're easy to see, and we can very easily turn them on and off whenever we want to show just the geometry or just the bones or just the character."
"The bonds in the carbon like to be at 109.5 degrees."
"If we draw it like this, with 1/2 in the center, and remember that Φ and ψ are just half of √5 away from that in either direction, that tells us this span is a whole √5."
"The soul is a geometry but it is incommensurable and it is not rational conventional platonic geometry."
"You need to find the area of the two equilateral triangles and the area of these rectangular faces."
"Matrices and rotations: the key connection."
"Sacred geometry is a science of proportion."
"We're going to talk about a certain philosophy that I've began to use with geometry nodes."
"Geometry nodes at the end of the day is basically visual programming."
"Geometry was so highly respected in the Platonist schools and regarded as a means by which the mind of a human being can be elevated from the psychic to the intellectual realm."
"Geometry was conceived precisely as an irreducible whole and so as something that pre-exists in a realm beyond the reach of space and time."
"Geometry that we now have set down as a standard as a real geometry has nothing to say anymore at all."
"Angles 2 and 6 are corresponding angles."
"Anytime two angles add up to 180, they're supplementary."
"...the tangential velocity of a point moving around on a unit sphere can be decomposed into two components one from projecting the position Vector onto the XY plane and rotating it 90° therein scaled by its horizontal angular velocity."