Common Core: 7.G.3 Show Suggested Learning Targets
What is a cross section? The following diagrams show the horizontal and vertical slices of a rectangular prism. Scroll down the page for examples and solutions.  How to describe the cross sections of a right rectangular prism by slicing at different angles? Example: A chef needs a piece of cheese for a new recipe. The chef makes a straight top to bottom slice from a block of cheese. How are the attributes of the piece of cheese and the attributes of the block of cheese alike? How are they different? Explain your reasoning. A cross section is the intersection of a three-dimensional figure and a plane. You can think of a cross section as a two-dimensional slice of the figure. A vertical slice can be parallel to the left and right faces. The cross section always has the same shape and dimensions as there faces. A vertical slice can also be parallel to the front and back faces. The cross section always has the same shape and dimensions as these faces. A horizontal slice is parallel to the bases. The cross section always has the same shape and dimensions as these faces. How to draw cross sections? Examples:
The following diagrams show the horizontal and vertical slices of a rectangular pyramid. Scroll down the page for examples and solutions. How to describe the cross sections of a right rectangular pyramid by slicing at different angles? Example: A waiter slices his restaurant’s world-famous meatloaf as shown for two diners to share. Could the waiter’s split be even? Is there a better way to make sure? Explain. If you make any horizontal slice of a rectangular pyramid, the resulting cross section, or slice, is a rectangle. The size of the rectangle depends on the distance of the slice from the base. If you make a vertical slice of a rectangular pyramid through the vertex, the resulting cross
section, or slice, is an isosceles triangle. The base of the triangle is equal in length to an
edge of the triangular base. The height of the triangle is equal to the height of the pyramid.
Examples:
Nets and Cross Sections of Solids Example:
Slicing 3-D Figures A cross section is the two-dimensional shape that results from cutting a three-dimensional with a plane. How to identify the face shape from cuts made parallel and perpendicular to the bases of right dimensional figures? Parallel cuts will take the shape of the base. Perpendicular cuts will take the shape of the lateral face. Cuts made at an angle through the right rectangular prism or pyramid will produce a parallelogram.
Cross Sections of 3 Dimensional Figures Example: The fruit to the right has been slice horizontally. Since the fruit represented is usually a sphere, the resulting cross section is a circle. Horizontal slice, vertical slice and angled slice of a rectangular pyramid. Horizontal slice, vertical slice and angled slice of a cylinder.
What is the shape of the cross section?
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problem and check your answer with the step-by-step explanations. We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page. Tesseract The animation shows a three-dimensional projection of a rotating tesseract, the four-dimensional equivalent of a cube.Basic DescriptionThe tesseract, or tetracube, is a shape inhabiting four spatial dimensions. More specifically, it is the four-dimensional hypercube. The sides of the four-dimensional tesseract are three-dimensional cubes. Instead of a cube’s eight corners, or vertices, a tesseract has sixteen. If you find this hard to picture, don’t worry. As inhabitants of a three-dimensional world, we cannot fully visualize objects in four spatial dimensions. However, we can develop a general understanding of the tesseract by learning its structure, examining representations of the shape in lower dimensions, and exploring the math behind it. ConstructionsSweepingThe tesseract is analogous to the cube in the same way that the cube is analogous to the square, the square to the line, and the line to the point. To begin thinking about the relationship between tesseracts and cubes, it is helpful to consider the relation of cubes to squares, squares to lines, and lines to points. Let’s start from the zero-dimensional point and build our way up to the four-dimensional tesseract. We form a one-dimensional line from a point by sweeping, or stretching, the point straight out in some direction. This is the first step shown in Image 1. Now imagine taking hold of this line and sweeping it out in a direction perpendicular to its length. If you sweep out a distance equal to the length of the line, you will form a two-dimensional square. This is shown in the second step of the diagram. Image 1. For a good interactive version, try this link. We can do the same sort of thing with our square to form a cube. Imagine pulling the square outward in a direction perpendicular to its surface. You will have swept out a three-dimensional cube, shown in the third step. Now we know the procedure to use to construct a tesseract from a cube. At each step so far, we took the original object and swept it out in a new direction perpendicular to every direction in the original object. We only have three spatial dimensions, and a cube inhabits all three, but try to imagine a new direction perpendicular all of the up-down, left-right, and back-forth directions of the cube. Stretch the cube out a distance equal to the length of one of its sides into this new, fourth direction and you will have swept out a tesseract. This is shown in the last panel of Image 1. In the diagram, the orange w direction is not actually perpendicular to the other three, but it is the best we can do in a three-dimensional world. In fact, even the blue z direction isn't actually perpendicular to the flat x and y directions in the diagram. We just know to interpret the fact that these directions are perpendicular in three dimensions from how they are drawn on the two-dimensional computer screen. This is an important fact to keep in mind when discussing the 4-D tesseract. We can't directly draw 4-D objects, but we can't directly draw 3-D objects either since drawings are two dimensional. So whenever we talk about a 3-D visualization of a 4-D object, we really mean a 2-D representation of a 3-D representation of the real 4-D object.
FoldingImagine folding the six squares in Image 2 into a closed, hollow object. This is another approach that we can use to construct a cube from squares. You take two-dimensional squares and use the third dimension to fold them into a cube. Each of the squares becomes a flat face of the cube. We can follow the same approach to construct a tesseract. Consider Image 3. It looks a lot like Image 2, except instead of a flat collection of six squares we have a three-dimensional collection of eight cubes. It is far more difficult to imagine than for the squares, but using the fourth dimension we could fold these eight cubes together to form a tesseract. Each cube becomes a side of the tesseract, analogous to the square faces of the cube, except three-dimensional. These eight cubic facets are oriented such that two parallel facets lie on opposing sides of the tesseract in each of the four spatial directions. Visualizing the TesseractVisualizing four dimensions isn’t easy when you live in three and use computer screens in two. In order to better understand the tesseract and interpret images like the one at the top of the page, it is helpful to consider how inhabitants of a two-dimensional world would go about understanding objects in three dimensions. Edwin Abbott’s book Flatland presents such an analogy. Inhabitants of Flatland see and move in just two dimensions. In their world, three-dimensional shapes cannot be seen all at once, just as we cannot fully visualize a tesseract. There are two main ways an inhabitant of a flat world could perceive the structure of a three-dimensional object. We can use analogous methods to picture the tesseract. Slices
If a three-dimensional shape were to pass through the two-dimensional world of flatland, the inhabitants would perceive a series of its slices. For a sphere, first a point would appear, then a gradually growing circle until the sphere was half-way through, and finally a circle that shrinks until it disappears altogether. An object like a cube would be more confusing to a flatlander, since its slices look different depending on how it is tilted as it passes through a flat plane. Consider Images 4 and 5, which show a cube passing through a two-dimensional plane and the corresponding slices at two different tilts. The right-side panel of each image is all a flatlander would be able to experience. As you can see, the slices in the two images look fairly different. This doesn't seem too strange to us, since we can see the cube in the first half of each image. But for a flatlander it might be hard to tell that the slices are from the same object. Just as flatlanders can only perceive two-dimensional slices of three-dimensional objects, we are limited to visualizing three-dimensional “slices” of the four-dimensional tesseract. Images 6 and 7 show the slices of a tesseract passing through three-dimensional space at two different tilts. The perspective is closely analogous to the flatlander’s view of a passing cube in the second halves of Images 4 and 5. Similar to Image 4’s depiction of a square being sliced parallel to one of its faces as it passes through a two-dimensional plane, image 6 shows a tesseract being sliced in three dimensions parallel to one of its cubical facets. As illustrated in the animations, slicing a cube this way yields a square while slicing a tesseract this way yields a cube. In Image 7 the tesseract is being sliced corner to corner as it passes through our three-dimensional view, analogous to how the tilted cube is being sliced in Image 5 as it passes through a two-dimensional plane. Note that only the light blue parts of these animations are the actual slices. The static backgrounds are just shadows or projections of the shapes being sliced. ProjectionsThe second way Flatlanders could try to understand a three-dimensional object is by looking at its projections onto their two-dimensional world. The easiest way to think about projections is probably as shadows. There are two main types of shadows, depending on the distance between the object casting a shadow and the source of light. These correspond to the two main types of projection that can be used to visualize objects in fewer dimensions than they inhabit. For objects held close to a light source, features that are farther away from the light appear smaller in the shadow than those that are near the light source. This kind of shadow depicting objects in perspective is called a Stereographic Projection. For objects very far away from a light source, the light rays are so close to parallel that features farther from the source cease to be reduced in size in the shadow. The limiting situation, a shadow cast by the exactly parallel light of an infinitely distant source, is called an Orthographic Projection. This type of projection makes for more symmetric images, but lacks the sense of depth provided by stereographic projection. Even if a flatlander was told which type of shadow they were looking at, it would still be quite a challenge for them to mentally translate the two dimensional projection into a three-dimensional shape . They would need to be told what motion and positioning in three dimensions looks like in a projection. Cube ShadowsConsider the shadow cast by the rotating cube beneath a nearby light source in the animation on the right, an example of stereographic projection. Note that although the cube may look solid in the animation, it casts a shadow as if its sides were semi-transparent, perhaps made of red glass, with edges made of some solid like wire. What is really a cube with six square faces appears in the shadow as a small square inside of a large square with four highly distorted squares in between. As the cube rotates, the side lengths and internal angles in the projection change; the distorted sides morph into squares and back as the inner and outer squares change places. We know these distortions are not actually occurring, and that as a part of the shadow grows the corresponding cube face is just rotating closer to the light source. The important features of the cube, like the number of faces and vertices, stay true to the actual three-dimensional object even in projection. These would all be important things for an inhabitant of Flatland trying to understand a cube to know. By analogy we can use these lessons about shadows to better visualize and understand the tesseract. Tesseract "Shadows"Image 9 The four-dimensional equivalent of a shadow is a three-dimensional projection, like the one shown in the animation at the top of the page and here in the still Image 9. As with the animation of the rotating cube, these are stereographic projections. While a cube with a facet directed towards a nearby light in three dimensions casts a shadow of a square within a square, a tesseract with a facet directed towards a nearby light source in four dimensions casts a three-dimensional “shadow” of a cube within a cube. Instead of the cube’s facet closest to the light source projected as an outer square, we have the facet of the tesseract closest to the four-dimensional light source projected as an outer cube. Similarly, instead of the facet of a cube farthest from the light projected as an inner square, the facet of a tesseract farthest from the light is projected as an inner cube. In the shadow of a cube, the four sides appear as highly distorted squares in between these inner and outer squares. In the 3-D tesseract projection in image 9, six of the tesseract's facets appear as highly distorted cubes occupying the space between the inner and outer cubes. In the 2-D shadow of the rotating cube we saw the inner square replace the outer square as the cube rotated through a third dimension. In the animation at the top of the page we observe the inner cube, really just a facet of the tesseract farther away from the light source in the fourth dimension, unfold to replace the outer cube as the tesseract completes a half turn through the fourth dimension. These images help us to interpret stereographic projections of a tesseract from one perspective, but we could always change the tesseract's orientation so that, say, a corner were facing the light source. The projection would look quite different. To explore what the projection of a differently oriented tesseract would look like, try out the interactive feature below. To view stereographic projections like the one in image 9 or this page's main animation, check the Perspective box. This provides a greater sense of depth, albeit with greater distortion of the true dimensions of the tesseract than with the default orthographic projection setting. This applet was created by Milosz Kosmider. For more visualizations of the tesseract, see the Related Links section at the bottom of the page. A More Mathematical Explanation[Click to view A More Mathematical Explanation] In the language of geometry, the tesseract is a type of regular polytope. Since it [...] Why It's InterestingToday the idea of more than three dimensions is fairly common. You can read about hyperspace in science fiction stories, four-dimensional space-time in physics textbooks, and a mind boggling 10 to 26 “curled up” dimensions in the writings of many modern scientists. But prior to the development of four-dimensional geometry and the popularization of the idea of dimensions by books like Abbott’s Flatland in the 1800s, the public, the physicists, and even most mathematicians did not pay much attention to the idea of four dimensions, much less 26.[1] The door to higher dimensionality opened when people started studying strange geometric shapes like the tesseract. The tesseract is probably the best known higher dimensional shape[2], and as such represents a kind of symbol of the expansion of the human imagination into higher dimensions. As Many Dimensions as You LikeA two-dimensional orthographic projection of the 10-dimensional hypercube. The visualization techniques and geometry we have been developing for tesseracts can be extended to help us understand other four-dimensional objects and even higher dimensional shapes. In three dimensions there are five regular polytopes, known as the Platonic Solids, which as the name suggests have been studied since the time of the ancient greeks. One of the first mathematicians to take four dimensional geometry seriously was Ludwig Schläfli. In the mid-1800s, Schläfli figured out that in four-dimensional space there are six regular polytopes[3]. The tesseract is one, as is an enormous shape with 600 faces, each one a three-dimensional tetrahedron. And why stop at four? We can mathematically analyze and even form visual projections of objects in five, six, or more dimensions. Instead of using triples or quadruples of coordinates, we can consider a space of arbitrarily many dimensions, consisting of all n-tuples of the form [Show more about Hypercubes] These are just the regular polytopes, shapes with all identical faces. Higher dimensions are home to innumerable irregular polytopes as well. Knowledge of relatively simple higher-dimensional shapes like the tesseract and how to wrap one's head around its four-dimensional structure would be essential for anyone interested in tackling those far stranger creatures. Higher Dimensions in PhysicsPhysics and mathematics borrow from each other all the time. Sometimes the mathematicians develop an idea that the physicists find useful later, and sometimes the physicists discover a phenomenon and end up developing exciting new mathematics to describe it. When the geometry of higher dimensions first started to be studied in the 1800s, it was generally regarded as purely abstract and mathematical. But with the development of Einstein's theories of relativity in the early 1900s and more recent developments in superstring theory, physicists have been taking the idea very seriously. As mathematician Ian Stewart says, "The potential importance of high-dimensional geometry for real physical space and time has suddenly become immense".[4] Space-timeIn some versions of string theory, six extra "curled up" dimensions are thought to take on a six-dimensional structure called a Calabi-Yau manifold, a section of which is shown here in a three-dimensional projection. Albert Einstein's theories of relativity treat space and time together, as a single four-dimensional entity called space-time. Space-time coordinates are quadruples of the form Dimensions in String TheoryIn modern String Theory, the fundamental components of the universe are not particles but tiny vibrating strings. Within the framework of the theory, the large variety of different types of particles we observe are composed of the same fundamental strings vibrating at distinct frequencies. While this model is quite successful in many respects, it requires our space-time universe to be either 10-dimensional or 26-dimensional, implying there are either six or 22 spatial dimensions that we don't know about.[1] While this may sound absurd, there is no discrepancy with our everyday experience if these extra dimensions are "curled up" too small for us to detect. Imagine ants confined to walking along a thin piece of thread. For all practical means and purposes, their little world is one-dimensional. But imagine that the thread were thicker, like a large rope. Suddenly besides just going backwards and forwards the ants can move side to side along the curvature of this thick rope. According to String Theory, we are like the ants on the thread, living in a world with extra dimensions too small to be noticed. Some physicists are considering an even stranger possibility, and suggest that the extra dimensions are quite large, so large that our four-dimensional world exists inside of a higher-dimensional reality.[2] Like inhabitants of Flatland who can't move in the third dimension, we would be prevented from moving in these hidden directions by our laws of physics. Other ApplicationsThe mathematics of higher dimensions has applications beyond just spatial and temporal dimensions. An n-dimensional space consists of a bunch of points, each of which is list of n numbers. We don’t have to think of these numbers as coordinates for positions in physical space. They could represent any variables. Consider a bicycle. We can describe the state of all the bicycle’s crucial components with six numbers: the angle between the handlebars and the frame, the angular positions of each of the two wheels, the positioning of the pedals’ axle, and the angular positions of each of the two pedals. A bicycle is of course a three-dimensional object. But we can describe any configuration of the bicycle with six numbers, numbers we can view as generalized coordinates, meaning that the state of the bicycle exists in an abstract, six-dimensional space. To get the hang of riding a bicycle, you need learn how these six numbers interact, not to mention the extra variables for motion and interaction with the road. This can be thought of as learning the six-dimensional geometry of “bicycle space”. This way of visualizing the state of a system in an imaginary space of as many dimensions as you have variables turns out to be quite useful, and is used by mathematicians, physicists, and even economists and biologists[4]. VirologyFor example, virologists find it useful to think of specific viruses as “points” in a space of DNA sequences. Each virus has a DNA sequence composed of a series of smaller molecular components called bases, represented by the four letters A, C, G, and T. The bases are the coordinates, and the DNA sequences are the points. Each coordinate is limited to being one of the four bases, just like how the coordinates for the vertices of a unit square, cube, or tesseract are limited to being either 0 or 1. The numbers 0 and 1 are like DNA bases, which makes unit hypercubes like DNA if it had only two bases. This allows scientists to think of the possible viral DNA sequences as vertices of a very high-dimensional hypercube-like object, a sort of hypercube with its interior filled with smaller hypercubes. Each vertex corresponds to a specific virus, and since edges connect two vertices which differ by exactly one position, each edge can be thought of as a point mutation changing one base in a virus’ DNA sequence. Because this hypercube-like shape has such a high dimension, each vertex connects to quite a lot of other vertices, meaning that the virus has the potential to mutate in an enormous number of different ways. Using geometry of higher dimensional objects, virologists can understand how quickly the variation in possible mutations grows with longer sequences of viral DNA.[4]
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What shape would you get if you took a slice of a cube that is perpendicular to its bottom base
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