Hello everyone. Welcome to our webinar for today. We are going to work on the 3D geometry. My name is Lucia and I have my TA Elita here. So first, the ground rules we already repeated many times. I know you are very familiar. We just quickly go through here. We will share the recording after the lesson. And the 4-step method. We want to reinforce every time because this is very helpful. The tools help you to use all the strategies you learned in the past lessons. And use this too. You combine all the strategies and skills. Help you to solve the 4-point or 5-point question. First, understand the question. And you know what question to ask for. And then you plan how to solve it. Pick one strategy or more. And try to identify the question. You know what kind of problem. And you know what strategy you want to use to solve the problem. So that's step two. Step three, you write down your solution step by step. We strongly recommend you write down your solution. But you do not need to write down very complicated steps. Very simple, easy for you to review your question later. So in the competition, prepare the sketch paper. And prepare pencil ready to solve the problem. So please do not just do the mental math. And also once you finish the solution. It's better you have a quick check with a fresh memory. Before you move on to the new question. But also you need to remember to manage your time. The competition is a time test. It's 75 minutes. You will have 24 questions. So you have to manage your time. Make sure you get time to finish all the questions. So the last lesson. Before the Thanksgiving break. We work on the make table. This is not an easy strategy. I know for the younger kids. Elita, what do you think? Making table is not easy for the grade 1 and grade 2 students. No, I think once you have the table, it's pretty simple. But just knowing what to put in the columns. And how to organize it can be really challenging for younger kids. Yeah, yeah. But we want to introduce this one to you. We could want to show you. How the table could help you to organize the data. Organize the information. Identify the relationships. So that's why we introduce this strategy. And give you some time to practice. And later with more practice. You will have better understanding. And you will know how to solve the problem. Use table. This is a very useful strategy. Especially in later. It's a very hard question. You have nowhere to go to solve it directly. Use table. Show you all the information. Given facts. So that's really helpful. Yeah, for you to analyze what you have in the question. So the table. We just quickly mentioned. You remember you make the columns. Make the rows. Yeah. So you give the name to the columns. Give the name to the rows. And then you fill in all the black cells. So once you have a table. That's very easy for you to locate all the information. Like if I ask you. So what's the favorite color for this? If we find it right away. Yellow. Yeah. If we write it in a paragraph. It's not easy for you to find right away. So that's why table is very clear. Especially show a big data set. And a lot of information. So today we're going to learn the 3D geometry. Yeah. The first question here. Yeah. The little theater. A little boy. Assembled a stacking toy. As in the picture. You can see. We look at it from above. So you have to know what they ask for. Yeah. You can see this is a stacking toy. Yeah. If you look from above. What you can see. Yeah. So that means you look down. Yeah. I think we can open the poll. Yeah. Yeah. The poll should be open now. Yeah. I see it. Yeah, I just thought there was another side that had answers on it. I think the answers are just all in the poll. And also today we will do a little experiment. So please prepare three colored pencil or markers, and also a scissor if available, but this is optional, yeah. Okay, I think I'll end the poll now. Yeah. So, yeah, so you can see the result, most of you picked the B3, that's correct. Congratulations. Yeah, let's see the solution for this one. So here, when you look from the above, you look down, you can imagine, you look down. Yeah, you will see a lot of ruins, is that right? Not from the side, from the side, you can see the separated ruins. But from the look down, you only see the circles, yeah, different circles, yeah. So what do you can see? You can see the color we paint, the top one, you definitely could see that. And the second one, it's bigger than the top one. So you can see it, because nothing block it. The next two, you cannot see them, because it's covered by the second one. And the longest one, nothing can block it, you can see it too. And the bottom one is blocked by the longest one, the biggest one. So now, you would have found the answer, three, yeah. Some of you answer six, yeah. So from the side, you could see six ruins, but they ask you, look down. You have to find the correct direction to work on this question. So today, because we work on the 3D shape, so we will need you to do the spatial thinking, and you need to see a solid object from different directions. So sometimes, the different direction will show you a different look, yeah. So like this question, yeah. So when you look down, only three, the bigger ruins, you can see, yeah. And the other three are blocked. So the correct answer would be three, yeah. Okay here, yeah, for the 3D shape, yeah, we already introduced the 3D shape. We call it a solid, yeah, solid shape, not like a 2D shape. It's a flat, and also 3D shape, from the name, it's a three dimensions. You could have length, width, and height, yeah. Unlike the 2D shape, yeah, so 3D shape, they have thickness and depth, yeah. You couldn't see, yeah, if you see the 2D shape, it's on the paper, yeah. You drew the square, drew a circle, yeah. So there is no depth. There's no height. But the 3D shape, yeah, you could find those. Because there are the three dimensions, so you could find the more, the properties related with 3D shape, like we call the edges. In the 2D shape, we call sides, yeah. Here we call edges. And also faces, you have different faces. And also the vertices, like corners. And the 3D shape solid, 2D shape flat. So you can just look the toy from the top. If you look down or from the side, you could have different looks. So that's, you have to pay attention when you work with the 3D shapes. So here, let's take a look at the first question. Elita, go ahead. Yeah, Mark always builds figures made up of eight identical wooden blocks. Which of the figures below was not made by Mark? So if it's made by Mark, it would have eight blocks. And if it's not made by Mark, then it wouldn't have eight blocks. So I think I can launch the poll because all of the options are in there. Yeah. There is only one question. Might not be clear. So pay attention, they ask what's not made by Mark. okay um i think that's enough people yeah uh we also want to mention this is a three-point question yeah usually three-point question is a little easier compared to the other two sections so you might solve this question a little faster than the other questions okay so you guys can see that most of you got it right so good job and uh you guys are pretty fast too so that's good so you can see that we count the blocks in every single auction so the first one in the first like row there's one block and then in the next row there's five and then in the third row there's two so if you add them up that's eight in total and then for the second one there's four blocks in front and then four blocks behind it so that also makes eight but for this one we see there's four blocks in front and then there's at least another um block other than the one that other than the ones that we see so you know there's definitely more than eight blocks most likely it looks like there's 12 blocks if there's just four i mean three sets of four right so that would make 12 which is not equal to eight so we can see that c is probably not made by mark because it doesn't have eight blocks and then if we want we can also check d we see there's two in front and then six behind it so that also makes eight yeah yeah this is good i think um the uh the blocks the cubes you will meet a lot in the questions so sometimes it's a single cube sometimes it's combined cube together so you should consider there is some hidden hidden blocks yeah make sure you count them all okay next question mary made a shape using some white cubes and 14 gray cubes how many of these gray cubes cannot be seen in the picture we can open the poll for this one too Okay, so let's share the result. Yeah, most of you got the correct answer. Yeah, six. Yeah, so let's look at the solution. Yeah, so here you want to make sure they give you 14 green cubes. Ask how many you cannot see. Yeah, so here you need to find how many you could see them. Yeah, so remember, you do not only count the gray faces because sometimes the one cube might show you different faces. Yeah, you can see the corner one. You can see the three faces of one cube. So you have to make sure you count the single small cubes, not count the faces. So after count, you will find that there are eight. Yeah, so use 14 minus eight. You could get six cubes. You cannot see them. Okay, yeah, next one. Five shapes are made by gluing cubes together face-to-face. Which shape uses the most cubes? Another one, count the cubes, yeah. Yeah, they really like cubes because they stack together. For this kind of question, you definitely need to count each of the options. And sometimes, when you count, you might find, oh, the first answer might be correct, but you cannot stop there. You need to count all of them, make sure. Some of the options, they are very similar. Sometimes you might miscount something, so please check all the answers. Okay, I think that's enough people. Yeah, very good. This one's great. Good job, everybody. It looks like you all got the right answer for this one. So the correct answer was E. And I'm guessing the way you guys all did it was you just counted every single option. So if we start from A, we see that that one has five cubes, right? One in front and then four behind it. And then for B, there's one, two, three, four, five, six. For C, we see that you can kind of count it in layers. The first layer would have one cube, the second layer would have five cubes, and then the third layer has one cube. So that's seven cubes in total. For D, we can do the same thing. The first layer has two cubes, the second layer has four, and the third layer has one. So again, that's seven in total. Lucia is kind of drawing out the layers for you guys. And then for E, the first layer has three cubes, the second layer has five. So that's eight, and eight is the biggest of all of those numbers. So that's why the answer is eight. Yeah, good job. I think for the counting cubes, you should all get good scores. Okay, next one. Don has two identical bricks. Please see the picture below. Which figure can he not build using these two bricks? When you read a question, you also need to pay attention to the keywords. If you do the paper test, you can underline the keywords on the test papers. Do not make a drawing on your answer card, because the answer card has to be scanned to get your score. So pay attention to that. If you could, please prepare three colored pencils or markers, and also a scissor. Yeah, so next question. We might do a, we will do a experiment. So, yeah, it's not this poll now. Okay. Yeah, so most of you pick the correct answer. Yeah, good job. Yeah, let's see the solution for this one. So, I want to mention you have to pay attention what they ask for. They said, not build, use these two bricks, and the two bricks, both with use the two cubes stick together. So that means you cannot break it to use it. So once you find all the combinations, you, you need to figure out. So which one you, you, you need to break to get the result. That's the answer. So also for those bricks, you could lie down, and you could rotate, so that could make different positions. Yeah, so you can see, A, you could get the vertical one and the lie down one, and B, you could get a two vertical one, and D, you could get a lie down one and also vertical one, and E, you could get a two vertical stick on. So let's see, you have to break one to make it. Yeah, this is correct answer. Okay, now we are going to talk about more about cubes. Yeah, so you already learned how to count the cubes, but you also need to learn the property about the cubes. So look here. You can see this kind of like the wrap. Yeah, wrap a box, like a wrap a present. Yeah, wrap a cube, the shape present. And then you would see you unwrap it. So once you unwrap it, we see this one, like a 2D shape. So this one what we call, we call the net. We call it the net of the cube. So remember that net of cube, that would be the 2D shape. Yeah. And also you need to know the cube, they have six faces. Yeah, six faces here. Yeah. And also the net when you open, you could open differently. So we only show you one way you open the net. Yeah, but there could be another way. So look what's the original cube, and then you open the wrap, and then you get the net here. So the next one, we would ask you. Yeah, so please make the, use the colors. Yeah, color pencil. And you color the opposite faces. Yeah, color opposite faces. So here, what's opposite faces? You could see here six faces. You have a top, bottom, left, right, front, and back. So what are those opposite faces? Top and bottom, left and right, front and back. That's opposite faces. So when you open it, so can you use a color pencil to paint all the six faces? And the same, the opposite face use the same color. And then you can use a scissor, cut it out, and you can fold it back to see if you color the correct faces. Now we give you two or three minutes. So when you make the, when you color this squares, you do not really need the color very perfectly. And you just, you know, quickly color, make the color. Yeah. And then you cut it out and fold it back. Yeah, like here, fold it back. And then you could see if you get the top, bottom, same color, left, right, same color, or front, back, same color. I'll give you two minutes to do. Let's open the poll for this one. Okay. So there are two answers. Yes or no. Can you get to the color? Yeah. Can you, do you color the correct faces? Yeah. Okay. Yeah. Yeah, you can use markers pen, the color pencils, whatever work, because with colorful, you couldn't know the opposite faces, yeah. Yeah. We gave us 30 more seconds and then we will share the result. So remember, in the real computation, you have sketch paper. Sometimes, you are not only drawing the picture on the sketch paper, you also can use sketch paper to fold the paper, fold the shape. Yeah, help you to solve the problem. That's allowed in the computation. Okay, so let's share the result. I think most of the students got the correct answer. Good job. Okay, so let's see here. Yeah, so whatever the color you paint, so you need to find the opposite faces. You could have three pair opposite faces, like the two green one, two red one, and two blue one. You might paint the same position with different color. It doesn't matter. Once you paint the same color, you'll get the correct answer. So let's look more about the cube, yeah, and its net. So you see the three pair of the opposite faces. We have a top, bottom, and we have left, right, and front, back. So when you unwrap the net, so you could see the back and the front. It's just the back, front, right, and the left. It's all around the bottom. And then you have a top on the other side. So there is one thing, it's very interesting. You want to get it, make sure, and understand. So you could see the bottom. We cannot see bottom here, yeah. But you know the bottom has four edges. And the four edges connected with left, right, back, and front. So because you have four edges, you should connect with four faces. That's your neighborhood faces. And you never connect with your opposite faces. You can see the top. Yeah, the bottom cannot connect with top. So this one you should remember, okay? Six faces for the cube. And every face has four neighborhood face. Use the edge to connect with the neighborhood faces. And you never connect with your opposite faces. You parallel with your opposite faces. Never touch each other. Now let's look at this question. This is a little tricky question, yeah. So one of the six stickers shown was placed on each of the six faces of a die. The die, can you see the shape they give to you? Sticker, yeah. Circle, triangle, star, arrow, square, and kangaroo. The die is shown in two positions. They give you two positions here. Which picture is on the face opposite the face with the kangaroo sticker? They ask, what is opposite face with kangaroo? Yeah, the answer, A is a circle, triangle, star, arrow, and square. Because it doesn't allow I insert two pictures, so I have to change it to the word. This is a hard question, yeah? Elita, what do you think? Yeah, I think this one's a little bit tricky to think about because you have to really understand how the cube works. Yeah. So if you could special thinking, you could match these two positions and you could find all the faces, the sticker, the position. Yeah. But you also have another strategy. Yeah, we just mentioned. Yeah, we just mentioned that every face has four neighborhood face and only do not connect with its opposite face. You could have either way to solve this problem. Yeah, this is a pretty tricky question. OK, let's share the result. OK. So the correct answer should be B. Yeah, this is not easy, but I will show you two ways to solve this problem. So let's see the first one. Yeah. So remember, I mentioned you have four neighborhood faces. You only not connect with your opposite face. So here, they give you two positions. And you could see two triangles over there. Is that right? Two triangles, position 1 and position 2. And then you could see triangle connected with what? Connected with square. Triangle also connected with star. Triangle also connected with circle and also connected with arrow. See? Yeah, the triangle connected with this four stickers except for kangaroo. So you never connect with your opposite face. So the answer, if for the triangle, the opposite face would be kangaroo. But for kangaroo, the opposite face would be triangle. So this is the answer. Yeah. So here, there is another approach. Yeah. We want to use the special thinking to figure it out, use a net to figure out how to match all the positions of the stickers. And you could see here, you could see the triangle. Yeah? The position 1, the triangle, can you see here? I just show you how you match them together. Yeah? You can see the triangle here. Yeah. So the triangle connected with the square and also the star. Yeah. And also, you open it. Yeah, you can see you open it. Yeah, you open the net. But also, you could rotate the second position. Can you see the edge I highlighted? So they are matching. You should match them together. And then you could rotate the second position. And then you match the triangle. And you could see the circle is on the bottom and the arrow on the right. So circle on the bottom would be opposite with the square. And the arrow on the right would be opposite with the front. Yeah, on the back, we are opposite with the front, the star. So triangle would definitely be opposite with the kangaroo. That's the only leftover sticker. So if by the special thinking, you can do this way, either way would work. But you have to understand. So any face, you connect it with the four neighborhood faces. Never connect it with your opposite face. OK, next one. Yeah, we will meet another problem, another type of problem. We paint the faces. So here, you could see the cube have six faces. We use a paint. We paint whatever the color. Yeah, so you paint the opposite faces. Or you can paint the same color. So here, we have the new question, what they ask for. Elita, yeah, could you go through this question? So it asks, if you paint the faces, which figure will need the least amount of paint? So when we paint every face, that means we paint the entire outside of the item. So the way I would try to think about this is from each angle. So how many faces facing forward need paint? How many from the top? How many from the bottom? How many from each side? And how many from the back? But this is kind of a hard question. So we'll see. Yeah, you have to count all of them to find the answer. Look down, look up. Look from left, look from right. And look from front, look from back. So six directions, yeah, from different angles. So why? Because why they ask which one has least amount of paint? So they all have five blocks. But why? Because you stick the blocks together. Yeah, when you stick blocks, you will lose some faces. Yeah, you do not lose the same, exactly same amount of faces. So that's why. Some have more faces left, some have less faces left. When you meet this kind of problem, it's better you check all the answers, and then you can compare which one has the least amount. Yeah, I think, Elita, you might need to go through this question from different angles, so that might helpful. Yeah. It looks like even if we give more time, it might not be easy to get the correct answer. Yeah. Let's share the pool. Yeah. Yeah. So the correct answer should be B, but this question is really difficult. So I understand why you guys might have trouble with it. So let's see how we do this one. Okay. So what we're trying to find is how many faces are in each figure, right? So for A, we see that if we look from the front, there would be five faces, right? Because yeah, one, two, three, four, five. Then if we look from the back, it would be the same because the front and the back should be identical. So it would be five faces for each. And then if we look from the top, you see we have three faces visible, one, two, three, right? And then the same thing for the bottom, there would be three faces visible. And then from each side, there would also be three faces visible. So if we think about it in total, we would have five from the front, five from the back, three from the top, three from the bottom, and then three from each side. So in total, that would be five times two plus three times four, which should give you 22, right? And then we can do the same thing for B. So it would have five in the front and five in the back, but we can't see the back. So we're just assuming it's the same as the front. And then we have two on each side. So yeah, two on the sides. And then three on the top and three on the bottom. So if we think about it, it would be five times two because five for the front, five for the back, three times two, three for the top, three for the bottom, plus two times two because two for each side. So in total, you should get 10 plus six plus four, which would be 20. And then for C, we can kind of do the same thing. I think C is probably the easiest one to calculate, but you can see there's five in the front, five in the back, five on top, five on bottom, and then two for each side. So it's five times four plus two, which gives you 22. And then for D, we can kind of do the same thing. It's five in the front, five in the back, three on top, three on the bottom, and then three on each side. So you get five times two plus three times four, which is 10 plus 12, which is 22. And then for E, it's five in the front, five in the back, three on the top, three on the bottom, and then three on both sides. So again, that's five times two plus three times four, which gives you 22. So you can see all of them have 22 except for B, which has 20. So 20 is the least out of all five of those numbers. So that's why the answer is B. But this one's really challenging, I think. Yeah, it takes a lot of time. You already, Mastcam rule, do not have this kind of longer time calculation needed. But for this question, you definitely need to find all the answers. You cannot just find out the first two and make assumption for the rest of them. So you have to check all the options. Okay, so there is another way you can count the faces. Remember, every cube has six faces. If you have two separate cube, it's very clear. You have six plus six, that's 12. But if you stick them together, can you see? If you have the two stick together, that means the left cube lose the right face. The right cube lose the left face. So with one glue, you will lose two faces. So you could figure out how many faces for this one, 12 minus 2. That's 10. Yeah. Or you can use the other from different angle to check the total faces. But you minus the sticky faces, it's more efficient, yeah? It's easy if you do the right calculation. And then you see the three cubes stick together. You have two sticky faces, two glue faces. So two glues means each of them lose two faces. And then total should be 18, 18 minus 4, you get 14 faces. So this is faster, yeah, with the calculation, it might be faster, yeah, to find the answer. So now let's look back this question again. Yeah, you can choose either approach. You want to count the different angle, or you want to use the subtraction, take off the sticky faces, both work. You know for the five cube, that's 30 faces in total. And then you can see that A, you have four glues over there. Yeah, we already mark over there, very easy to count. So that means you lose eight faces. And for B, you have five glues over there, you lose 10 faces. And C, D, E all have four glues, you lose eight faces. That's very clear. Yeah, which one has less faces? You lose more, you have less. So that would be B. So which one you like? You can make your choice. Elita, which one you like? I think this one is definitely faster. I would use this one. But I feel like you could also mess up using this one. Yeah, if you do not count the correct glue faces, if you did the wrong calculation, do the wrong calculation, yeah. So either approaches, you can choose. Okay, let's see the next question. Paul and John were building using identical cube blocks. Paul made the building show in the picture one. So this is a Paul. And the picture two show Paul's building as seen from above. You know, you look down, you get picture two. So picture three shows John's building. You look from the top. So now they said, note the number in each square indicate how many blocks are placed one on top of another in that place. So like you have two, that means you have a stack of two. You can see here. You can see on the back. So there are two blocks stacked together. And then on the very right, you'll see three. So here, that's a three stacked together. So this is how you can get the different numbers. If you have one, that's only just one layer. Yeah. So here they ask, which of the answers show John's building? Yeah. So you need to match ABCD, which one match with picture three. from the top. Remember, look from different direction, look from different angles. You might see the different look. Pay attention, you need to look down. Oh, I don't think you can fix your answer. Once you submit, it's a final. Yeah, I think so. But it doesn't matter if you could get it wrong. No worries, okay. So if you want to change the answer, you write down on your paper. So later when we share the answer with you, so you can compare if you find the correct answer. So all the answers for us as anonymous. We do not see who answer correct or wrong. We just know the total number. Okay, I think that's enough people. Yeah, so the correct answer is B. Yeah, so here, when we look down, yeah, with those, the four shapes over here, we look for joints, yeah. They give example for pulse, yeah. So we already know how to figure out the different numbers stand for. So for joints, you would see on the very left, yeah, you would have two columns. Both has two layers. Yeah, you would see from A, B, C, D. It's very easy for you to eliminate the first one. Yeah, the A. The A has four layers. Yeah, so that's definitely wrong. And also you check B, C, D, both correct. So you need to check the rest parts. So you would see you have three, one, and then you have two, and then on the very right, you have three. So on the very right, you have three layers. Yeah, so you could eliminate D. You only have B and C left over. And then you could see the C, you have actual one over there. Yeah, pay attention. This is the very, the small difference. Sometimes you might overlook. So pay attention to all the details. So the C is not correct because you have actual one cube. And then you would find C's, B's correct answer. Good job. Okay, so this is a 3D geometry. Yeah, so when you're involved with the 3D shapes, you might need to consider, look from the different angles and consider all the properties we talked about, like the cube. We show you the net. We show you the opposite faces, and you should understand the opposite faces never touch each other. And for the cube, like we mentioned, the six faces, eight edges. There is eight edges. And also we have eight corners, and we have 12 edges. The edges, that's the line you connect the two faces. Yeah, we have six faces, eight corners. Yeah, you can see the cube, yeah. So the cube have eight corners and also the 12 edges, yeah. Okay, so we could work on the bonus question now. Yeah, Elita. Okay, so Ania makes a large cube from 27 small white cubes. She paints all of the faces of the large cube green. Then Ania moves a small cube from four corners as shown. While the paint is still wet, she stamps each of the new faces onto a piece of paper. How many of the following stamps can Ania make? Maybe you explain a little bit. Yeah, so you guys can see that a stamp would have a green square there if that face of the cube has a green square. But since she removed some of the cubes from the corners, not all of the corners would be there, right? So you have to choose how many corners are missing and in what orientation they're missing from. I'll just go over the first one. So the first stamp can be made, wait. Yeah, I think the first stamp should be able to be made from the bottom of this cube because it's missing two of the adjacent corners. So you can see that it's missing one in there and then one there. So you should be able to make that stamp. So now you have to go through the rest of them and check if those stamps are possible given what we know about this cube. Yeah. Thank you, Elita. Very clear, yeah. You check all the six faces from six directions. Check off them. We worked a lot on cubes today. I really like cubes. It really stacks spheres. Yeah. So everything has to be a cube. please check from all the six angles and to see. They already give you the five five stamps and ask you how many of them you can make. Okay I think enough is answered. Yeah. So this question is really hard but I think you guys did pretty good on it. The correct answer was D which was the most popular answer. So four of these can be made. So like I said the first one can be made from the or well I guess now the black one can be made from this side here because you can see we're missing three three corners so that's how you get the black one and then the red one can be made from the top because we're missing two opposing corners so they're on opposite diagonals and then the blue one can be made from that side because it's only missing one corner this green one can be it's the same as the red one it's missing two corners two opposite corners and then the yellow one is on the back like I said because it's missing two adjacent corners so the corners are on the same edge but this one this other green one can't be made because there's no face that's missing all four corners you can see it every face is either missing one two or three corners but never all four so we can make four of the shapes just not all five of them. So for this one we recommend you to check all the to check all the directions yeah Elita can you go through so the top bottom which which faces they would show the you know the which stamp yeah so this first one can be seen on the bottom or on the back because you'll notice both of the bottom and the back are missing two of the adjacent corners so two corners that same share the same edge and then the next shape or the next stamp is made from the right side so from this side where it's missing three of the corners you can see you can make that shape and then the third stamp is made using either the top or the front because both of those are missing two opposite corners so corners that are on opposite sides of the square they're they don't share an edge and then the fourth stamp can be made from the left side where it's only missing one corner and then obviously the fifth stamp can't be made because no no faces are missing all four corners yeah thank you Elita thank you everyone that's today's lesson and then next week we will have our final lesson so we will send you the recording shortly after so hope you have a best the rest of the weekend and see you next week thank you thank you Elita thank you everyone bye

3D geometry

problem-solving

cubes

spatial thinking

nets

visible faces

cube configurations

geometry challenges

visualization techniques