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I-have-a-theory-this-may-not-be-true-and-I-cannot-prove-it-I-think-If-you-want-to-draw-a-closed-shape-in-x-th-dimention-the-minimum-number-of-vertex-you-will-need-is-x-1-like-if-you-want-to-

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Question Number 223210 by fantastic last updated on 17/Jul/25
I have a theory. this may not be true and I cannot prove it . I think If you want to draw a closed shape in x^(th) dimention the minimum number of vertex you will need is x+1 like if you want to draw a closed shape in 2^(nd) dimention you will need atleast 3 vertex(2+1){triangle} you can not draw a closed shape in two dimention using 2 vertex Similarly if you want to draw a closed shape in 3rd fimention you will need atleast 4 vertex(3+1){tetrahedron} you can not make a closed shape in 3dimention using 3 or 2 vertex Does that mean you will need atleast 5 vertex to draw a 4d object?? I dont know. It is just an assumption based on some similarities I noticed I cannot give you any proof of my theory
$${I}\:{have}\:{a}\:{theory}.\:{this}\:{may}\:{not}\:{be}\:{true}\:{and} \\ $$$${I}\:{cannot}\:{prove}\:{it}\:.\:{I}\:{think} \\ $$$${If}\:{you}\:{want}\:{to}\:{draw} \\ $$$$\:{a}\:{closed}\:{shape}\:{in}\:{x}^{{th}} \:{dimention} \\ $$$${the}\:{minimum}\:{number}\:{of} \\ $$$$\:{vertex}\:{you}\:{will}\:{need}\:{is}\:{x}+\mathrm{1} \\ $$$$ \\ $$$$\:{like}\:{if}\:{you}\:{want}\:{to}\:{draw}\:{a}\:{closed}\:{shape} \\ $$$${in}\:\mathrm{2}^{{nd}} \:{dimention}\:{you}\:{will}\:{need}\:{atleast} \\ $$$$\mathrm{3}\:{vertex}\left(\mathrm{2}+\mathrm{1}\right)\left\{{triangle}\right\} \\ $$$${you}\:{can}\:{not}\:{draw}\:{a}\:{closed}\:{shape}\:{in}\:{two} \\ $$$${dimention}\:{using}\:\mathrm{2}\:{vertex} \\ $$$${Similarly}\:{if}\:{you}\:\:{want}\:{to}\:{draw} \\ $$$${a}\:{closed}\:{shape}\:{in}\:\mathrm{3}{rd}\:{fimention} \\ $$$${you}\:{will}\:{need}\:{atleast}\:\mathrm{4}\:{vertex}\left(\mathrm{3}+\mathrm{1}\right)\left\{{tetrahedron}\right\} \\ $$$${you}\:{can}\:{not}\:{make}\:{a}\:{closed}\:{shape}\: \\ $$$${in}\:\mathrm{3}{dimention}\:{using}\:\mathrm{3}\:{or}\:\mathrm{2}\:{vertex} \\ $$$${Does}\:{that}\:{mean}\:{you}\:{will}\:{need}\: \\ $$$${atleast}\:\mathrm{5}\:{vertex}\:{to}\:{draw}\:{a}\:\mathrm{4}{d}\:{object}?? \\ $$$${I}\:{dont}\:{know}. \\ $$$${It}\:{is}\:{just}\:{an}\:{assumption}\:{based}\:{on} \\ $$$${some}\:{similarities}\:{I}\:{noticed} \\ $$$${I}\:{cannot}\:{give}\:{you}\:{any}\:{proof}\:{of}\:{my}\:{theory}\: \\ $$
Answered by Ghisom last updated on 17/Jul/25
you′re unprecise I can draw a triangle in R^3 but I know what you mean, it′s trivial: you need 2 vertices for a line in R^2 3 vertices for a plane in R^3 4 verrices for a “space” in R^4 n verices for a n−1 dimensional body in R^n ⇒ for an n−dimensional body in R^n you need at least n+1 vertices
$$\mathrm{you}'\mathrm{re}\:\mathrm{unprecise} \\ $$$$\mathrm{I}\:\mathrm{can}\:\mathrm{draw}\:\mathrm{a}\:\mathrm{triangle}\:\mathrm{in}\:\mathbb{R}^{\mathrm{3}} \\ $$$$\mathrm{but}\:\mathrm{I}\:\mathrm{know}\:\mathrm{what}\:\mathrm{you}\:\mathrm{mean},\:\mathrm{it}'\mathrm{s}\:\mathrm{trivial}: \\ $$$$\mathrm{you}\:\mathrm{need} \\ $$$$\mathrm{2}\:\mathrm{vertices}\:\mathrm{for}\:\mathrm{a}\:\mathrm{line}\:\mathrm{in}\:\mathbb{R}^{\mathrm{2}} \\ $$$$\mathrm{3}\:\mathrm{vertices}\:\mathrm{for}\:\mathrm{a}\:\mathrm{plane}\:\mathrm{in}\:\mathbb{R}^{\mathrm{3}} \\ $$$$\mathrm{4}\:\mathrm{verrices}\:\mathrm{for}\:\mathrm{a}\:“\mathrm{space}''\:\mathrm{in}\:\mathbb{R}^{\mathrm{4}} \\ $$$${n}\:\mathrm{verices}\:\mathrm{for}\:\mathrm{a}\:{n}−\mathrm{1}\:\mathrm{dimensional}\:\mathrm{body}\:\mathrm{in}\:\mathbb{R}^{{n}} \\ $$$$\Rightarrow \\ $$$$\mathrm{for}\:\mathrm{an}\:{n}−\mathrm{dimensional}\:\mathrm{body}\:\mathrm{in}\:\mathbb{R}^{{n}} \:\mathrm{you} \\ $$$$\mathrm{need}\:\mathrm{at}\:\mathrm{least}\:{n}+\mathrm{1}\:\mathrm{vertices} \\ $$

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