Few years ago, by making some kind of introspection into basic numbers, i've found some interesting things in multiple of nine. So far I have not found an answer to my questions, so I allow myself to invoke you to clarify my thoughts. Let me briefly describe the procedure for visualizing the idea.
Construction
1. make a $10\times11$ matrix of multiple of nine (from $9$ to $990$)
$$\mathtt{\small\color{black}{009\quad018\quad027\quad036\quad045\quad054\quad063\quad072\quad081\quad090\quad099\\[1.5ex]108\quad117\quad126\quad135\quad144\quad153\quad162\quad171\quad180\quad189\quad198\\[1.5ex]207\quad216\quad225\quad234\quad243\quad252\quad261\quad270\quad279\quad288\quad297\\[1.5ex]306\quad315\quad324\quad333\quad342\quad351\quad360\quad369\quad378\quad387\quad396\\[1.5ex]405\quad414\quad423\quad432\quad441\quad450\quad459\quad468\quad477\quad486\quad495\\[1.5ex]504\quad513\quad522\quad531\quad540\quad549\quad558\quad567\quad576\quad585\quad594\\[1.5ex]603\quad612\quad621\quad630\quad639\quad648\quad657\quad666\quad675\quad684\quad693\\[1.5ex]702\quad711\quad720\quad729\quad738\quad747\quad756\quad765\quad774\quad783\quad792\\[1.5ex]801\quad810\quad819\quad828\quad837\quad846\quad855\quad864\quad873\quad882\quad891\\[1.5ex]900\quad909\quad918\quad927\quad936\quad945\quad954\quad963\quad972\quad981\quad990\\[1.5ex]}}$$
2. shift each line as well
$$\mathtt{\small\color{black}{009\quad018\quad027\quad036\quad045\quad054\quad063\quad072\quad081\quad090\quad099\\[1.5ex]\quad\;108\quad117\quad126\quad135\quad144\quad153\quad162\quad171\quad180\quad189\quad198\\[1.5ex]\quad\;\quad\;207\quad216\quad225\quad234\quad243\quad252\quad261\quad270\quad279\quad288\quad297\\[1.5ex]\quad\;\quad\;\quad\;306\quad315\quad324\quad333\quad342\quad351\quad360\quad369\quad378\quad387\quad396\\[1.5ex]\quad\;\quad\;\quad\;\quad\;405\quad414\quad423\quad432\quad441\quad450\quad459\quad468\quad477\quad486\quad495\\[1.5ex]\quad\;\quad\;\quad\;\quad\;\quad\;504\quad513\quad522\quad531\quad540\quad549\quad558\quad567\quad576\quad585\quad594\\[1.5ex]\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;603\quad612\quad621\quad630\quad639\quad648\quad657\quad666\quad675\quad684\quad693\\[1.5ex]\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;702\quad711\quad720\quad729\quad738\quad747\quad756\quad765\quad774\quad783\quad792\\[1.5ex]\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;801\quad810\quad819\quad828\quad837\quad846\quad855\quad864\quad873\quad882\quad891\\[1.5ex]\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;900\quad909\quad918\quad927\quad936\quad945\quad954\quad963\quad972\quad981\quad990\\[1.5ex]}}$$
3. split the matrix to produce two mirrored triangles
$$\mathtt{\small\color{black}{009\quad018\quad027\quad036\quad045\quad054\quad063\quad072\quad081\quad090\quad\qquad099\\[1.5ex]\quad108\quad117\quad126\quad135\quad144\quad153\quad162\quad171\quad180\quad\qquad189\quad198\\[1.5ex]\qquad\;207\quad216\quad225\quad234\quad243\quad252\quad261\quad270\quad\qquad279\quad288\quad297\\[1.5ex]\qquad\quad\;306\quad315\quad324\quad333\quad342\quad351\quad360\quad\qquad369\quad378\quad387\quad396\\[1.5ex]\qquad\qquad\;\;405\quad414\quad423\quad432\quad441\quad450\quad\qquad459\quad468\quad477\quad486\quad495\\[1.5ex]\qquad\qquad\quad\;\;504\quad513\quad522\quad531\quad540\quad\qquad549\quad558\quad567\quad576\quad585\quad594\\[1.5ex]\qquad\qquad\qquad\;\;\;603\quad612\quad621\quad630\quad\qquad639\quad648\quad657\quad666\quad675\quad684\quad693\\[1.5ex]\qquad\quad\qquad\qquad\;\;\;702\quad711\quad720\quad\qquad729\quad738\quad747\quad756\quad765\quad774\quad783\quad792\\[1.5ex]\qquad\qquad\qquad\qquad\;\;\;\;801\quad810\quad\qquad819\quad828\quad837\quad846\quad855\quad864\quad873\quad882\quad891\\[1.5ex]\qquad\qquad\qquad\qquad\qquad900\quad\qquad909\quad918\quad927\quad936\quad945\quad954\quad963\quad972\quad981\quad990\\[1.5ex]}}$$
First Observation
The matrix is composed of $110$ numbers for a sum of $54945$ where
- The left triangle has $55$ numbers for a sum of $18315$ ($\frac{1}{3}$ of total)
- The right triangle has $55$ numbers for a sum of $36630$ ($\frac{2}{3}$ of total)
The two triangles are mirrored
- The left triangle contains $30$ even number and $25$ odd number where
- The right triangle contains $25$ even number and $30$ odd number
The two triangles central pivots
- The left pivot is $333$ and correspond to the mean ($\frac{18315}{55}=333$)
- The right pivot is $666$ and correspond to the mean ($\frac{36630}{55}=666$)
Second Observation
$$\mathtt{\small\color{silver}{\color{black}{009}\quad\color{gray}{018}\quad\color{gray}{027}\quad\color{gray}{036}\quad\color{gray}{045}\quad\color{gray}{054}\quad\color{gray}{063}\quad\color{gray}{072}\quad\color{gray}{081}\quad\color{black}{090}\quad\qquad\color{black}{099}\\[1.5ex]\quad\color{gray}{108}\quad117\quad126\quad135\quad144\quad153\quad162\quad171\quad\color{gray}{180}\quad\qquad\color{gray}{189}\quad\color{gray}{198}\\[1.5ex]\qquad\;\color{gray}{207}\quad216\quad225\quad234\quad243\quad252\quad261\quad\color{gray}{270}\quad\qquad\color{gray}{279}\quad288\quad\color{gray}{297}\\[1.5ex]\qquad\quad\;\color{gray}{306}\quad315\quad324\quad\color{green}{333}\quad342\quad351\quad\color{gray}{360}\quad\qquad\color{gray}{369}\quad378\quad387\quad\color{gray}{396}\\[1.5ex]\qquad\qquad\;\;\color{gray}{405}\quad414\quad423\quad432\quad441\quad\color{gray}{450}\quad\qquad\color{gray}{459}\quad468\quad477\quad486\quad\color{gray}{495}\\[1.5ex]\qquad\qquad\quad\;\;\color{gray}{504}\quad513\quad522\quad531\quad\color{gray}{540}\quad\qquad\color{gray}{549}\quad558\quad567\quad576\quad585\quad\color{gray}{594}\\[1.5ex]\qquad\qquad\qquad\;\;\;\color{gray}{603}\quad612\quad621\quad\color{gray}{630}\quad\qquad\color{gray}{639}\quad648\quad657\quad\color{green}{666}\quad675\quad684\quad\color{gray}{693}\\[1.5ex]\qquad\qquad\qquad\quad\;\;\;\color{gray}{702}\quad711\quad\color{gray}{720}\quad\qquad\color{gray}{729}\quad738\quad747\quad756\quad765\quad774\quad783\quad\color{gray}{792}\\[1.5ex]\qquad\qquad\qquad\qquad\;\;\;\;\color{gray}{801}\quad\color{gray}{810}\quad\qquad\color{gray}{819}\quad828\quad837\quad846\quad855\quad864\quad873\quad882\quad\color{gray}{891}\\[1.5ex]\qquad\qquad\qquad\qquad\qquad\color{black}{900}\quad\qquad\color{black}{909}\quad\color{gray}{918}\quad\color{gray}{927}\quad\color{gray}{936}\quad\color{gray}{945}\quad\color{gray}{954}\quad\color{gray}{963}\quad\color{gray}{972}\quad\color{gray}{981}\quad\color{black}{990}\\[1.5ex]}}$$
- Left triangle $009 + 090 + 900 = 999$ which is triple of central point ($3\times333 = 999$)
- Right triangle $099 + 909 + 990 = 1998$ which is triple of central point ($3\times666 = 1998$)
The same fact is true for the following triplets:
$$\begin{array}{r|r} Left \quad\quad\quad\quad & Right \quad\quad\quad\quad\\ \hline 009 + 090 + 900 = 999 & 099 + 909 + 990 = 1998 \\ 018 + 180 + 801 = 999 & 198 + 819 + 981 = 1998 \\ 027 + 270 + 702 = 999 & 297 + 729 + 972 = 1998 \\ 036 + 360 + 603 = 999 & 396 + 639 + 963 = 1998 \\ 045 + 450 + 504 = 999 & 495 + 549 + 954 = 1998 \\ 054 + 540 + 405 = 999 & 594 + 459 + 945 = 1998 \\ 063 + 630 + 306 = 999 & 693 + 369 + 936 = 1998 \\ 072 + 720 + 207 = 999 & 792 + 279 + 927 = 1998 \\ 081 + 810 + 108 = 999 & 891 + 189 + 918 = 1998 \\ \hdashline 090 + 900 + 009 = 999 & 990 + 099 + 909 = 1998 \\ \end{array}$$
The sequencing produce a rotating triangle on the enclosure.
This is always true for the inner permutations sequences.
$$\mathtt{\small\color{silver}{009\quad018\quad027\quad036\quad045\quad054\quad063\quad072\quad081\quad090\quad\qquad099\\[1.5ex]\quad108\quad\color{black}{117}\quad\color{gray}{126}\quad\color{gray}{135}\quad\color{gray}{144}\quad\color{gray}{153}\quad\color{gray}{162}\quad\color{black}{171}\quad180\quad\qquad189\quad198\\[1.5ex]\qquad\;207\quad\color{gray}{216}\quad225\quad234\quad243\quad252\quad\color{gray}{261}\quad270\quad\qquad279\quad\color{black}{288}\quad297\\[1.5ex]\qquad\quad\;306\quad\color{gray}{315}\quad324\quad\color{green}{333}\quad342\quad\color{gray}{351}\quad360\quad\qquad369\quad\color{gray}{378}\quad\color{gray}{387}\quad396\\[1.5ex]\qquad\qquad\;\;405\quad\color{gray}{414}\quad423\quad432\quad\color{gray}{441}\quad450\quad\qquad459\quad\color{gray}{468}\quad477\quad\color{gray}{486}\quad495\\[1.5ex]\qquad\qquad\quad\;\;504\quad\color{gray}{513}\quad522\quad\color{gray}{531}\quad540\quad\qquad549\quad\color{gray}{558}\quad567\quad576\quad\color{gray}{585}\quad594\\[1.5ex]\qquad\qquad\qquad\;\;\;603\quad\color{gray}{612}\quad\color{gray}{621}\quad630\quad\qquad639\quad\color{gray}{648}\quad657\quad\color{green}{666}\quad675\quad\color{gray}{684}\quad693\\[1.5ex]\qquad\qquad\qquad\quad\;\;\;702\quad\color{black}{711}\quad720\quad\qquad729\quad\color{gray}{738}\quad747\quad756\quad765\quad774\quad\color{gray}{783}\quad792\\[1.5ex]\qquad\qquad\qquad\qquad\;\;\;\;801\quad810\quad\qquad819\quad\color{black}{828}\quad\color{gray}{837}\quad\color{gray}{846}\quad\color{gray}{855}\quad\color{gray}{864}\quad\color{gray}{873}\quad\color{black}{882}\quad891\\[1.5ex]\qquad\qquad\qquad\qquad\qquad900\quad\qquad909\quad918\quad927\quad936\quad945\quad954\quad963\quad972\quad981\quad990\\[1.5ex]}}$$
$$\begin{array}{r|r} Left \quad\quad\quad\quad & Right \quad\quad\quad\quad\\ \hline 117 + 171 + 711 = 999 & 288 + 828 + 882 = 1998 \\ 126 + 261 + 612 = 999 & 387 + 738 + 873 = 1998 \\ 135 + 351 + 513 = 999 & 486 + 648 + 864 = 1998 \\ 144 + 441 + 414 = 999 & 585 + 558 + 855 = 1998 \\ 153 + 531 + 315 = 999 & 684 + 468 + 846 = 1998 \\ 162 + 621 + 216 = 999 & 783 + 378 + 837 = 1998 \\ \hdashline 171 + 711 + 117 = 999 & 882 + 288 + 828 = 1998 \\ \end{array}$$
$$\mathtt{\small\color{silver}{009\quad018\quad027\quad036\quad045\quad054\quad063\quad072\quad081\quad090\quad\qquad099\\[1.5ex]\quad108\quad117\quad126\quad135\quad144\quad153\quad162\quad171\quad180\quad\qquad189\quad198\\[1.5ex]\qquad\;207\quad216\quad\color{black}{225}\quad\color{gray}{234}\quad\color{gray}{243}\quad\color{black}{252}\quad261\quad270\quad\qquad279\quad288\quad297\\[1.5ex]\qquad\quad\;306\quad315\quad\color{gray}{324}\quad\color{green}{333}\quad\color{gray}{342}\quad351\quad360\quad\qquad369\quad378\quad387\quad396\\[1.5ex]\qquad\qquad\;\;405\quad414\quad\color{gray}{423}\quad\color{gray}{432}\quad441\quad450\quad\qquad459\quad468\quad\color{black}{477}\quad486\quad495\\[1.5ex]\qquad\qquad\quad\;\;504\quad513\quad\color{black}{522}\quad531\quad540\quad\qquad549\quad558\quad\color{gray}{567}\quad\color{gray}{576}\quad585\quad594\\[1.5ex]\qquad\qquad\qquad\;\;\;603\quad612\quad621\quad630\quad\qquad639\quad648\quad\color{gray}{657}\quad\color{green}{666}\quad\color{gray}{675}\quad684\quad693\\[1.5ex]\qquad\qquad\qquad\quad\;\;\;702\quad711\quad720\quad\qquad729\quad738\quad\color{black}{747}\quad\color{gray}{756}\quad\color{gray}{765}\quad\color{black}{774}\quad783\quad792\\[1.5ex]\qquad\qquad\qquad\qquad\;\;\;\;801\quad810\quad\qquad819\quad828\quad837\quad846\quad855\quad864\quad873\quad882\quad891\\[1.5ex]\qquad\qquad\qquad\qquad\qquad900\quad\qquad909\quad918\quad927\quad936\quad945\quad954\quad963\quad972\quad981\quad990\\[1.5ex]}}$$
$$\begin{array}{r|r} Left \quad\quad\quad\quad & Right \quad\quad\quad\quad\\ \hline 225 + 252 + 522 = 999 & 477 + 747 + 774 = 1998 \\ 234 + 342 + 423 = 999 & 576 + 657 + 765 = 1998 \\ 243 + 432 + 324 = 999 & 675 + 567 + 756 = 1998 \\ \hdashline 252 + 522 + 225 = 999 & 774 + 477 + 747 = 1998 \\ \end{array}$$
The number of permutations triplets from enclosure (outer) to surrounding central number (inner)
$$\bbox[12px,border:1px solid black]{ \begin{array}{r|r} Outer & 9 \\ Center & 6 \\ Inner & 3 \\ \end{array} }$$
Third Observation
Theses observations are not only limited to triangular shapes, we can observe regular hexagonal shapes framing the central number (here, from inner to outer). The permutations can be associated by symmetricals couples to produce the double of the central point ($2\times333$ & $2\times666$)
$$\mathtt{\small\color{silver}{009\quad018\quad027\quad036\quad045\quad054\quad063\quad072\quad081\quad090\quad\qquad099\\[1.5ex]\quad108\quad117\quad126\quad135\quad144\quad153\quad162\quad171\quad180\quad\qquad189\quad198\\[1.5ex]\qquad\;207\quad216\quad225\quad\color{black}{234}\quad\color{black}{243}\quad252\quad261\quad270\quad\qquad279\quad288\quad297\\[1.5ex]\qquad\quad\;306\quad315\quad\color{black}{324}\quad\color{green}{333}\quad\color{black}{342}\quad351\quad360\quad\qquad369\quad378\quad387\quad396\\[1.5ex]\qquad\qquad\;\;405\quad414\quad\color{black}{423}\quad\color{black}{432}\quad441\quad450\quad\qquad459\quad468\quad477\quad486\quad495\\[1.5ex]\qquad\qquad\quad\;\;504\quad513\quad522\quad531\quad540\quad\qquad549\quad558\quad\color{black}{567}\quad\color{black}{576}\quad585\quad594\\[1.5ex]\qquad\qquad\qquad\;\;\;603\quad612\quad621\quad630\quad\qquad639\quad648\quad\color{black}{657}\quad\color{green}{666}\quad\color{black}{675}\quad684\quad693\\[1.5ex]\qquad\qquad\qquad\quad\;\;\;702\quad711\quad720\quad\qquad729\quad738\quad747\quad\color{black}{756}\quad\color{black}{765}\quad774\quad783\quad792\\[1.5ex]\qquad\qquad\qquad\qquad\;\;\;\;801\quad810\quad\qquad819\quad828\quad837\quad846\quad855\quad864\quad873\quad882\quad891\\[1.5ex]\qquad\qquad\qquad\qquad\qquad900\quad\qquad909\quad918\quad927\quad936\quad945\quad954\quad963\quad972\quad981\quad990\\[1.5ex]}}$$
$$\begin{array}{r|r|r} Left & 234 + 243 + 324 + 342 + 423 + 432 = 1998 & 6 \times 333 = 1998 \\ \hline Right & 567 + 576 + 657 + 675 + 756 + 765 = 3996 & 6 \times 666 = 3996 \\ \end{array}$$
$$\begin{array}{r|r} Left \quad\quad & Right \quad\quad\\ \hline 234 + 432 = 666 & 567 + 765 = 1332 \\ 243 + 423 = 666 & 576 + 756 = 1332 \\ 324 + 342 = 666 & 657 + 675 = 1332 \\ \end{array}$$
$$\mathtt{\small\color{silver}{009\quad018\quad027\quad036\quad045\quad054\quad063\quad072\quad081\quad090\quad\qquad099\\[1.5ex]\quad108\quad117\quad126\quad\color{black}{135}\quad144\quad\color{black}{153}\quad162\quad171\quad180\quad\qquad189\quad198\\[1.5ex]\qquad\;207\quad216\quad225\quad234\quad243\quad252\quad261\quad270\quad\qquad279\quad288\quad297\\[1.5ex]\qquad\quad\;306\quad\color{black}{315}\quad324\quad\color{green}{333}\quad342\quad\color{black}{351}\quad360\quad\qquad369\quad378\quad387\quad396\\[1.5ex]\qquad\qquad\;\;405\quad414\quad423\quad432\quad441\quad450\quad\qquad459\quad\color{black}{468}\quad477\quad\color{black}{486}\quad495\\[1.5ex]\qquad\qquad\quad\;\;504\quad\color{black}{513}\quad522\quad\color{black}{531}\quad540\quad\qquad549\quad558\quad567\quad576\quad585\quad594\\[1.5ex]\qquad\qquad\qquad\;\;\;603\quad612\quad621\quad630\quad\qquad639\quad\color{black}{648}\quad657\quad\color{green}{666}\quad675\quad\color{black}{684}\quad693\\[1.5ex]\qquad\qquad\qquad\quad\;\;\;702\quad711\quad720\quad\qquad729\quad738\quad747\quad756\quad765\quad774\quad783\quad792\\[1.5ex]\qquad\qquad\qquad\qquad\;\;\;\;801\quad810\quad\qquad819\quad828\quad837\quad\color{black}{846}\quad855\quad\color{black}{864}\quad873\quad882\quad891\\[1.5ex]\qquad\qquad\qquad\qquad\qquad900\quad\qquad909\quad918\quad927\quad936\quad945\quad954\quad963\quad972\quad981\quad990\\[1.5ex]}}$$
$$\begin{array}{r|r|r} Left & 135 + 153 + 315 + 351 + 513 + 531 = 1998 & 6 \times 333 = 1998 \\ \hline Right & 468 + 486 + 648 + 684 + 846 + 864 = 3996 & 6 \times 666 = 3996 \\ \end{array}$$
$$\begin{array}{r|r} Left \quad\quad & Right \quad\quad\\ \hline 135 + 531 = 666 & 468 + 864 = 1332 \\ 153 + 513 = 666 & 486 + 846 = 1332 \\ 315 + 351 = 666 & 648 + 684 = 1332 \\ \end{array}$$
$$\mathtt{\small\color{silver}{009\quad018\quad027\quad\color{black}{036}\quad045\quad054\quad\color{black}{063}\quad072\quad081\quad090\quad\qquad099\\[1.5ex]\quad108\quad117\quad126\quad135\quad144\quad153\quad162\quad171\quad180\quad\qquad189\quad198\\[1.5ex]\qquad\;207\quad216\quad225\quad234\quad243\quad252\quad261\quad270\quad\qquad279\quad288\quad297\\[1.5ex]\qquad\quad\;\color{black}{306}\quad315\quad324\quad\color{green}{333}\quad342\quad351\quad\color{black}{360}\quad\qquad\color{black}{369}\quad378\quad387\quad\color{black}{396}\\[1.5ex]\qquad\qquad\;\;405\quad414\quad423\quad432\quad441\quad450\quad\qquad459\quad468\quad477\quad486\quad495\\[1.5ex]\qquad\qquad\quad\;\;504\quad513\quad522\quad531\quad540\quad\qquad549\quad558\quad567\quad576\quad585\quad594\\[1.5ex]\qquad\qquad\qquad\;\;\;\color{black}{603}\quad612\quad621\quad\color{black}{630}\quad\qquad\color{black}{639}\quad648\quad657\quad\color{green}{666}\quad675\quad684\quad\color{black}{693}\\[1.5ex]\qquad\qquad\qquad\quad\;\;\;702\quad711\quad720\quad\qquad729\quad738\quad747\quad756\quad765\quad774\quad783\quad792\\[1.5ex]\qquad\qquad\qquad\qquad\;\;\;\;801\quad810\quad\qquad819\quad828\quad837\quad846\quad855\quad864\quad873\quad882\quad891\\[1.5ex]\qquad\qquad\qquad\qquad\qquad900\quad\qquad909\quad918\quad927\quad\color{black}{936}\quad945\quad954\quad\color{black}{963}\quad972\quad981\quad990\\[1.5ex]}}$$
$$\begin{array}{r|r|r} Left & 036 + 063 + 306 + 360 + 603 + 630 = 1998 & 6 \times 333 = 1998 \\ \hline Right & 369 + 396 + 639 + 693 + 936 + 963 = 3996 & 6 \times 666 = 3996 \\ \end{array}$$
$$\begin{array}{r|r} Left \quad\quad & Right \quad\quad\\ \hline 036 + 630 = 666 & 369 + 963 = 1332 \\ 063 + 603 = 666 & 396 + 936 = 1332 \\ 306 + 360 = 666 & 639 + 693 = 1332 \\ \end{array}$$
Fourth Observation
The others sequences of permutations that produces the same effect, but the shapes are non-regular.
$$\mathtt{\small\color{silver}{009\quad\color{black}{018}\quad027\quad036\quad045\quad054\quad063\quad072\quad\color{black}{081}\quad090\quad\qquad099\\[1.5ex]\quad\color{black}{108}\quad117\quad126\quad135\quad144\quad153\quad162\quad171\quad\color{black}{180}\quad\qquad\color{black}{189}\quad\color{black}{198}\\[1.5ex]\qquad\;207\quad216\quad225\quad234\quad243\quad252\quad261\quad270\quad\qquad279\quad288\quad297\\[1.5ex]\qquad\quad\;306\quad315\quad324\quad\color{green}{333}\quad342\quad351\quad360\quad\qquad369\quad378\quad387\quad396\\[1.5ex]\qquad\qquad\;\;405\quad414\quad423\quad432\quad441\quad450\quad\qquad459\quad468\quad477\quad486\quad495\\[1.5ex]\qquad\qquad\quad\;\;504\quad513\quad522\quad531\quad540\quad\qquad549\quad558\quad567\quad576\quad585\quad594\\[1.5ex]\qquad\qquad\qquad\;\;\;603\quad612\quad621\quad630\quad\qquad639\quad648\quad657\quad\color{green}{666}\quad675\quad684\quad693\\[1.5ex]\qquad\qquad\qquad\quad\;\;\;702\quad711\quad720\quad\qquad729\quad738\quad747\quad756\quad765\quad774\quad783\quad792\\[1.5ex]\qquad\qquad\qquad\qquad\;\;\;\;\color{black}{801}\quad\color{black}{810}\quad\qquad\color{black}{819}\quad828\quad837\quad846\quad855\quad864\quad873\quad882\quad\color{black}{891}\\[1.5ex]\qquad\qquad\qquad\qquad\qquad900\quad\qquad909\quad\color{black}{918}\quad927\quad936\quad945\quad954\quad963\quad972\quad\color{black}{981}\quad990\\[1.5ex]}}$$
$$\mathtt{\small\color{silver}{009\quad018\quad\color{black}{027}\quad036\quad045\quad054\quad063\quad\color{black}{072}\quad081\quad090\quad\qquad099\\[1.5ex]\quad108\quad117\quad126\quad135\quad144\quad153\quad162\quad171\quad180\quad\qquad189\quad198\\[1.5ex]\qquad\;\color{black}{207}\quad216\quad225\quad234\quad243\quad252\quad261\quad\color{black}{270}\quad\qquad\color{black}{279}\quad288\quad\color{black}{297}\\[1.5ex]\qquad\quad\;306\quad315\quad324\quad\color{green}{333}\quad342\quad351\quad360\quad\qquad369\quad378\quad387\quad396\\[1.5ex]\qquad\qquad\;\;405\quad414\quad423\quad432\quad441\quad450\quad\qquad459\quad468\quad477\quad486\quad495\\[1.5ex]\qquad\qquad\quad\;\;504\quad513\quad522\quad531\quad540\quad\qquad549\quad558\quad567\quad576\quad585\quad594\\[1.5ex]\qquad\qquad\qquad\;\;\;603\quad612\quad621\quad630\quad\qquad639\quad648\quad657\quad\color{green}{666}\quad675\quad684\quad693\\[1.5ex]\qquad\qquad\qquad\quad\;\;\;\color{black}{702}\quad711\quad\color{black}{720}\quad\qquad\color{black}{729}\quad738\quad747\quad756\quad765\quad774\quad783\quad\color{black}{792}\\[1.5ex]\qquad\qquad\qquad\qquad\;\;\;\;801\quad810\quad\qquad819\quad828\quad837\quad846\quad855\quad864\quad873\quad882\quad891\\[1.5ex]\qquad\qquad\qquad\qquad\qquad900\quad\qquad909\quad918\quad\color{black}{927}\quad936\quad945\quad954\quad963\quad\color{black}{972}\quad981\quad990\\[1.5ex]}}$$
$$\mathtt{\small\color{silver}{009\quad018\quad027\quad036\quad\color{black}{045}\quad\color{black}{054}\quad063\quad072\quad081\quad090\quad\qquad099\\[1.5ex]\quad108\quad117\quad126\quad135\quad144\quad153\quad162\quad171\quad180\quad\qquad189\quad198\\[1.5ex]\qquad\;207\quad216\quad225\quad234\quad243\quad252\quad261\quad270\quad\qquad279\quad288\quad297\\[1.5ex]\qquad\quad\;306\quad315\quad324\quad\color{green}{333}\quad342\quad351\quad360\quad\qquad369\quad378\quad387\quad396\\[1.5ex]\qquad\qquad\;\;\color{black}{405}\quad414\quad423\quad432\quad441\quad\color{black}{450}\quad\qquad\color{black}{459}\quad468\quad477\quad486\quad\color{black}{495}\\[1.5ex]\qquad\qquad\quad\;\;\color{black}{504}\quad513\quad522\quad531\quad\color{black}{540}\quad\qquad\color{black}{549}\quad558\quad567\quad576\quad585\quad\color{black}{594}\\[1.5ex]\qquad\qquad\qquad\;\;\;603\quad612\quad621\quad630\quad\qquad639\quad648\quad657\quad\color{green}{666}\quad675\quad684\quad693\\[1.5ex]\qquad\qquad\qquad\quad\;\;\;702\quad711\quad720\quad\qquad729\quad738\quad747\quad756\quad765\quad774\quad783\quad792\\[1.5ex]\qquad\qquad\qquad\qquad\;\;\;\;801\quad810\quad\qquad819\quad828\quad837\quad846\quad855\quad864\quad873\quad882\quad891\\[1.5ex]\qquad\qquad\qquad\qquad\qquad900\quad\qquad909\quad918\quad927\quad936\quad\color{black}{945}\quad\color{black}{954}\quad963\quad972\quad981\quad990\\[1.5ex]}}$$
Maybe this is based on a naïve approach, but I'm highly interested by the math behind this relation between number and geometry.
Thank you very much for your time.
Maxime Coorevits
The first explaination about this interestings organization of the permutation comes with the observation of the each digits of the numbers separatly:
1st digit
$$\mathtt{\small\color{silver}{\color{black}{0}09\quad\color{black}{0}18\quad\color{black}{0}27\quad\color{black}{0}36\quad\color{black}{0}45\quad\color{black}{0}54\quad\color{black}{0}63\quad\color{black}{0}72\quad\color{black}{0}81\quad\color{black}{0}90\quad\qquad\color{black}{0}99\\[1.5ex]\quad\color{black}{1}08\quad\color{black}{1}17\quad\color{black}{1}26\quad\color{black}{1}35\quad\color{black}{1}44\quad\color{black}{1}53\quad\color{black}{1}62\quad\color{black}{1}71\quad\color{black}{1}80\quad\qquad\color{black}{1}89\quad\color{black}{1}98\\[1.5ex]\qquad\;\color{black}{2}07\quad\color{black}{2}16\quad\color{black}{2}25\quad\color{black}{2}34\quad\color{black}{2}43\quad\color{black}{2}52\quad\color{black}{2}61\quad\color{black}{2}70\quad\qquad\color{black}{2}79\quad\color{black}{2}88\quad\color{black}{2}97\\[1.5ex]\qquad\quad\;\color{black}{3}06\quad\color{black}{3}15\quad\color{black}{3}24\quad\color{black}{3}33\quad\color{black}{3}42\quad\color{black}{3}51\quad\color{black}{3}60\quad\qquad\color{black}{3}69\quad\color{black}{3}78\quad\color{black}{3}87\quad\color{black}{3}96\\[1.5ex]\qquad\qquad\;\;\color{black}{4}05\quad\color{black}{4}14\quad\color{black}{4}23\quad\color{black}{4}32\quad\color{black}{4}41\quad\color{black}{4}50\quad\qquad\color{black}{4}59\quad\color{black}{4}68\quad\color{black}{4}77\quad\color{black}{4}86\quad\color{black}{4}95\\[1.5ex]\qquad\qquad\quad\;\;\color{black}{5}04\quad\color{black}{5}13\quad\color{black}{5}22\quad\color{black}{5}31\quad\color{black}{5}40\quad\qquad\color{black}{5}49\quad\color{black}{5}58\quad\color{black}{5}67\quad\color{black}{5}76\quad\color{black}{5}85\quad\color{black}{5}94\\[1.5ex]\qquad\qquad\qquad\;\;\;\color{black}{6}03\quad\color{black}{6}12\quad\color{black}{6}21\quad\color{black}{6}30\quad\qquad\color{black}{6}39\quad\color{black}{6}48\quad\color{black}{6}57\quad\color{black}{6}66\quad\color{black}{6}75\quad\color{black}{6}84\quad\color{black}{6}93\\[1.5ex]\qquad\qquad\qquad\quad\;\;\;\color{black}{7}02\quad\color{black}{7}11\quad\color{black}{7}20\quad\qquad\color{black}{7}29\quad\color{black}{7}38\quad\color{black}{7}47\quad\color{black}{7}56\quad\color{black}{7}65\quad\color{black}{7}74\quad\color{black}{7}83\quad\color{black}{7}92\\[1.5ex]\qquad\qquad\qquad\qquad\;\;\;\;\color{black}{8}01\quad\color{black}{8}10\quad\qquad\color{black}{8}19\quad\color{black}{8}28\quad\color{black}{8}37\quad\color{black}{8}46\quad\color{black}{8}55\quad\color{black}{8}64\quad\color{black}{8}73\quad\color{black}{8}82\quad\color{black}{8}91\\[1.5ex]\qquad\qquad\qquad\qquad\qquad\color{black}{9}00\quad\qquad\color{black}{9}09\quad\color{black}{9}18\quad\color{black}{9}27\quad\color{black}{9}36\quad\color{black}{9}45\quad\color{black}{9}54\quad\color{black}{9}63\quad\color{black}{9}72\quad\color{black}{9}81\quad\color{black}{9}90\\[1.5ex]}}$$
2nd digit
$$\mathtt{\small\color{silver}{0\color{black}{0}9\quad0\color{black}{1}8\quad0\color{black}{2}7\quad0\color{black}{3}6\quad0\color{black}{4}5\quad0\color{black}{5}4\quad0\color{black}{6}3\quad0\color{black}{7}2\quad0\color{black}{8}1\quad0\color{black}{9}0\quad\qquad0\color{black}{9}9\\[1.5ex]\quad1\color{black}{0}8\quad1\color{black}{1}7\quad1\color{black}{2}6\quad1\color{black}{3}5\quad1\color{black}{4}4\quad1\color{black}{5}3\quad1\color{black}{6}2\quad1\color{black}{7}1\quad1\color{black}{8}0\quad\qquad1\color{black}{8}9\quad1\color{black}{9}8\\[1.5ex]\qquad\;2\color{black}{0}7\quad2\color{black}{1}6\quad2\color{black}{2}5\quad2\color{black}{3}4\quad2\color{black}{4}3\quad2\color{black}{5}2\quad2\color{black}{6}1\quad2\color{black}{7}0\quad\qquad2\color{black}{7}9\quad2\color{black}{8}8\quad2\color{black}{9}7\\[1.5ex]\qquad\quad\;3\color{black}{0}6\quad3\color{black}{1}5\quad3\color{black}{2}4\quad3\color{black}{3}3\quad3\color{black}{4}2\quad3\color{black}{5}1\quad3\color{black}{6}0\quad\qquad3\color{black}{6}9\quad3\color{black}{7}8\quad3\color{black}{8}7\quad3\color{black}{9}6\\[1.5ex]\qquad\qquad\;\;4\color{black}{0}5\quad4\color{black}{1}4\quad4\color{black}{2}3\quad4\color{black}{3}2\quad4\color{black}{4}1\quad4\color{black}{5}0\quad\qquad4\color{black}{5}9\quad4\color{black}{6}8\quad4\color{black}{7}7\quad4\color{black}{8}6\quad4\color{black}{9}5\\[1.5ex]\qquad\qquad\quad\;\;5\color{black}{0}4\quad5\color{black}{1}3\quad5\color{black}{2}2\quad5\color{black}{3}1\quad5\color{black}{4}0\quad\qquad5\color{black}{4}9\quad5\color{black}{5}8\quad5\color{black}{6}7\quad5\color{black}{7}6\quad5\color{black}{8}5\quad5\color{black}{9}4\\[1.5ex]\qquad\qquad\qquad\;\;\;6\color{black}{0}3\quad6\color{black}{1}2\quad6\color{black}{2}1\quad6\color{black}{3}0\quad\qquad6\color{black}{3}9\quad6\color{black}{4}8\quad6\color{black}{5}7\quad6\color{black}{6}6\quad6\color{black}{7}5\quad6\color{black}{8}4\quad6\color{black}{9}3\\[1.5ex]\qquad\qquad\qquad\quad\;\;\;7\color{black}{0}2\quad7\color{black}{1}1\quad7\color{black}{2}0\quad\qquad7\color{black}{2}9\quad7\color{black}{3}8\quad7\color{black}{4}7\quad7\color{black}{5}6\quad7\color{black}{6}5\quad7\color{black}{7}4\quad7\color{black}{8}3\quad7\color{black}{9}2\\[1.5ex]\qquad\qquad\qquad\qquad\;\;\;\;8\color{black}{0}1\quad8\color{black}{1}0\quad\qquad8\color{black}{1}9\quad8\color{black}{2}8\quad8\color{black}{3}7\quad8\color{black}{4}6\quad8\color{black}{5}5\quad8\color{black}{6}4\quad8\color{black}{7}3\quad8\color{black}{8}2\quad8\color{black}{9}1\\[1.5ex]\qquad\qquad\qquad\qquad\qquad9\color{black}{0}0\quad\qquad9\color{black}{0}9\quad9\color{black}{1}8\quad9\color{black}{2}7\quad9\color{black}{3}6\quad9\color{black}{4}5\quad9\color{black}{5}4\quad9\color{black}{6}3\quad9\color{black}{7}2\quad9\color{black}{8}1\quad9\color{black}{9}0\\[1.5ex]}}$$
3rd digit
$$\mathtt{\small\color{silver}{00\color{black}{9}\quad01\color{black}{8}\quad02\color{black}{7}\quad03\color{black}{6}\quad04\color{black}{5}\quad05\color{black}{4}\quad06\color{black}{3}\quad07\color{black}{2}\quad08\color{black}{1}\quad09\color{black}{0}\quad\qquad09\color{black}{9}\\[1.5ex]\quad10\color{black}{8}\quad11\color{black}{7}\quad12\color{black}{6}\quad13\color{black}{5}\quad14\color{black}{4}\quad15\color{black}{3}\quad16\color{black}{2}\quad17\color{black}{1}\quad18\color{black}{0}\quad\qquad18\color{black}{9}\quad19\color{black}{8}\\[1.5ex]\qquad\;20\color{black}{7}\quad21\color{black}{6}\quad22\color{black}{5}\quad23\color{black}{4}\quad24\color{black}{3}\quad25\color{black}{2}\quad26\color{black}{1}\quad27\color{black}{0}\quad\qquad27\color{black}{9}\quad28\color{black}{8}\quad29\color{black}{7}\\[1.5ex]\qquad\quad\;30\color{black}{6}\quad31\color{black}{5}\quad32\color{black}{4}\quad33\color{black}{3}\quad34\color{black}{2}\quad35\color{black}{1}\quad36\color{black}{0}\quad\qquad36\color{black}{9}\quad37\color{black}{8}\quad38\color{black}{7}\quad39\color{black}{6}\\[1.5ex]\qquad\qquad\;\;40\color{black}{5}\quad41\color{black}{4}\quad42\color{black}{3}\quad43\color{black}{2}\quad44\color{black}{1}\quad45\color{black}{0}\quad\qquad45\color{black}{9}\quad46\color{black}{8}\quad47\color{black}{7}\quad48\color{black}{6}\quad49\color{black}{5}\\[1.5ex]\qquad\qquad\quad\;\;50\color{black}{4}\quad51\color{black}{3}\quad52\color{black}{2}\quad53\color{black}{1}\quad54\color{black}{0}\quad\qquad54\color{black}{9}\quad55\color{black}{8}\quad56\color{black}{7}\quad57\color{black}{6}\quad58\color{black}{5}\quad59\color{black}{4}\\[1.5ex]\qquad\qquad\qquad\;\;\;60\color{black}{3}\quad61\color{black}{2}\quad62\color{black}{1}\quad63\color{black}{0}\quad\qquad63\color{black}{9}\quad64\color{black}{8}\quad65\color{black}{7}\quad66\color{black}{6}\quad67\color{black}{5}\quad68\color{black}{4}\quad69\color{black}{3}\\[1.5ex]\qquad\qquad\qquad\quad\;\;\;70\color{black}{2}\quad71\color{black}{1}\quad72\color{black}{0}\quad\qquad72\color{black}{9}\quad73\color{black}{8}\quad74\color{black}{7}\quad75\color{black}{6}\quad76\color{black}{5}\quad77\color{black}{4}\quad78\color{black}{3}\quad79\color{black}{2}\\[1.5ex]\qquad\qquad\qquad\qquad\;\;\;\;80\color{black}{1}\quad81\color{black}{0}\quad\qquad81\color{black}{9}\quad82\color{black}{8}\quad83\color{black}{7}\quad84\color{black}{6}\quad85\color{black}{5}\quad86\color{black}{4}\quad87\color{black}{3}\quad88\color{black}{2}\quad89\color{black}{1}\\[1.5ex]\qquad\qquad\qquad\qquad\qquad90\color{black}{0}\quad\qquad90\color{black}{9}\quad91\color{black}{8}\quad92\color{black}{7}\quad93\color{black}{6}\quad94\color{black}{5}\quad95\color{black}{4}\quad96\color{black}{3}\quad97\color{black}{2}\quad98\color{black}{1}\quad99\color{black}{0}\\[1.5ex]}}$$
$$\bbox[12px,border:1px solid black]{ \begin{array}{r|r} Digit & Direction \\ \hline 1st & \Downarrow \quad\quad \\ 2nd & \Rightarrow \quad\quad \\ 3rd & \Leftarrow \quad\quad \\ \end{array} }$$
We've noticed that the diagonal division of the original matrix produce two regular groups of weight:
$$\mathtt{\small\color{silver}{\color{blue}{009}\quad\color{blue}{018}\quad\color{blue}{027}\quad\color{blue}{036}\quad\color{blue}{045}\quad\color{blue}{054}\quad\color{blue}{063}\quad\color{blue}{072}\quad\color{blue}{081}\quad\color{blue}{090}\quad\color{red}{099}\\[1.5ex]\color{blue}{108}\quad\color{blue}{117}\quad\color{blue}{126}\quad\color{blue}{135}\quad\color{blue}{144}\quad\color{blue}{153}\quad\color{blue}{162}\quad\color{blue}{171}\quad\color{blue}{180}\quad\color{red}{189}\quad\color{red}{198}\\[1.5ex]\color{blue}{207}\quad\color{blue}{216}\quad\color{blue}{225}\quad\color{blue}{234}\quad\color{blue}{243}\quad\color{blue}{252}\quad\color{blue}{261}\quad\color{blue}{270}\quad\color{red}{279}\quad\color{red}{288}\quad\color{red}{297}\\[1.5ex]\color{blue}{306}\quad\color{blue}{315}\quad\color{blue}{324}\quad\color{blue}{333}\quad\color{blue}{342}\quad\color{blue}{351}\quad\color{blue}{360}\quad\color{red}{369}\quad\color{red}{378}\quad\color{red}{387}\quad\color{red}{396}\\[1.5ex]\color{blue}{405}\quad\color{blue}{414}\quad\color{blue}{423}\quad\color{blue}{432}\quad\color{blue}{441}\quad\color{blue}{450}\quad\color{red}{459}\quad\color{red}{468}\quad\color{red}{477}\quad\color{red}{486}\quad\color{red}{495}\\[1.5ex]\color{blue}{504}\quad\color{blue}{513}\quad\color{blue}{522}\quad\color{blue}{531}\quad\color{blue}{540}\quad\color{red}{549}\quad\color{red}{558}\quad\color{red}{567}\quad\color{red}{576}\quad\color{red}{585}\quad\color{red}{594}\\[1.5ex]\color{blue}{603}\quad\color{blue}{612}\quad\color{blue}{621}\quad\color{blue}{630}\quad\color{red}{639}\quad\color{red}{648}\quad\color{red}{657}\quad\color{red}{666}\quad\color{red}{675}\quad\color{red}{684}\quad\color{red}{693}\\[1.5ex]\color{blue}{702}\quad\color{blue}{711}\quad\color{blue}{720}\quad\color{red}{729}\quad\color{red}{738}\quad\color{red}{747}\quad\color{red}{756}\quad\color{red}{765}\quad\color{red}{774}\quad\color{red}{783}\quad\color{red}{792}\\[1.5ex]\color{blue}{801}\quad\color{blue}{810}\quad\color{red}{819}\quad\color{red}{828}\quad\color{red}{837}\quad\color{red}{846}\quad\color{red}{855}\quad\color{red}{864}\quad\color{red}{873}\quad\color{red}{882}\quad\color{red}{891}\\[1.5ex]\color{blue}{900}\quad\color{red}{909}\quad\color{red}{918}\quad\color{red}{927}\quad\color{red}{936}\quad\color{red}{945}\quad\color{red}{954}\quad\color{red}{963}\quad\color{red}{972}\quad\color{red}{981}\quad\color{red}{990}\\[1.5ex]}}$$
$$\bbox[12px,border:1px solid black]{ \begin{array}{r|r} \huge{\frac{1}{3}} & \huge{\frac{2}{3}} \\ \end{array} }$$
By applying a $180^\text{o}$ rotation to the right triangle and sum the two triangle, the resulted triangle is fully filled with $999$ numbers.
$$\mathtt{\color{black}{009\quad018\quad027\quad036\quad045\quad054\quad063\quad072\quad081\quad090\\[1.5ex]\quad\;108\quad117\quad126\quad135\quad144\quad153\quad162\quad171\quad180\\[1.5ex]\quad\;\quad\;207\quad216\quad225\quad234\quad243\quad252\quad261\quad270\\[1.5ex]\quad\;\quad\;\quad\;306\quad315\quad324\quad333\quad342\quad351\quad360\\[1.5ex]\quad\;\quad\;\quad\;\quad\;405\quad414\quad423\quad432\quad441\quad450\\[1.5ex]\quad\;\quad\;\quad\;\quad\;\quad\;504\quad513\quad522\quad531\quad540\\[1.5ex]\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;603\quad612\quad621\quad630\\[1.5ex]\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;702\quad711\quad720\\[1.5ex]\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;801\quad810\\[1.5ex]\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;900\\[1.5ex]}}$$ $$\huge{+}$$ $$\mathtt{\color{black}{990\quad981\quad972\quad963\quad954\quad945\quad936\quad927\quad918\quad909\\[1.5ex]\quad\;891\quad882\quad873\quad864\quad855\quad846\quad837\quad828\quad819\\[1.5ex]\quad\;\quad\;792\quad783\quad774\quad765\quad756\quad747\quad738\quad729\\[1.5ex]\quad\;\quad\;\quad\;693\quad684\quad675\quad666\quad657\quad648\quad639\\[1.5ex]\quad\;\quad\;\quad\;\quad\;594\quad585\quad576\quad567\quad558\quad549\\[1.5ex]\quad\;\quad\;\quad\;\quad\;\quad\;495\quad486\quad477\quad468\quad459\\[1.5ex]\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;396\quad387\quad378\quad369\\[1.5ex]\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;297\quad288\quad279\\[1.5ex]\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;198\quad189\\[1.5ex]\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;099\\[1.5ex]}}$$ $$\huge{=}$$ $$\mathtt{\color{black}{999\quad999\quad999\quad999\quad999\quad999\quad999\quad999\quad999\quad999\\[1.5ex]\quad\;999\quad999\quad999\quad999\quad999\quad999\quad999\quad999\quad999\\[1.5ex]\quad\;\quad\;999\quad999\quad999\quad999\quad999\quad999\quad999\quad999\\[1.5ex]\quad\;\quad\;\quad\;999\quad999\quad999\quad999\quad999\quad999\quad999\\[1.5ex]\quad\;\quad\;\quad\;\quad\;999\quad999\quad999\quad999\quad999\quad999\\[1.5ex]\quad\;\quad\;\quad\;\quad\;\quad\;999\quad999\quad999\quad999\quad999\\[1.5ex]\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;999\quad999\quad999\quad999\\[1.5ex]\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;999\quad999\quad999\\[1.5ex]\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;999\quad999\\[1.5ex]\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;\quad\;999\\[1.5ex]}}$$